weqweasdas/qwen15b_train_simple_subset5k_for_difficulty_transition

problem stringlengths 30 2.03k	gt stringlengths 1 130
Let $r$ be a nonzero real number. The values of $z$ which satisfy the equation \[ r^4z^4 + (10r^6-2r^2)z^2-16r^5z+(9r^8+10r^4+1) = 0 \] are plotted on the complex plane (i.e. using the real part of each root as the x-coordinate and the imaginary part as the y-coordinate). Show that the area of the convex quadrilateral with these points as vertices is independent of $r$, and find this area.	8
Find all strictly increasing sequences of positive integers $a_1, a_2, \ldots$ with $a_1=1$, satisfying $$3(a_1+a_2+\ldots+a_n)=a_{n+1}+\ldots+a_{2n}$$ for all positive integers $n$.	a_n = 2n - 1
7. The minor axis of the ellipse $\rho=\frac{1}{2-\cos \theta}$ is equal to 保留了源文本的换行和格式。	\frac{2\sqrt{3}}{3}
15. Right trapezoid $A B C D$, the upper base is 1, the lower base is 7, connecting point $E$ on side $A B$ and point $F$ on side $D C$, forming a line segment $E F$ parallel to $A D$ and $B C$ that divides the area of the right trapezoid into two equal parts, then the length of line segment $E F$ is $\qquad$	5
Initially Alex, Betty, and Charlie had a total of $444$ peanuts. Charlie had the most peanuts, and Alex had the least. The three numbers of peanuts that each person had formed a geometric progression. Alex eats $5$ of his peanuts, Betty eats $9$ of her peanuts, and Charlie eats $25$ of his peanuts. Now the three numbers of peanuts each person has forms an arithmetic progression. Find the number of peanuts Alex had initially.	108
8. Given $x, y \in \mathbf{R}$, for any $n \in \mathbf{Z}_{+}$, $n x+\frac{1}{n} y \geqslant 1$. Then the minimum value of $41 x+2 y$ is $\qquad$	9
3. The area of the closed figure enclosed by the curve $x^{2}+2 y^{2} \leqslant 4\|y\|$ is $\qquad$	2 \sqrt{2} \pi
16) How many of the following statements about natural numbers are true? (i) Given any two consecutive odd numbers, at least one is prime. (ii) Given any three consecutive odd numbers, at least two are prime. (iii) Given any four consecutive odd numbers, at least one is not prime. (iv) Given any five consecutive odd numbers, at least two are not prime. (A) None (B) one (C) two (D) three (E) four.	B
[b]H[/b]orizontal parallel segments $AB=10$ and $CD=15$ are the bases of trapezoid $ABCD$. Circle $\gamma$ of radius $6$ has center within the trapezoid and is tangent to sides $AB$, $BC$, and $DA$. If side $CD$ cuts out an arc of $\gamma$ measuring $120^{\circ}$, find the area of $ABCD$.	\frac{225}{2}
7. If $p$ and $q$ are both prime numbers, the number of divisors $d(a)$ of the natural number $a=p^{\alpha} q^{\beta}$ is given by the formula $$ d(a)=(\alpha+1)(\beta+1) $$ For example, $12=2^{2} \times 3^{1}$, so the number of divisors of 12 is $$ d(12)=(2+1)(1+1)=6, $$ with the divisors being $1, 2, 3, 4, 6$, and 12. According to the given formula, please answer: Among the divisors of $20^{30}$ that are less than $20^{15}$, how many are not divisors of $20^{15}$? $\qquad$	450
3. Four identical small rectangles are put together to form a large rectangle as shown. The length of a shorter side of each small rectangle is $10 \mathrm{~cm}$. What is the length of a longer side of the large rectangle? A $50 \mathrm{~cm}$ B $40 \mathrm{~cm}$ C $30 \mathrm{~cm}$ D $20 \mathrm{~cm}$ E $10 \mathrm{~cm}$	40
4. Given the quadratic function $y=x^{2}-x+a$ whose graph intersects the $x$-axis at two distinct points, the sum of the distances from these points to the origin does not exceed 5. Then the range of values for $a$ is $\qquad$ .	-6 \leqslant a < \frac{1}{4}
3. Let trapezium have bases $A B$ and $C D$, measuring $5 \mathrm{~cm}$ and $1 \mathrm{~cm}$, respectively. Let $M$ and $N$ be points on $A D$ and $B C$ such that the segment $M N$ is parallel to the base $A B$, and the area of quadrilateral $A B N M$ is twice the area of quadrilateral $C D M N$. How many centimeters does the segment $M N$ measure?	3
4. There are 8 cards; one side of each is clean, while the other side has the letters: И, Я, Л, З, Г, О, О, О printed on them. The cards are placed on the table with the clean side up, shuffled, and then sequentially flipped over one by one. What is the probability that the letters will form the word ЗОоЛОГИЯ when they appear in sequence?	\frac{1}{6720}
1. Given a quadratic trinomial $f(x)$ such that the equation $(f(x))^{3}-4 f(x)=0$ has exactly three solutions. How many solutions does the equation $(f(x))^{2}=1$ have?	2
2. "Real numbers $a=b=c$" is the ( ) condition for the inequality $a^{3}+b^{3}+c^{3} \geqslant 3 a b c$ to hold with equality. (A) Sufficient but not necessary (B) Sufficient and necessary (C) Necessary but not sufficient (D) Neither sufficient nor necessary	A
1. 2. 1 * Let $A=\{1,2,3, m\}, B=\left\{4,7, n^{4}, n^{2}+3 n\right\}$, the correspondence rule $f: a \rightarrow$ $b=pa+q$ is a one-to-one mapping from $A$ to $B$. It is known that $m, n$ are positive integers, and the image of 1 is 4, the preimage of 7 is 2. Find the values of $p, q, m, n$.	(3,1,5,2)
Find the smallest positive integer that, when multiplied by 1999, the last four digits of the resulting number are 2001.	5999
6、Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=1, a_{n+1}=\frac{(n+1) a_{n}}{2 n+a_{n}}\left(n \in N_{+}\right)$. Then $\sum_{k=1}^{2017} \frac{k}{a_{k}}=$	2^{2018}-2019
Let $d(m)$ denote the number of positive integer divisors of a positive integer $m$. If $r$ is the number of integers $n \leqslant 2023$ for which $\sum_{i=1}^{n} d(i)$ is odd. , find the sum of digits of $r.$	18
In a $12\times 12$ grid, colour each unit square with either black or white, such that there is at least one black unit square in any $3\times 4$ and $4\times 3$ rectangle bounded by the grid lines. Determine, with proof, the minimum number of black unit squares.	12
9.077. $2^{x+2}-2^{x+3}-2^{x+4}>5^{x+1}-5^{x+2}$.	x\in(0;\infty)
7. It is known that for some natural numbers $a, b$, the number $N=\frac{a^{2}+b^{2}}{a b-1}$ is also natural. Find all possible values of $N$. --- The provided text has been translated into English while preserving the original formatting and structure.	5
\%EA360 * Find the number of positive integer solutions $(m, n, r)$ for the indeterminate equation $$ m n+n r+m r=2(m+n+r) $$	7
