weqweasdas/dapo_and_openr1_can_be_evaluated_by_daporm_deduplicate

problem stringlengths 16 7.47k	answer stringlengths 1 261
Prove that number $1$ can be represented as a sum of a finite number $n$ of real numbers, less than $1,$ not necessarily distinct, which contain in their decimal representation only the digits $0$ and/or $7.$ Which is the least possible number $n$?	8
A right-angled triangle has side lengths that are integers. What could be the last digit of the area's measure, if the length of the hypotenuse is not divisible by 5?	0
Let $a_n\ (n\geq 1)$ be the value for which $\int_x^{2x} e^{-t^n}dt\ (x\geq 0)$ is maximal. Find $\lim_{n\to\infty} \ln a_n.$	-\ln 2
6. As shown in Figure 2, let $P$ be a point inside the equilateral $\triangle ABC$ with side length 12. Draw perpendiculars from $P$ to the sides $BC$, $CA$, and $AB$, with the feet of the perpendiculars being $D$, $E$, and $F$ respectively. Given that $PD: PE: PF = 1: 2: 3$. Then, the area of quadrilateral $BDPF$ is	11 \sqrt{3}
Problem 6. (8 points) In the plane, there is a non-closed, non-self-intersecting broken line consisting of 31 segments (adjacent segments do not lie on the same straight line). For each segment, the line defined by it is constructed. It is possible for some of the 31 constructed lines to coincide. What is the minimum number of different lines that can be obtained? Answer. 9.	9
Four, (50 points) In an $n \times n$ grid, fill each cell with one of the numbers 1 to $n^{2}$. If no matter how you fill it, there must be two adjacent cells where the difference between the two numbers is at least 1011, find the minimum value of $n$. --- The translation preserves the original text's formatting and structure.	2020
On a board, the numbers from 1 to 2009 are written. A couple of them are erased and instead of them, on the board is written the remainder of the sum of the erased numbers divided by 13. After a couple of repetition of this erasing, only 3 numbers are left, of which two are 9 and 999. Find the third number.	8
4. The number of real solutions to the equation $\left\|x^{2}-3 x+2\right\|+\left\|x^{2}+2 x-3\right\|=11$ is ( ). (A) 0 (B) 1 (C) 2 (D) 4	C
## Problem Statement Calculate the limit of the numerical sequence: $\lim _{n \rightarrow \infty} \frac{(n+1)^{4}-(n-1)^{4}}{(n+1)^{3}+(n-1)^{3}}$	4
6. Let $[x]$ denote the greatest integer not exceeding the real number $x$, $$ \begin{array}{c} S=\left[\frac{1}{1}\right]+\left[\frac{2}{1}\right]+\left[\frac{1}{2}\right]+\left[\frac{2}{2}\right]+\left[\frac{3}{2}\right]+ \\ {\left[\frac{4}{2}\right]+\left[\frac{1}{3}\right]+\left[\frac{2}{3}\right]+\left[\frac{3}{3}\right]+\left[\frac{4}{3}\right]+} \\ {\left[\frac{5}{3}\right]+\left[\frac{6}{3}\right]+\cdots} \end{array} $$ up to 2016 terms, where, for a segment with denominator $k$, there are $2 k$ terms $\left[\frac{1}{k}\right],\left[\frac{2}{k}\right], \cdots,\left[\frac{2 k}{k}\right]$, and only the last segment may have fewer than $2 k$ terms. Then the value of $S$ is	1078
19. Given $m \in\{11,13,15,17,19\}$, $n \in\{1999,2000, \cdots, 2018\}$. Then the probability that the unit digit of $m^{n}$ is 1 is ( ). (A) $\frac{1}{5}$ (B) $\frac{1}{4}$ (C) $\frac{3}{10}$ (D) $\frac{7}{20}$ (E) $\frac{2}{5}$	E
1. A line $l$ intersects a hyperbola $c$, then the maximum number of intersection points is ( ). A. 1 B. 2 C. 3 D. 4	B
4. As shown in Figure 1, in the right triangular prism $A B C-A_{1} B_{1} C_{1}$, $A A_{1}=A B=A C$, and $M$ and $Q$ are the midpoints of $C C_{1}$ and $B C$ respectively. If for any point $P$ on the line segment $A_{1} B_{1}$, $P Q \perp A M$, then $\angle B A C$ equals ( ). (A) $30^{\circ}$ (B) $45^{\circ}$ (C) $60^{\circ}$ (D) $90^{\circ}$	D