2. Solve the equation $$ \sqrt{\frac{x-3}{11}}+\sqrt{\frac{x-4}{10}}=\sqrt{\frac{x-11}{3}}+\sqrt{\frac{x-10}{4}} $$	14
$3+$ [ The transfer helps solve the task_ ] On the side AB of the square ABCD, an equilateral triangle AKB was constructed (outside). Find the radius of the circle circumscribed around triangle CKD, if $\mathrm{AB}=1$. #	1
1. If $x+y+z=0$, simplify the expression $$ \frac{x^{7}+y^{7}+z^{7}}{x y z\left(x^{4}+y^{4}+z^{4}\right)} $$ Hint: first calculate $(x+y)^{4}$ and $(x+y)^{6}$.	\frac{7}{2}
4. Given that the complex number $z$ satisfies $z^{3}+z=2\|z\|^{2}$. Then all possible values of $z$ are $\qquad$	0, 1, -1 \pm 2i
3. On the diagonal $BD$ of square $ABCD$, take two points $E$ and $F$, such that the extension of $AE$ intersects side $BC$ at point $M$, and the extension of $AF$ intersects side $CD$ at point $N$, with $CM = CN$. If $BE = 3$, $EF = 4$, what is the length of the diagonal of this square?	10
[ Special cases of parallelepipeds (other).] Area of the section The base of a right parallelepiped is a rhombus, the area of which is equal to $Q$. The areas of the diagonal sections are $S 1$ and $S 2$. Find the volume of the parallelepiped.	\sqrt{\frac{QS_{1}S_{2}}{2}}
Example 3 Given real numbers $x_{1}, x_{2}, \cdots, x_{10}$ satisfy $\sum_{i=1}^{10}\left\|x_{i}-1\right\| \leqslant 4, \sum_{i=1}^{10}\left\|x_{i}-2\right\| \leqslant 6$. Find the average value $\bar{x}$ of $x_{1}, x_{2}, \cdots, x_{10}$. (2012, Zhejiang Province High School Mathematics Competition)	1.4
In 1998, the population of Canada was 30.3 million. Which of the options below represents the population of Canada in 1998? (a) 30300000 (b) 303000000 (c) 30300 (d) 303300 (e) 30300000000	30300000
Let $S$ be the set of all partitions of $2000$ (in a sum of positive integers). For every such partition $p$, we deﬁne $f (p)$ to be the sum of the number of summands in $p$ and the maximal summand in $p$. Compute the minimum of $f (p)$ when $p \in S .$	90
Let $x_0,x_1,x_2,\dots$ be the sequence such that $x_0=1$ and for $n\ge 0,$ \[x_{n+1}=\ln(e^{x_n}-x_n)\] (as usual, the function $\ln$ is the natural logarithm). Show that the infinite series \[x_0+x_1+x_2+\cdots\] converges and find its sum.	e - 1
Given $u_0,u_1$ with $0<u_0,u_1<1$, define the sequence $(u_n)$ recurrently by the formula $$u_{n+2}=\frac12\left(\sqrt{u_{n+1}}+\sqrt{u_n}\right).$$(a) Prove that the sequence $u_n$ is convergent and find its limit. (b) Prove that, starting from some index $n_0$, the sequence $u_n$ is monotonous.	1
5. What is the total area in $\mathrm{cm}^{2}$ of the shaded region? A 50 B 80 C 100 D 120 E 150	100
Given a right triangle $A B C$. From the vertex $B$ of the right angle, a median $B D$ is drawn. Let $K$ be the point of tangency of side $A D$ of triangle $A B D$ with the inscribed circle of this triangle. Find the acute angles of triangle $A B C$, if $K$ bisects $A D$.	30,60
3. 1. 4 * The terms of the sequence $\left\{a_{n}\right\}$ are positive, and the sum of the first $n$ terms is $S_{n}$, satisfying $S_{n}=\frac{1}{2}\left(a_{n}+\frac{1}{a_{n}}\right)$, find the general term of the sequence $\left\{a_{n}\right\}$.	a_{n}=\sqrt{n}-\sqrt{n-1}
Example 19 (Problem 1506 from "Mathematics Bulletin") In $\triangle A B C$, $A B=A C$, the angle bisector of $\angle B$ intersects $A C$ at $D$, and $B C=B D+A D$. Find $\angle A$. Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.	100
16. If $y+4=(x-2)^{2}, x+4=(y-2)^{2}$, and $x \neq y$, then the value of $x^{2}+y^{2}$ is ( ). (A) 10 (B) 15 (C) 20 (D) 25 (E) 30	15
6. Solve the equation $\sqrt{2 x^{2}+3 x+2}-\sqrt{2 x^{2}+3 x-5}=1$.	2,-\frac{7}{2}
An integer's product of all its positive integer divisors is $2^{120} \cdot 3^{60} \cdot 5^{90}$. What could this number be?	18000
(1) Given the set $M=\{1,3,5,7,9\}$, if the non-empty set $A$ satisfies: the elements of $A$ each increased by 4 form a subset of $M$, and the elements of $A$ each decreased by 4 also form a subset of $M$, then $A=$ $\qquad$ .	{5}
Place a triangle under a magnifying glass that enlarges by 5 times, the perimeter is $\qquad$ times that of the original triangle, and the area is $\qquad$ times that of the original triangle.	5,25
10. The Kingdom of Geometry awards badges to outstanding citizens of 2021. The badges are rectangular (including squares), and these rectangular badges are all different, but their area values are 2021 times their perimeter values, and both length and width are natural numbers. Therefore, the maximum number of different badges for the year 2021 is $\qquad$ kinds.	14
Example 3. The circumference of a circle is measured to be 10.7 meters, find its radius.	1.70
10. The function $\operatorname{SPF}(n)$ denotes the sum of the prime factors of $n$, where the prime factors are not necessarily distinct. For example, $120=2^{3} \times 3 \times 5$, so $\operatorname{SPF}(120)=2+2+2+3+5=14$. Find the value of $\operatorname{SPF}\left(2^{22}-4\right)$.	100
## Task A-2.7. In a basketball tournament, each team plays exactly twice against each of the other teams. A win brings 2 points, a loss 0 points, and there are no draws. Determine all natural numbers $n$ for which there exists a basketball tournament with $n$ teams where one team, the tournament winner, has 26 points, and exactly two teams have the lowest number of points, which is 20 points.	12
T10. Find all positive integers $a$ such that the quadratic equation $a x^{2}+2(2 a-1) x+4(a-3)=0$ has at least one integer root. (3rd Zu Chongzhi Cup Junior High School Mathematics Invitational Competition)	a=1,3,6,10
4. As shown in Figure 1, in the Cartesian coordinate system $x O y$, $A$ and $B$ are two points on the graph of $y=\frac{1}{x}$ in the first quadrant. If $\triangle A O B$ is an equilateral triangle, then its area $S$ is ( ). (A) $2 \sqrt{3}$ (B) 3 (C) $\sqrt{3}$ (D) $\frac{\sqrt{3}}{2}$	\sqrt{3}
1. [4] If $A=10^{9}-987654321$ and $B=\frac{123456789+1}{10}$, what is the value of $\sqrt{A B}$ ?	12345679
1. Let $\mathbf{Z}$ denote the set of all integers. Find all functions $f: \mathbf{Z} \rightarrow \mathbf{Z}$, such that for any integers $a, b$, we have $$ f(2 a)+2 f(b)=f(f(a+b)) . $$	f(x)=0
The centers of three circles, each touching the other two externally, are located at the vertices of a right triangle. These circles are internally tangent to a fourth circle. Find the radius of the fourth circle if the perimeter of the right triangle is $2 p$. #	p