The equation of line $\ell_1$ is $24x-7y = 319$ and the equation of line $\ell_2$ is $12x-5y = 125$. Let $a$ be the number of positive integer values $n$ less than $2023$ such that for both $\ell_1$ and $\ell_2$ there exists a lattice point on that line that is a distance of $n$ from the point $(20,23)$. Determine $a$. [i]Proposed by Christopher Cheng[/i] [hide=Solution][i]Solution. [/i] $\boxed{6}$ Note that $(20,23)$ is the intersection of the lines $\ell_1$ and $\ell_2$. Thus, we only care about lattice points on the the two lines that are an integer distance away from $(20,23)$. Notice that $7$ and $24$ are part of the Pythagorean triple $(7,24,25)$ and $5$ and $12$ are part of the Pythagorean triple $(5,12,13)$. Thus, points on $\ell_1$ only satisfy the conditions when $n$ is divisible by $25$ and points on $\ell_2$ only satisfy the conditions when $n$ is divisible by $13$. Therefore, $a$ is just the number of positive integers less than $2023$ that are divisible by both $25$ and $13$. The LCM of $25$ and $13$ is $325$, so the answer is $\boxed{6}$.[/hide]	6
1B. If for the non-zero real numbers $a, b$ and $c$ the equalities $a^{2}+a=b^{2}, b^{2}+b=c^{2}$ and $c^{2}+c=a^{2}$ hold, determine the value of the expression $(a-b)(b-c)(c-a)$.	1
2. As shown in Figure 1, the side length of rhombus $A B C D$ is $a$, and $O$ is a point on the diagonal $A C$, with $O A=a, O B=$ $O C=O D=1$. Then $a$ equals ( ). (A) $\frac{\sqrt{5}+1}{2}$ (B) $\frac{\sqrt{5}-1}{2}$ (C) 1 (D) 2	A
1. Given $a, b>0, a \neq 1$, and $a^{b}=\log _{a} b$, then the value of $a^{a^{b}}-\log _{a} \log _{a} b^{a}$ is	-1
There is a regular $17$-gon $\mathcal{P}$ and its circumcircle $\mathcal{Y}$ on the plane. The vertices of $\mathcal{P}$ are coloured in such a way that $A,B \in \mathcal{P}$ are of different colour, if the shorter arc connecting $A$ and $B$ on $\mathcal{Y}$ has $2^k+1$ vertices, for some $k \in \mathbb{N},$ including $A$ and $B.$ What is the least number of colours which suffices?	4
Find all triples $(m,p,q)$ where $ m $ is a positive integer and $ p , q $ are primes. \[ 2^m p^2 + 1 = q^5 \]	(1, 11, 3)
2. In the complex plane, there are 7 points corresponding to the 7 roots of the equation $x^{7}=$ $-1+\sqrt{3} i$. Among the four quadrants where these 7 points are located, only 1 point is in ( ). (A) the I quadrant (B) the II quadrant (C) the III quadrant (D) the IV quadrant	C
Of the 36 students in Richelle's class, 12 prefer chocolate pie, 8 prefer apple, and 6 prefer blueberry. Half of the remaining students prefer cherry pie and half prefer lemon. For Richelle's pie graph showing this data, how many degrees should she use for cherry pie? $ \text{(A)}\ 10\qquad\text{(B)}\ 20\qquad\text{(C)}\ 30\qquad\text{(D)}\ 50\qquad\text{(E)}\ 72 $	50
Father played chess with uncle. For a won game, the winner received 8 crowns from the opponent, and for a draw, nobody got anything. Uncle won four times, there were five draws, and in the end, father earned 24 crowns. How many games did father play with uncle? (M. Volfová)	16
## Problem 4 Given the numbers $1,2,3, \ldots, 1000$. Find the largest number $m$ with the property that by removing any $m$ numbers from these 1000 numbers, among the $1000-m$ remaining numbers, there exist two such that one divides the other. Selected problems by Prof. Cicortas Marius Note: a) The actual working time is 3 hours. b) All problems are mandatory. c) Each problem is graded from 0 to 7. ## NATIONAL MATHEMATICS OLYMPIAD Local stage - 15.02.2014 ## Grade IX ## Grading Rubric	499
4. Let $A$ and $B$ be $n$-digit numbers, where $n$ is odd, which give the same remainder $r \neq 0$ when divided by $k$. Find at least one number $k$, which does not depend on $n$, such that the number $C$, obtained by appending the digits of $A$ and $B$, is divisible by $k$.	11
Example: Given the radii of the upper and lower bases of a frustum are 3 and 6, respectively, and the height is $3 \sqrt{3}$, the radii $O A$ and $O B$ of the lower base are perpendicular, and $C$ is a point on the generatrix $B B^{\prime}$ such that $B^{\prime} C: C B$ $=1: 2$. Find the shortest distance between points $A$ and $C$ on the lateral surface of the frustum.	4 \sqrt{13-6 \sqrt{2}}
6. Let set $A=\{1,2,3,4,5,6\}$, and a one-to-one mapping $f: A \rightarrow A$ satisfies that for any $x \in A$, $f(f(f(x)))$ $=x$. Then the number of mappings $f$ that satisfy the above condition is ( ). (A) 40 (B) 41 (C) 80 (D) 81	D
5. For a convex $n$-sided polygon, if circles are constructed with each side as the diameter, the convex $n$-sided polygon must be covered by these $n$ circles. Then the maximum value of $n$ is: ( ). (A) 3 . (B) 4 . (C) 5 . (D) Greater than 5 .	B
1. As shown in Figure 1, in the Cartesian coordinate system, the graph of the quadratic function $y=a x^{2}+m c(a \neq$ $0)$ passes through three vertices $A$, $B$, and $C$ of the square $A B O C$, and $a c=-2$. Then the value of $m$ is ( ). (A) 1 (B) -1 (C) 2 (D) -2	A
Let $a_1,a_2,a_3,a_4,a_5$ be distinct real numbers. Consider all sums of the form $a_i + a_j$ where $i,j \in \{1,2,3,4,5\}$ and $i \neq j$. Let $m$ be the number of distinct numbers among these sums. What is the smallest possible value of $m$?	7
7. A circle with a radius of 1 has six points, these six points divide the circle into six equal parts. Take three of these points as vertices to form a triangle. If the triangle is neither equilateral nor isosceles, then the area of this triangle is ( ). (A) $\frac{\sqrt{3}}{3}$ (B) $\frac{\sqrt{3}}{2}$ (C) 1 (D) $\sqrt{2}$ (E) 2	B
15.15 A paper punch can be placed at any point in the plane, and when it operates, it can punch out points at an irrational distance from it. What is the minimum number of paper punches needed to punch out all points in the plane? (51st Putnam Mathematical Competition, 1990)	3
1. A three-digit number is 29 times the sum of its digits. Then this three-digit number is $\qquad$	261
The area of the region in the $xy$-plane satisfying the inequality \[\min_{1 \le n \le 10} \max\left(\frac{x^2+y^2}{4n^2}, \, 2 - \frac{x^2+y^2}{4n^2-4n+1}\right) \le 1\] is $k\pi$, for some integer $k$. Find $k$. [i]Proposed by Michael Tang[/i]	210
2. On a line, several points were marked, including points $A$ and $B$. All possible segments with endpoints at the marked points are considered. Vasya calculated that point $A$ is inside 50 of these segments, and point $B$ is inside 56 segments. How many points were marked? (The endpoints of a segment are not considered its internal points.)	16
Lines $l_1^{}$ and $l_2^{}$ both pass through the origin and make first-quadrant angles of $\frac{\pi}{70}$ and $\frac{\pi}{54}$ radians, respectively, with the positive x-axis. For any line $l^{}_{}$, the transformation $R(l)^{}_{}$ produces another line as follows: $l^{}_{}$ is reflected in $l_1^{}$, and the resulting line is reflected in $l_2^{}$. Let $R^{(1)}(l)=R(l)^{}_{}$ and $R^{(n)}(l)^{}_{}=R\left(R^{(n-1)}(l)\right)$. Given that $l^{}_{}$ is the line $y=\frac{19}{92}x^{}_{}$, find the smallest positive integer $m^{}_{}$ for which $R^{(m)}(l)=l^{}_{}$.	945