4. As shown in Figure 3, in $\triangle A B C$, $M$ is the midpoint of side $B C$, and $M D \perp A B, M E \perp A C$, with $D$ and $E$ being the feet of the perpendiculars. If $B D=2, C E=1$, and $D E \parallel B C$, then $D M^{2}$ equals $\qquad$	1
16. Given functions $f(x), g(x)(x \in \mathrm{R})$, let the solution set of the inequality $\|f(x)\|+\|g(x)\|<a (a>0)$ be $M$, and the solution set of the inequality $\|f(x)+g(x)\|<a (a>0)$ be $N$, then the relationship between the solution sets $M$ and $N$ is ( ). A. $M \subseteq N$ B. $M=N$ C. $N \varsubsetneqq M$ D. $M \varsubsetneqq N$	A
1. How many positive integers less than 2019 are divisible by either 18 or 21 , but not both?	176
13. There is a sequence of numbers, the first 4 numbers are $2,0,1,8$, starting from the 5th number, each number is the unit digit of the sum of the previous 4 adjacent numbers, will the sequence $2,0,1, 7$ appear in this sequence of numbers?	No
Example 5.17 Given non-negative real numbers $a, b, c$ satisfy $a b + b c + c a + 6 a b c = 9$. Determine the maximum value of $k$ such that the following inequality always holds. $$a + b + c + k a b c \geqslant k + 3$$	3
4. Find the range of real numbers $a$ such that the inequality $$ \begin{array}{l} \sin 2 \theta-(2 \sqrt{2}+a \sqrt{2}) \sin \left(\theta+\frac{\pi}{4}\right)-2 \sqrt{2} \sec \left(\theta-\frac{\pi}{4}\right) \\ >-3-2 a \end{array} $$ holds for all $\theta \in\left[0, \frac{\pi}{2}\right]$. (2000, Tianjin High School Mathematics Competition)	a>3
In $\triangle A B C$, $A B$ is the longest side, $\sin A \sin B=\frac{2-\sqrt{3}}{4}$, then the maximum value of $\cos A \cos B$ is Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.	\frac{2+\sqrt{3}}{4}
1. Let $A$ and $B$ be two moving points on the ellipse $\frac{x^{2}}{2}+y^{2}=1$, and $O$ be the origin. Also, $\overrightarrow{O A} \cdot \overrightarrow{O B}=0$. Let point $P$ be on $AB$, and $O P \perp A B$. Find the value of $\|O P\|$.	\frac{\sqrt{6}}{3}
4. (15 points) Identical gases are in two thermally insulated vessels with volumes $V_{1}=2$ L and $V_{2}=3$ L. The pressures of the gases are $p_{1}=3$ atm and $p_{2}=4$ atm, and their temperatures are $T_{1}=400$ K and $T_{2}=500$ K, respectively. The gases are mixed. Determine the temperature that will be established in the vessels.	462\mathrm{~K}
Let $ABCDE$ be a convex pentagon such that $AB = BC = CD$ and $\angle BDE = \angle EAC = 30 ^{\circ}$. Find the possible values of $\angle BEC$. [i]Proposed by Josef Tkadlec (Czech Republic)[/i]	60^\circ
Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length $1$. The polygons meet at a point $A$ in such a way that the sum of the three interior angles at $A$ is $360^{\circ}$. Thus the three polygons form a new polygon with $A$ as an interior point. What is the largest possible perimeter that this polygon can have? $\mathrm{(A) \ }12 \qquad \mathrm{(B) \ }14 \qquad \mathrm{(C) \ }18 \qquad \mathrm{(D) \ }21 \qquad \mathrm{(E) \ } 24$	21
3. Vasya thought of two numbers. Their sum equals their product and equals their quotient. What numbers did Vasya think of?	\frac{1}{2},-1
One hundred and one of the squares of an $n\times n$ table are colored blue. It is known that there exists a unique way to cut the table to rectangles along boundaries of its squares with the following property: every rectangle contains exactly one blue square. Find the smallest possible $n$.	101
72. Given three integers $x, y, z$ satisfying $x+y+z=100$, and $x<y<2z$, then the minimum value of $z$ is	21
Let $ n\left(A\right)$ be the number of distinct real solutions of the equation $ x^{6} \minus{}2x^{4} \plus{}x^{2} \equal{}A$. When $ A$ takes every value on real numbers, the set of values of $ n\left(A\right)$ is $\textbf{(A)}\ \left\{0,1,2,3,4,5,6\right\} \\ \textbf{(B)}\ \left\{0,2,4,6\right\} \\ \textbf{(C)}\ \left\{0,3,4,6\right\} \\ \textbf{(D)}\ \left\{0,2,3,4,6\right\} \\ \textbf{(E)}\ \left\{0,2,3,4\right\}$	D
Example 7 Find all positive integers $n$ that satisfy the following condition: there exist two complete residue systems modulo $n$, $a_{i}$ and $b_{i} (1 \leqslant i \leqslant n)$, such that $a_{i} b_{i} (1 \leqslant i \leqslant n)$ is also a complete residue system modulo $n$. [2]	1,2
Say a positive integer $n>1$ is $d$-coverable if for each non-empty subset $S\subseteq \{0, 1, \ldots, n-1\}$, there exists a polynomial $P$ with integer coefficients and degree at most $d$ such that $S$ is exactly the set of residues modulo $n$ that $P$ attains as it ranges over the integers. For each $n$, find the smallest $d$ such that $n$ is $d$-coverable, or prove no such $d$ exists. [i]Proposed by Carl Schildkraut[/i]	d = n-1
7.041. $\lg \sqrt{5^{x(13-x)}}+11 \lg 2=11$.	2;11
12. In the park, there are two rivers $O M$ and $O N$ converging at point $O$ (as shown in Figure 6, $\angle M O N=60^{\circ}$. On the peninsula formed by the two rivers, there is an ancient site $P$. It is planned to build a small bridge $Q$ and $R$ on each of the two rivers, and to construct three small roads to connect the two bridges $Q$, $R$, and the ancient site $P$. If the distance from the ancient site $P$ to the two rivers is $50 \sqrt{3} \mathrm{~m}$ each, then the minimum value of the sum of the lengths of the three small roads is $\qquad$ m.	300
## Task 3 Write under each number its double: $5 \quad 6 \quad 14$	10\quad12\quad28
C2. A circle lies within a rectangle and touches three of its edges, as shown. The area inside the circle equals the area inside the rectangle but outside the circle. What is the ratio of the length of the rectangle to its width?	\pi:2
9. Given the function $$ f(x)=\left\{\begin{array}{ll} 3 x-1 & x \leqslant 1 ; \\ \frac{2 x+3}{x-1} & x>1 \end{array}\right. $$ If the graph of the function $y=g(x)$ is symmetric to the graph of the function $y=f^{-1}(x+1)$ about the line $y=x$, then the value of $g(11)$ is	\frac{3}{2}
[ Examples and counterexamples. Constructions ] [ Partitions into pairs and groups; bijections ] [ Decimal number system ] A subset $X$ of the set of "two-digit" numbers $00,01, \ldots, 98,99$ is such that in any infinite sequence of digits, there are two adjacent digits that form a number in $X$. What is the smallest number of elements that $X$ can contain?	55