2、D Teacher has five vases, these five vases are arranged in a row from shortest to tallest, the height difference between adjacent vases is 2 centimeters, and the tallest vase is exactly equal to the sum of the heights of the two shortest vases, then the total height of the five vases is _ centimeters	50
5. Let the base of the right quadrilateral prism $A^{\prime} B^{\prime} C^{\prime} D^{\prime}-A B C D$ be a rhombus, with an area of $2 \sqrt{3} \mathrm{~cm}^{2}, \angle A B C=60^{\circ}, E$ and $F$ be points on the edges $C C^{\prime}$ and $B B^{\prime}$, respectively, such that $E C=B C=$ $2 F B$. Then the volume of the quadrilateral pyramid $A-B C E F$ is ( ). (A) $\sqrt{3} \mathrm{~cm}^{3}$ (B) $\sqrt{5} \mathrm{~cm}^{3}$ (C) $6 \mathrm{~cm}^{3}$ (D) $\dot{9} \mathrm{~cm}^{3}$	A
3. In circle $\odot O$, the radius $r=5 \mathrm{~cm}$, $A B$ and $C D$ are two parallel chords, and $A B=8 \mathrm{~cm}, C D=6 \mathrm{~cm}$. Then the length of $A C$ has ( ). (A) 1 solution (B) 2 solutions (C) 3 solutions (D) 4 solutions	C
4. Zhao, Qian, Sun, and Li each prepared a gift to give to one of the other three classmates when the new semester started. It is known that Zhao's gift was not given to Qian, Sun did not receive Li's gift, Sun does not know who Zhao gave the gift to, and Li does not know who Qian received the gift from. So, Qian gave the gift to ( ). (A) Zhao (B) Sun (C) Li (D) None of the above	B
3. As shown in Figure 1, $EF$ is the midline of $\triangle ABC$, $O$ is a point on $EF$, and satisfies $OE=2OF$. Then the ratio of the area of $\triangle ABC$ to the area of $\triangle AOC$ is ( ). (A) 2 (B) $\frac{3}{2}$ (C) $\frac{5}{3}$ (D) 3	D
2. Let $x$ be a positive integer, and $y$ is obtained from $x$ when the first digit of $x$ is moved to the last place. Determine the smallest number $x$ for which $3 x=y$.	142857
18. Let $a_{k}$ be the coefficient of $x^{k}$ in the expansion of $$ (x+1)+(x+1)^{2}+(x+1)^{3}+(x+1)^{4}+\cdots+(x+1)^{99} \text {. } $$ Determine the value of $\left\lfloor a_{4} / a_{3}\right\rfloor$.	19
Henry starts with a list of the first 1000 positive integers, and performs a series of steps on the list. At each step, he erases any nonpositive integers or any integers that have a repeated digit, and then decreases everything in the list by 1. How many steps does it take for Henry's list to be empty? [i]Proposed by Michael Ren[/i]	11
\section*{Problem \(6-330916=331016\)} It is known that \(2^{10}=1024\). Formulate a computer program that can determine the smallest natural exponent \(p>10\) for which the number \(2^p\) also ends in the digits ...024! Explain why the program you have formulated solves this problem! Hint: Note that for the numbers involved in the calculations, there are limitations on the number of digits when using typical computer usage.	110
rainbow is the name of a bird. this bird has $n$ colors and it's colors in two consecutive days are not equal. there doesn't exist $4$ days in this bird's life like $i,j,k,l$ such that $i<j<k<l$ and the bird has the same color in days $i$ and $k$ and the same color in days $j$ and $l$ different from the colors it has in days $i$ and $k$. what is the maximum number of days rainbow can live in terms of $n$?	2n - 1
【Question 6】Using 2 unit squares (unit squares) can form a 2-connected square, which is commonly known as a domino. Obviously, dominoes that can coincide after translation, rotation, or symmetry transformation should be considered as the same one, so there is only one domino. Similarly, the different 3-connected squares formed by 3 unit squares are only 2. Using 4 unit squares to form different 4-connected squares, there are 5. Then, using 5 unit squares to form different 5-connected squares, there are $\qquad$. (Note: The blank space at the end is left as in the original text for the answer to be filled in.)	12
3. Given a convex quadrilateral $ABCD$ with area $P$. We extend side $AB$ beyond $B$ to $A_1$ such that $\overline{AB}=\overline{BA_1}$, then $BC$ beyond $C$ to $B_1$ such that $\overline{BC}=\overline{CB_1}$, then $CD$ beyond $D$ to $C_1$ so that $\overline{CD}=\overline{DC_1}$, and $DA$ beyond $A$ to $D_1$ such that $\overline{DA}=\overline{AD_1}$. What is the area of quadrilateral $A_1B_1C_1D_1$?	5P
B. As shown in Figure 2, in the square $A B C D$ with side length 1, $E$ and $F$ are points on $B C$ and $C D$ respectively, and $\triangle A E F$ is an equilateral triangle. Then the area of $\triangle A E F$ is	2\sqrt{3}-3
1. If $x-y=12$, find the value of $x^{3}-y^{3}-36 x y$. (1 mark) If $x-y=12$, find the value of $x^{3}-y^{3}-36 x y$. (1 mark)	1728
1. If $\log _{4}(x+2 y)+\log _{4}(x-2 y)=1$, then the minimum value of $\|x\|-\|y\|$ is $\qquad$	\sqrt{3}
5. Given a sphere with a radius of 6. Then the maximum volume of a regular tetrahedron inscribed in the sphere is ( ). (A) $32 \sqrt{3}$ (B) $54 \sqrt{3}$ (C) $64 \sqrt{3}$ (D) $72 \sqrt{3}$	C
Let $n$ be a nonzero natural number, and $x_1, x_2,..., x_n$ positive real numbers that $ \frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}= n$. Find the minimum value of the expression $x_1 +\frac{x_2^2}{2}++\frac{x_3^3}{3}+...++\frac{x_n^n}{n}$.	1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}
8. Let $a>0, b>0$. Then the inequality that does not always hold is ( ). (A) $\frac{2}{\frac{1}{a}+\frac{1}{b}} \geqslant \sqrt{a b}$ (B) $\frac{1}{a}+\frac{1}{b} \geqslant \frac{4}{a+b}$ (C) $\sqrt{\|a-b\|} \geqslant \sqrt{a}-\sqrt{b}$ (D) $a^{2}+b^{2}+1>a+b$	A
A cuboid has an integer volume. Three of the faces have different areas, namely $7, 27$, and $L$. What is the smallest possible integer value for $L$?	21
11. The solution set of the equation $x^{2}\|x\|+\|x\|^{2}-x^{2}-\|x\|=0$ in the complex number set corresponds to a figure in the complex plane which is ( ). A. Several points and a line B. Unit circle and a line C. Several lines D. Origin and unit circle	D
\section*{Problem 1 - 101211} In a parent-teacher meeting, exactly 18 fathers and exactly 24 mothers were present, with at least one parent of each student in the class attending. Of exactly 10 boys and exactly 8 girls, both parents were present for each. For exactly 4 boys and exactly 3 girls, only the mother was present, while for exactly 1 boy and exactly 1 girl, only the father was present. Determine the number of all those children in this class who have siblings in the same class! (There are no children in this class who have step-parents or step-siblings.)	4
Let $k$ and $a$ are positive constants. Denote by $V_1$ the volume of the solid generated by a rotation of the figure enclosed by the curve $C: y=\frac{x}{x+k}\ (x\geq 0)$, the line $x=a$ and the $x$-axis around the $x$-axis, and denote by $V_2$ that of the solid by a rotation of the figure enclosed by the curve $C$, the line $y=\frac{a}{a+k}$ and the $y$-axis around the $y$-axis. Find the ratio $\frac{V_2}{V_1}.$	k
Problem 5.4. At the end-of-the-year school dance, there were twice as many boys as girls. Masha counted that there were 8 fewer girls, besides herself, than boys. How many boys came to the dance?	14
Suppose the side lengths of triangle $ABC$ are the roots of polynomial $x^3 - 27x^2 + 222x - 540$. What is the product of its inradius and circumradius?	10
5. The number of $n$ that makes each interior angle of a regular $n$-sided polygon an even number of degrees is ( ). (A) 15 (B) 16 (C) 17 (D) 18	B