3. Four balls with equal radii $r$ are given, and they touch each other pairwise. A fifth ball is described around them. Find the radius of the fifth ball.	\frac{r}{2}(2+\sqrt{6})
3. As shown in Figure 1, the incircle of $\triangle A B C$ is $\odot O$. A line parallel to $B C$ is drawn through $O$, intersecting $A B$ and $A C$ at points $D$ and $E$, respectively. If the lengths of the three sides of $\triangle A B C$, $B C$, $C A$, and $A B$, are 8, 7, and 5, respectively, then the length of $D E$ is $\qquad$	\frac{24}{5}
## Task 2 Solve the equations. $$ 6538+1603+x=14000 \quad y-835-642=526 $$	5859;2003
26th Putnam 1965 Problem A1 How many positive integers divide at least one of 10 40 and 20 30 ? Solution	2301
5. We have cards with numbers 5, 6, 7, ..., 55 (each card has one number). What is the maximum number of cards we can select so that the sum of the numbers on any two selected cards is not a palindrome? (A palindrome is a number that reads the same backward as forward.)	25
4. Let $S_{n}$ be the sum of the first $n$ terms of a geometric sequence. If $x=S_{n}^{2}+S_{2 n}^{2}, y=S_{n}\left(S_{2 n}+S_{3 n}\right)$, then $x-y(\quad)$. (A) is 0 (B) is a positive number (C) is a negative number (D) is sometimes positive and sometimes negative	A
23. In our school netball league a team gains a certain whole number of points if it wins a game, a lower whole number of points if it draws a game and no points if it loses a game. After 10 games my team has won 7 games, drawn 3 and gained 44 points. My sister's team has won 5 games, drawn 2 and lost 3 . How many points has her team gained? A 28 B 29 C 30 D 31 E 32	31
3. As in Figure 1, in quadrilateral $A B C D$, it is known that $$ \begin{array}{l} \text { } \angle A C B=\angle B A D \\ =105^{\circ}, \angle A B C= \\ \angle A D C=45^{\circ} \text {. Then } \\ \angle C A D=(\quad) . \end{array} $$ (A) $65^{\circ}$ (B) $70^{\circ}$ (C) $75^{\circ}$ (D) $80^{\circ}$	C
Three regular solids: tetrahedron, hexahedron, and octahedron have equal surface areas. What is the ratio of their volumes?	1:\sqrt[4]{3}:\sqrt[4]{4}
10. (5 points) A rectangular prism, if the length is reduced by 2 cm, the width and height remain unchanged, the volume decreases by 48 cubic cm; if the width is increased by 3 cm, the length and height remain unchanged, the volume increases by 99 cubic cm; if the height is increased by 4 cm, the length and width remain unchanged, the volume increases by 352 cubic cm. The surface area of the original rectangular prism is $\qquad$ square cm.	290
[ Counting in two ways ] [ Different tasks on cutting ] Inside a square, 100 points are marked. The square is divided into triangles in such a way that the vertices of the triangles are only the 100 marked points and the vertices of the square, and for each triangle in the partition, each marked point either lies outside this triangle or is its vertex (such partitions are called triangulations). Find the number of triangles in the partition.	202
4. There is 25 ml of a $70\%$ acetic acid solution and 500 ml of a $5\%$ acetic acid solution. Find the maximum volume of a $9\%$ acetic acid solution that can be obtained from the available solutions (no dilution with water is allowed). #	406.25
A3. On the diagonal $B D$ of the rectangle $A B C D$, there are points $E$ and $F$ such that the lines $A E$ and $C F$ are perpendicular to the diagonal $B D$ (see figure). The area of triangle $A B E$ is equal to $\frac{1}{3}$ of the area of rectangle $A B C D$. What is the ratio $\|A D\|:\|A B\|$? (A) $2: \sqrt{5}$ (B) $1: \sqrt{2}$ (C) $2: 3$ (D) $1: \sqrt{3}$ (E) $1: 2$ ![](https://cdn.mathpix.com/cropped/2024_06_07_e347356273e0c1397da7g-14.jpg?height=280&width=354&top_left_y=1419&top_left_x=1545)	1:\sqrt{2}
How many times does the digit 0 appear in the integer equal to $20^{10}$ ?	11
8. Given $x, y>0$, and $x+2 y=2$. Then the minimum value of $\frac{x^{2}}{2 y}+\frac{4 y^{2}}{x}$ is . $\qquad$	2
## Task 2 - 160522 Two young pioneers in their rowing boat covered a distance of $1 \mathrm{~km}$ and $200 \mathrm{~m}$ downstream in 10 minutes. How much time did they need to row the same distance upstream, if they covered on average $40 \mathrm{~m}$ less per minute than on the way downstream?	15
The $25$ member states of the European Union set up a committee with the following rules: 1) the committee should meet daily; 2) at each meeting, at least one member should be represented; 3) at any two different meetings, a different set of member states should be represented; 4) at $n^{th}$ meeting, for every $k<n$, the set of states represented should include at least one state that was represented at the $k^{th}$ meeting. For how many days can the committee have its meetings?	2^{24}
3.1. Kolya had 10 sheets of paper. On the first step, he chooses one sheet and divides it into two parts. On the second step, he chooses one sheet from the available ones and divides it into 3 parts, on the third step, he chooses one sheet from the available ones and divides it into 4, and so on. After which step will the number of sheets first exceed 500?	31
Let's find two numbers whose sum is 316 and whose least common multiple is 4560.	240,76
1. If the function $f(x)=3 \cos \left(\omega x+\frac{\pi}{6}\right)-\sin \left(\omega x-\frac{\pi}{3}\right)(\omega>0)$ has the smallest positive period of $\pi$, then the maximum value of $f(x)$ in the interval $\left[0, \frac{\pi}{2}\right]$ is $\qquad$ .	2\sqrt{3}
Example 3 Solve the equation $5^{x+1}=3^{x^{2}-1}$.	-1or\log_{3}5+1
For each prime $p$, let $\mathbb S_p = \{1, 2, \dots, p-1\}$. Find all primes $p$ for which there exists a function $f\colon \mathbb S_p \to \mathbb S_p$ such that \[ n \cdot f(n) \cdot f(f(n)) - 1 \; \text{is a multiple of} \; p \] for all $n \in \mathbb S_p$. [i]Andrew Wen[/i]	2
G7.1 Let $p, q, r$ be the three sides of triangle $P Q R$. If $p^{4}+q^{4}+r^{4}=2 r^{2}\left(p^{2}+q^{2}\right)$, find $a$, where $a=\cos ^{2} R$ and $R$ denotes the angle opposite $r$.	\frac{1}{2}
Question 4: Let $\mathrm{A}=[-2,4), \mathrm{B}=\left\{\mathrm{x} \mid \mathrm{x}^{2}-\mathrm{ax}-4 \leq 0\right\}$, if $\mathrm{B} \subseteq \mathrm{A}$, then the range of the real number $\mathrm{a}$ is $\qquad$ -	[0,3)

Let $r$ be a nonzero real number. The values of $z$ which satisfy the equation \[ r^4z^4 + (10r^6-2r^2)z^2-16r^5z+(9r^8+10r^4+1) = 0 \] are plotted on the complex plane (i.e. using the real part of each root as the x-coordinate and the imaginary part as the y-coordinate). Show that the area of the convex quadrilateral with these points as vertices is independent of $r$, and find this area.

8

Find all strictly increasing sequences of positive integers $a_1, a_2, \ldots$ with $a_1=1$, satisfying $$3(a_1+a_2+\ldots+a_n)=a_{n+1}+\ldots+a_{2n}$$ for all positive integers $n$.

a_n = 2n - 1

7. The minor axis of the ellipse $\rho=\frac{1}{2-\cos \theta}$ is equal to 保留了源文本的换行和格式。

\frac{2\sqrt{3}}{3}

15. Right trapezoid $A B C D$, the upper base is 1, the lower base is 7, connecting point $E$ on side $A B$ and point $F$ on side $D C$, forming a line segment $E F$ parallel to $A D$ and $B C$ that divides the area of the right trapezoid into two equal parts, then the length of line segment $E F$ is $\qquad$

5

Initially Alex, Betty, and Charlie had a total of $444$ peanuts. Charlie had the most peanuts, and Alex had the least. The three numbers of peanuts that each person had formed a geometric progression. Alex eats $5$ of his peanuts, Betty eats $9$ of her peanuts, and Charlie eats $25$ of his peanuts. Now the three numbers of peanuts each person has forms an arithmetic progression. Find the number of peanuts Alex had initially.

108

8. Given $x, y \in \mathbf{R}$, for any $n \in \mathbf{Z}_{+}$, $n x+\frac{1}{n} y \geqslant 1$. Then the minimum value of $41 x+2 y$ is $\qquad$

9

3. The area of the closed figure enclosed by the curve $x^{2}+2 y^{2} \leqslant 4|y|$ is $\qquad$

2 \sqrt{2} \pi

16) How many of the following statements about natural numbers are true? (i) Given any two consecutive odd numbers, at least one is prime. (ii) Given any three consecutive odd numbers, at least two are prime. (iii) Given any four consecutive odd numbers, at least one is not prime. (iv) Given any five consecutive odd numbers, at least two are not prime. (A) None (B) one (C) two (D) three (E) four.

B

[b]H[/b]orizontal parallel segments $AB=10$ and $CD=15$ are the bases of trapezoid $ABCD$. Circle $\gamma$ of radius $6$ has center within the trapezoid and is tangent to sides $AB$, $BC$, and $DA$. If side $CD$ cuts out an arc of $\gamma$ measuring $120^{\circ}$, find the area of $ABCD$.

\frac{225}{2}

7. If $p$ and $q$ are both prime numbers, the number of divisors $d(a)$ of the natural number $a=p^{\alpha} q^{\beta}$ is given by the formula $$ d(a)=(\alpha+1)(\beta+1) $$ For example, $12=2^{2} \times 3^{1}$, so the number of divisors of 12 is $$ d(12)=(2+1)(1+1)=6, $$ with the divisors being $1, 2, 3, 4, 6$, and 12. According to the given formula, please answer: Among the divisors of $20^{30}$ that are less than $20^{15}$, how many are not divisors of $20^{15}$? $\qquad$

450

3. Four identical small rectangles are put together to form a large rectangle as shown. The length of a shorter side of each small rectangle is $10 \mathrm{~cm}$. What is the length of a longer side of the large rectangle? A $50 \mathrm{~cm}$ B $40 \mathrm{~cm}$ C $30 \mathrm{~cm}$ D $20 \mathrm{~cm}$ E $10 \mathrm{~cm}$

40

4. Given the quadratic function $y=x^{2}-x+a$ whose graph intersects the $x$-axis at two distinct points, the sum of the distances from these points to the origin does not exceed 5. Then the range of values for $a$ is $\qquad$ .

-6 \leqslant a < \frac{1}{4}

3. Let trapezium have bases $A B$ and $C D$, measuring $5 \mathrm{~cm}$ and $1 \mathrm{~cm}$, respectively. Let $M$ and $N$ be points on $A D$ and $B C$ such that the segment $M N$ is parallel to the base $A B$, and the area of quadrilateral $A B N M$ is twice the area of quadrilateral $C D M N$. How many centimeters does the segment $M N$ measure?

3

4. There are 8 cards; one side of each is clean, while the other side has the letters: И, Я, Л, З, Г, О, О, О printed on them. The cards are placed on the table with the clean side up, shuffled, and then sequentially flipped over one by one. What is the probability that the letters will form the word ЗОоЛОГИЯ when they appear in sequence?

\frac{1}{6720}

1. Given a quadratic trinomial $f(x)$ such that the equation $(f(x))^{3}-4 f(x)=0$ has exactly three solutions. How many solutions does the equation $(f(x))^{2}=1$ have?

2

2. "Real numbers $a=b=c$" is the ( ) condition for the inequality $a^{3}+b^{3}+c^{3} \geqslant 3 a b c$ to hold with equality. (A) Sufficient but not necessary (B) Sufficient and necessary (C) Necessary but not sufficient (D) Neither sufficient nor necessary

A

1. 2. 1 * Let $A=\{1,2,3, m\}, B=\left\{4,7, n^{4}, n^{2}+3 n\right\}$, the correspondence rule $f: a \rightarrow$ $b=pa+q$ is a one-to-one mapping from $A$ to $B$. It is known that $m, n$ are positive integers, and the image of 1 is 4, the preimage of 7 is 2. Find the values of $p, q, m, n$.

(3,1,5,2)

Find the smallest positive integer that, when multiplied by 1999, the last four digits of the resulting number are 2001.

5999

6、Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=1, a_{n+1}=\frac{(n+1) a_{n}}{2 n+a_{n}}\left(n \in N_{+}\right)$. Then $\sum_{k=1}^{2017} \frac{k}{a_{k}}=$

2^{2018}-2019

Let $d(m)$ denote the number of positive integer divisors of a positive integer $m$. If $r$ is the number of integers $n \leqslant 2023$ for which $\sum_{i=1}^{n} d(i)$ is odd. , find the sum of digits of $r.$

18

In a $12\times 12$ grid, colour each unit square with either black or white, such that there is at least one black unit square in any $3\times 4$ and $4\times 3$ rectangle bounded by the grid lines. Determine, with proof, the minimum number of black unit squares.

12

9.077. $2^{x+2}-2^{x+3}-2^{x+4}>5^{x+1}-5^{x+2}$.

x\in(0;\infty)

7. It is known that for some natural numbers $a, b$, the number $N=\frac{a^{2}+b^{2}}{a b-1}$ is also natural. Find all possible values of $N$. --- The provided text has been translated into English while preserving the original formatting and structure.

5

\%EA360 * Find the number of positive integer solutions $(m, n, r)$ for the indeterminate equation $$ m n+n r+m r=2(m+n+r) $$

7

2. Solve the equation $$ \sqrt{\frac{x-3}{11}}+\sqrt{\frac{x-4}{10}}=\sqrt{\frac{x-11}{3}}+\sqrt{\frac{x-10}{4}} $$

14

$3+$ [ The transfer helps solve the task_ ] On the side AB of the square ABCD, an equilateral triangle AKB was constructed (outside). Find the radius of the circle circumscribed around triangle CKD, if $\mathrm{AB}=1$. #

1

1. If $x+y+z=0$, simplify the expression $$ \frac{x^{7}+y^{7}+z^{7}}{x y z\left(x^{4}+y^{4}+z^{4}\right)} $$ Hint: first calculate $(x+y)^{4}$ and $(x+y)^{6}$.

\frac{7}{2}

4. Given that the complex number $z$ satisfies $z^{3}+z=2|z|^{2}$. Then all possible values of $z$ are $\qquad$

0, 1, -1 \pm 2i

3. On the diagonal $BD$ of square $ABCD$, take two points $E$ and $F$, such that the extension of $AE$ intersects side $BC$ at point $M$, and the extension of $AF$ intersects side $CD$ at point $N$, with $CM = CN$. If $BE = 3$, $EF = 4$, what is the length of the diagonal of this square?

10

[ Special cases of parallelepipeds (other).] Area of the section The base of a right parallelepiped is a rhombus, the area of which is equal to $Q$. The areas of the diagonal sections are $S 1$ and $S 2$. Find the volume of the parallelepiped.

\sqrt{\frac{QS_{1}S_{2}}{2}}

Example 3 Given real numbers $x_{1}, x_{2}, \cdots, x_{10}$ satisfy $\sum_{i=1}^{10}\left|x_{i}-1\right| \leqslant 4, \sum_{i=1}^{10}\left|x_{i}-2\right| \leqslant 6$. Find the average value $\bar{x}$ of $x_{1}, x_{2}, \cdots, x_{10}$. (2012, Zhejiang Province High School Mathematics Competition)

1.4

In 1998, the population of Canada was 30.3 million. Which of the options below represents the population of Canada in 1998? (a) 30300000 (b) 303000000 (c) 30300 (d) 303300 (e) 30300000000

30300000

Let $S$ be the set of all partitions of $2000$ (in a sum of positive integers). For every such partition $p$, we deﬁne $f (p)$ to be the sum of the number of summands in $p$ and the maximal summand in $p$. Compute the minimum of $f (p)$ when $p \in S .$

90

Let $x_0,x_1,x_2,\dots$ be the sequence such that $x_0=1$ and for $n\ge 0,$ \[x_{n+1}=\ln(e^{x_n}-x_n)\] (as usual, the function $\ln$ is the natural logarithm). Show that the infinite series \[x_0+x_1+x_2+\cdots\] converges and find its sum.

e - 1

Given $u_0,u_1$ with $0<u_0,u_1<1$, define the sequence $(u_n)$ recurrently by the formula $$u_{n+2}=\frac12\left(\sqrt{u_{n+1}}+\sqrt{u_n}\right).$$(a) Prove that the sequence $u_n$ is convergent and find its limit. (b) Prove that, starting from some index $n_0$, the sequence $u_n$ is monotonous.

1

5. What is the total area in $\mathrm{cm}^{2}$ of the shaded region? A 50 B 80 C 100 D 120 E 150

100

Given a right triangle $A B C$. From the vertex $B$ of the right angle, a median $B D$ is drawn. Let $K$ be the point of tangency of side $A D$ of triangle $A B D$ with the inscribed circle of this triangle. Find the acute angles of triangle $A B C$, if $K$ bisects $A D$.

30,60

3. 1. 4 * The terms of the sequence $\left\{a_{n}\right\}$ are positive, and the sum of the first $n$ terms is $S_{n}$, satisfying $S_{n}=\frac{1}{2}\left(a_{n}+\frac{1}{a_{n}}\right)$, find the general term of the sequence $\left\{a_{n}\right\}$.

a_{n}=\sqrt{n}-\sqrt{n-1}

Example 19 (Problem 1506 from "Mathematics Bulletin") In $\triangle A B C$, $A B=A C$, the angle bisector of $\angle B$ intersects $A C$ at $D$, and $B C=B D+A D$. Find $\angle A$. Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.

100

16. If $y+4=(x-2)^{2}, x+4=(y-2)^{2}$, and $x \neq y$, then the value of $x^{2}+y^{2}$ is ( ). (A) 10 (B) 15 (C) 20 (D) 25 (E) 30

15

6. Solve the equation $\sqrt{2 x^{2}+3 x+2}-\sqrt{2 x^{2}+3 x-5}=1$.

2,-\frac{7}{2}

An integer's product of all its positive integer divisors is $2^{120} \cdot 3^{60} \cdot 5^{90}$. What could this number be?

18000

(1) Given the set $M=\{1,3,5,7,9\}$, if the non-empty set $A$ satisfies: the elements of $A$ each increased by 4 form a subset of $M$, and the elements of $A$ each decreased by 4 also form a subset of $M$, then $A=$ $\qquad$ .

{5}

Place a triangle under a magnifying glass that enlarges by 5 times, the perimeter is $\qquad$ times that of the original triangle, and the area is $\qquad$ times that of the original triangle.

5,25

10. The Kingdom of Geometry awards badges to outstanding citizens of 2021. The badges are rectangular (including squares), and these rectangular badges are all different, but their area values are 2021 times their perimeter values, and both length and width are natural numbers. Therefore, the maximum number of different badges for the year 2021 is $\qquad$ kinds.

14

Example 3. The circumference of a circle is measured to be 10.7 meters, find its radius.

1.70

10. The function $\operatorname{SPF}(n)$ denotes the sum of the prime factors of $n$, where the prime factors are not necessarily distinct. For example, $120=2^{3} \times 3 \times 5$, so $\operatorname{SPF}(120)=2+2+2+3+5=14$. Find the value of $\operatorname{SPF}\left(2^{22}-4\right)$.

100

## Task A-2.7. In a basketball tournament, each team plays exactly twice against each of the other teams. A win brings 2 points, a loss 0 points, and there are no draws. Determine all natural numbers $n$ for which there exists a basketball tournament with $n$ teams where one team, the tournament winner, has 26 points, and exactly two teams have the lowest number of points, which is 20 points.

12

T10. Find all positive integers $a$ such that the quadratic equation $a x^{2}+2(2 a-1) x+4(a-3)=0$ has at least one integer root. (3rd Zu Chongzhi Cup Junior High School Mathematics Invitational Competition)

a=1,3,6,10

4. As shown in Figure 1, in the Cartesian coordinate system $x O y$, $A$ and $B$ are two points on the graph of $y=\frac{1}{x}$ in the first quadrant. If $\triangle A O B$ is an equilateral triangle, then its area $S$ is ( ). (A) $2 \sqrt{3}$ (B) 3 (C) $\sqrt{3}$ (D) $\frac{\sqrt{3}}{2}$

\sqrt{3}

1. [4] If $A=10^{9}-987654321$ and $B=\frac{123456789+1}{10}$, what is the value of $\sqrt{A B}$ ?

12345679

1. Let $\mathbf{Z}$ denote the set of all integers. Find all functions $f: \mathbf{Z} \rightarrow \mathbf{Z}$, such that for any integers $a, b$, we have $$ f(2 a)+2 f(b)=f(f(a+b)) . $$

f(x)=0

The centers of three circles, each touching the other two externally, are located at the vertices of a right triangle. These circles are internally tangent to a fourth circle. Find the radius of the fourth circle if the perimeter of the right triangle is $2 p$. #

p

4. As shown in Figure 3, in $\triangle A B C$, $M$ is the midpoint of side $B C$, and $M D \perp A B, M E \perp A C$, with $D$ and $E$ being the feet of the perpendiculars. If $B D=2, C E=1$, and $D E \parallel B C$, then $D M^{2}$ equals $\qquad$

1

16. Given functions $f(x), g(x)(x \in \mathrm{R})$, let the solution set of the inequality $|f(x)|+|g(x)|<a (a>0)$ be $M$, and the solution set of the inequality $|f(x)+g(x)|<a (a>0)$ be $N$, then the relationship between the solution sets $M$ and $N$ is ( ). A. $M \subseteq N$ B. $M=N$ C. $N \varsubsetneqq M$ D. $M \varsubsetneqq N$

A

1. How many positive integers less than 2019 are divisible by either 18 or 21 , but not both?

176

13. There is a sequence of numbers, the first 4 numbers are $2,0,1,8$, starting from the 5th number, each number is the unit digit of the sum of the previous 4 adjacent numbers, will the sequence $2,0,1, 7$ appear in this sequence of numbers?

No

Example 5.17 Given non-negative real numbers $a, b, c$ satisfy $a b + b c + c a + 6 a b c = 9$. Determine the maximum value of $k$ such that the following inequality always holds. $$a + b + c + k a b c \geqslant k + 3$$

3

4. Find the range of real numbers $a$ such that the inequality $$ \begin{array}{l} \sin 2 \theta-(2 \sqrt{2}+a \sqrt{2}) \sin \left(\theta+\frac{\pi}{4}\right)-2 \sqrt{2} \sec \left(\theta-\frac{\pi}{4}\right) \\ >-3-2 a \end{array} $$ holds for all $\theta \in\left[0, \frac{\pi}{2}\right]$. (2000, Tianjin High School Mathematics Competition)

a>3

In $\triangle A B C$, $A B$ is the longest side, $\sin A \sin B=\frac{2-\sqrt{3}}{4}$, then the maximum value of $\cos A \cos B$ is Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.

\frac{2+\sqrt{3}}{4}

1. Let $A$ and $B$ be two moving points on the ellipse $\frac{x^{2}}{2}+y^{2}=1$, and $O$ be the origin. Also, $\overrightarrow{O A} \cdot \overrightarrow{O B}=0$. Let point $P$ be on $AB$, and $O P \perp A B$. Find the value of $|O P|$.

\frac{\sqrt{6}}{3}

4. (15 points) Identical gases are in two thermally insulated vessels with volumes $V_{1}=2$ L and $V_{2}=3$ L. The pressures of the gases are $p_{1}=3$ atm and $p_{2}=4$ atm, and their temperatures are $T_{1}=400$ K and $T_{2}=500$ K, respectively. The gases are mixed. Determine the temperature that will be established in the vessels.

462\mathrm{~K}

Let $ABCDE$ be a convex pentagon such that $AB = BC = CD$ and $\angle BDE = \angle EAC = 30 ^{\circ}$. Find the possible values of $\angle BEC$. [i]Proposed by Josef Tkadlec (Czech Republic)[/i]

60^\circ

Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length $1$. The polygons meet at a point $A$ in such a way that the sum of the three interior angles at $A$ is $360^{\circ}$. Thus the three polygons form a new polygon with $A$ as an interior point. What is the largest possible perimeter that this polygon can have? $\mathrm{(A) \ }12 \qquad \mathrm{(B) \ }14 \qquad \mathrm{(C) \ }18 \qquad \mathrm{(D) \ }21 \qquad \mathrm{(E) \ } 24$

21

3. Vasya thought of two numbers. Their sum equals their product and equals their quotient. What numbers did Vasya think of?

\frac{1}{2},-1

One hundred and one of the squares of an $n\times n$ table are colored blue. It is known that there exists a unique way to cut the table to rectangles along boundaries of its squares with the following property: every rectangle contains exactly one blue square. Find the smallest possible $n$.

101

72. Given three integers $x, y, z$ satisfying $x+y+z=100$, and $x<y<2z$, then the minimum value of $z$ is

21

Let $ n\left(A\right)$ be the number of distinct real solutions of the equation $ x^{6} \minus{}2x^{4} \plus{}x^{2} \equal{}A$. When $ A$ takes every value on real numbers, the set of values of $ n\left(A\right)$ is $\textbf{(A)}\ \left\{0,1,2,3,4,5,6\right\} \\ \textbf{(B)}\ \left\{0,2,4,6\right\} \\ \textbf{(C)}\ \left\{0,3,4,6\right\} \\ \textbf{(D)}\ \left\{0,2,3,4,6\right\} \\ \textbf{(E)}\ \left\{0,2,3,4\right\}$

D

Example 7 Find all positive integers $n$ that satisfy the following condition: there exist two complete residue systems modulo $n$, $a_{i}$ and $b_{i} (1 \leqslant i \leqslant n)$, such that $a_{i} b_{i} (1 \leqslant i \leqslant n)$ is also a complete residue system modulo $n$. [2]

1,2

Say a positive integer $n>1$ is $d$-coverable if for each non-empty subset $S\subseteq \{0, 1, \ldots, n-1\}$, there exists a polynomial $P$ with integer coefficients and degree at most $d$ such that $S$ is exactly the set of residues modulo $n$ that $P$ attains as it ranges over the integers. For each $n$, find the smallest $d$ such that $n$ is $d$-coverable, or prove no such $d$ exists. [i]Proposed by Carl Schildkraut[/i]

d = n-1

7.041. $\lg \sqrt{5^{x(13-x)}}+11 \lg 2=11$.

2;11

12. In the park, there are two rivers $O M$ and $O N$ converging at point $O$ (as shown in Figure 6, $\angle M O N=60^{\circ}$. On the peninsula formed by the two rivers, there is an ancient site $P$. It is planned to build a small bridge $Q$ and $R$ on each of the two rivers, and to construct three small roads to connect the two bridges $Q$, $R$, and the ancient site $P$. If the distance from the ancient site $P$ to the two rivers is $50 \sqrt{3} \mathrm{~m}$ each, then the minimum value of the sum of the lengths of the three small roads is $\qquad$ m.

300

## Task 3 Write under each number its double: $5 \quad 6 \quad 14$

10\quad12\quad28

C2. A circle lies within a rectangle and touches three of its edges, as shown. The area inside the circle equals the area inside the rectangle but outside the circle. What is the ratio of the length of the rectangle to its width?

\pi:2

9. Given the function $$ f(x)=\left\{\begin{array}{ll} 3 x-1 & x \leqslant 1 ; \\ \frac{2 x+3}{x-1} & x>1 \end{array}\right. $$ If the graph of the function $y=g(x)$ is symmetric to the graph of the function $y=f^{-1}(x+1)$ about the line $y=x$, then the value of $g(11)$ is

\frac{3}{2}

[ Examples and counterexamples. Constructions ] [ Partitions into pairs and groups; bijections ] [ Decimal number system ] A subset $X$ of the set of "two-digit" numbers $00,01, \ldots, 98,99$ is such that in any infinite sequence of digits, there are two adjacent digits that form a number in $X$. What is the smallest number of elements that $X$ can contain?

55

3. Four balls with equal radii $r$ are given, and they touch each other pairwise. A fifth ball is described around them. Find the radius of the fifth ball.

\frac{r}{2}(2+\sqrt{6})

3. As shown in Figure 1, the incircle of $\triangle A B C$ is $\odot O$. A line parallel to $B C$ is drawn through $O$, intersecting $A B$ and $A C$ at points $D$ and $E$, respectively. If the lengths of the three sides of $\triangle A B C$, $B C$, $C A$, and $A B$, are 8, 7, and 5, respectively, then the length of $D E$ is $\qquad$

\frac{24}{5}

## Task 2 Solve the equations. $$ 6538+1603+x=14000 \quad y-835-642=526 $$

5859;2003

26th Putnam 1965 Problem A1 How many positive integers divide at least one of 10 40 and 20 30 ? Solution

2301

5. We have cards with numbers 5, 6, 7, ..., 55 (each card has one number). What is the maximum number of cards we can select so that the sum of the numbers on any two selected cards is not a palindrome? (A palindrome is a number that reads the same backward as forward.)

25

4. Let $S_{n}$ be the sum of the first $n$ terms of a geometric sequence. If $x=S_{n}^{2}+S_{2 n}^{2}, y=S_{n}\left(S_{2 n}+S_{3 n}\right)$, then $x-y(\quad)$. (A) is 0 (B) is a positive number (C) is a negative number (D) is sometimes positive and sometimes negative

A

23. In our school netball league a team gains a certain whole number of points if it wins a game, a lower whole number of points if it draws a game and no points if it loses a game. After 10 games my team has won 7 games, drawn 3 and gained 44 points. My sister's team has won 5 games, drawn 2 and lost 3 . How many points has her team gained? A 28 B 29 C 30 D 31 E 32

31

3. As in Figure 1, in quadrilateral $A B C D$, it is known that $$ \begin{array}{l} \text { } \angle A C B=\angle B A D \\ =105^{\circ}, \angle A B C= \\ \angle A D C=45^{\circ} \text {. Then } \\ \angle C A D=(\quad) . \end{array} $$ (A) $65^{\circ}$ (B) $70^{\circ}$ (C) $75^{\circ}$ (D) $80^{\circ}$

C

Three regular solids: tetrahedron, hexahedron, and octahedron have equal surface areas. What is the ratio of their volumes?

1:\sqrt[4]{3}:\sqrt[4]{4}

10. (5 points) A rectangular prism, if the length is reduced by 2 cm, the width and height remain unchanged, the volume decreases by 48 cubic cm; if the width is increased by 3 cm, the length and height remain unchanged, the volume increases by 99 cubic cm; if the height is increased by 4 cm, the length and width remain unchanged, the volume increases by 352 cubic cm. The surface area of the original rectangular prism is $\qquad$ square cm.

290

[ Counting in two ways ] [ Different tasks on cutting ] Inside a square, 100 points are marked. The square is divided into triangles in such a way that the vertices of the triangles are only the 100 marked points and the vertices of the square, and for each triangle in the partition, each marked point either lies outside this triangle or is its vertex (such partitions are called triangulations). Find the number of triangles in the partition.

202

4. There is 25 ml of a $70\%$ acetic acid solution and 500 ml of a $5\%$ acetic acid solution. Find the maximum volume of a $9\%$ acetic acid solution that can be obtained from the available solutions (no dilution with water is allowed). #

406.25

A3. On the diagonal $B D$ of the rectangle $A B C D$, there are points $E$ and $F$ such that the lines $A E$ and $C F$ are perpendicular to the diagonal $B D$ (see figure). The area of triangle $A B E$ is equal to $\frac{1}{3}$ of the area of rectangle $A B C D$. What is the ratio $|A D|:|A B|$? (A) $2: \sqrt{5}$ (B) $1: \sqrt{2}$ (C) $2: 3$ (D) $1: \sqrt{3}$ (E) $1: 2$ ![](https://cdn.mathpix.com/cropped/2024_06_07_e347356273e0c1397da7g-14.jpg?height=280&width=354&top_left_y=1419&top_left_x=1545)

1:\sqrt{2}

How many times does the digit 0 appear in the integer equal to $20^{10}$ ?

11

8. Given $x, y>0$, and $x+2 y=2$. Then the minimum value of $\frac{x^{2}}{2 y}+\frac{4 y^{2}}{x}$ is . $\qquad$

2

## Task 2 - 160522 Two young pioneers in their rowing boat covered a distance of $1 \mathrm{~km}$ and $200 \mathrm{~m}$ downstream in 10 minutes. How much time did they need to row the same distance upstream, if they covered on average $40 \mathrm{~m}$ less per minute than on the way downstream?

15

The $25$ member states of the European Union set up a committee with the following rules: 1) the committee should meet daily; 2) at each meeting, at least one member should be represented; 3) at any two different meetings, a different set of member states should be represented; 4) at $n^{th}$ meeting, for every $k<n$, the set of states represented should include at least one state that was represented at the $k^{th}$ meeting. For how many days can the committee have its meetings?

2^{24}

3.1. Kolya had 10 sheets of paper. On the first step, he chooses one sheet and divides it into two parts. On the second step, he chooses one sheet from the available ones and divides it into 3 parts, on the third step, he chooses one sheet from the available ones and divides it into 4, and so on. After which step will the number of sheets first exceed 500?

31

Let's find two numbers whose sum is 316 and whose least common multiple is 4560.

240,76

1. If the function $f(x)=3 \cos \left(\omega x+\frac{\pi}{6}\right)-\sin \left(\omega x-\frac{\pi}{3}\right)(\omega>0)$ has the smallest positive period of $\pi$, then the maximum value of $f(x)$ in the interval $\left[0, \frac{\pi}{2}\right]$ is $\qquad$ .

2\sqrt{3}

Example 3 Solve the equation $5^{x+1}=3^{x^{2}-1}$.

-1or\log_{3}5+1

For each prime $p$, let $\mathbb S_p = \{1, 2, \dots, p-1\}$. Find all primes $p$ for which there exists a function $f\colon \mathbb S_p \to \mathbb S_p$ such that \[ n \cdot f(n) \cdot f(f(n)) - 1 \; \text{is a multiple of} \; p \] for all $n \in \mathbb S_p$. [i]Andrew Wen[/i]

2

G7.1 Let $p, q, r$ be the three sides of triangle $P Q R$. If $p^{4}+q^{4}+r^{4}=2 r^{2}\left(p^{2}+q^{2}\right)$, find $a$, where $a=\cos ^{2} R$ and $R$ denotes the angle opposite $r$.

\frac{1}{2}

Question 4: Let $\mathrm{A}=[-2,4), \mathrm{B}=\left\{\mathrm{x} \mid \mathrm{x}^{2}-\mathrm{ax}-4 \leq 0\right\}$, if $\mathrm{B} \subseteq \mathrm{A}$, then the range of the real number $\mathrm{a}$ is $\qquad$ -

[0,3)