Let $A$, $B$, $C$, $D$ be four points on a line in this order. Suppose that $AC = 25$, $BD = 40$, and $AD = 57$. Compute $AB \cdot CD + AD \cdot BC$. [i]Proposed by Evan Chen[/i]	1000
\section*{Problem 4 - 261014} Jürgen claims that there is a positional system with base \(m\) in which the following calculation is correct: \begin{tabular}{lllllll} & 7 & 0 & 1 &. & 3 & 4 \\ \hline 2 & 5 & 0 & 3 & & & \\ & 3 & 4 & 0 & 4 & & \\ \hline 3 & 0 & 4 & 3 & 4 & & \end{tabular} Determine all natural numbers \(m\) for which this is true! Hint: In a positional system with base \(m\), there are exactly the digits \(0,1, \ldots, m-2, m-1\). Each natural number is represented as a sum of products of a power of \(m\) with one of the digits; the powers are ordered by decreasing exponents. The sequence of digits is then written as it is known for \(m=10\) in the decimal notation of natural numbers.	8
22. Consider a list of six numbers. When the largest number is removed from the list, the average is decreased by 1 . When the smallest number is removed, the average is increased by 1 . When both the largest and the smallest numbers are removed, the average of the remaining four numbers is 20 . Find the product of the largest and the smallest numbers.	375
$9 \cdot 23$ The largest of the following four numbers is (A) $\operatorname{tg} 48^{\circ}+\operatorname{ctg} 48^{\circ}$. (B) $\sin 48^{\circ}+\cos 48^{\circ}$. (C) $\operatorname{tg} 48^{\circ}+\cos 48^{\circ}$. (D) $\operatorname{ctg} 48^{\circ}+\sin 48^{\circ}$. (Chinese Junior High School Mathematics League, 1988)	A
10. On a plane, 2011 points are marked. We will call a pair of marked points $A$ and $B$ isolated if all other points are strictly outside the circle constructed on $A B$ as its diameter. What is the smallest number of isolated pairs that can exist?	2010
From the identity $$ \int_{0}^{\pi \slash 2} \log \sin 2x \, dx = \int_{0}^{\pi \slash 2} \log \sin x \, dx + \int_{0}^{\pi \slash 2} \log \cos x \, dx +\int_{0}^{\pi \slash 2} \log 2 \, dx, $$ deduce the value of $\int_{0}^{\pi \slash 2} \log \sin x \, dx.$	-\frac{\pi}{2} \log 2
Example 1 (An Ancient Chinese Mathematical Problem) Emperor Taizong of Tang ordered the counting of soldiers: if 1,001 soldiers make up one battalion, then one person remains; if 1,002 soldiers make up one battalion, then four people remain. This time, the counting of soldiers has at least $\qquad$ people.	1000000
1. Let $\mathbb{N}$ be the set of all natural numbers and $S=\left\{(a, b, c, d) \in \mathbb{N}^{4}: a^{2}+b^{2}+c^{2}=d^{2}\right\}$. Find the largest positive integer $m$ such that $m$ divides $a b c d$ for all $(a, b, c, d) \in S$.	12
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
Florián was thinking about what bouquet he would have tied for his mom for Mother's Day. In the florist's, according to the price list, he calculated that whether he buys 5 classic gerberas or 7 mini gerberas, the bouquet, when supplemented with a decorative ribbon, would cost the same, which is 295 crowns. However, if he bought only 2 mini gerberas and 1 classic gerbera without any additional items, he would pay 102 crowns. How much does one ribbon cost? (L. Šimůnek)	85
6. Given the function $f(x)=\|\| x-1\|-1\|$, if the equation $f(x)=m(m \in \mathbf{R})$ has exactly 4 distinct real roots $x_{1}, x_{2}, x_{3}, x_{4}$, then the range of $x_{1} x_{2} x_{3} x_{4}$ is $\qquad$ .	(-3,0)
Let's find those prime numbers $p$ for which the number $p^{2}+11$ has exactly 6 positive divisors.	3
4. In Rt $\triangle A B C$, $C D$ is the altitude to the hypotenuse $A B$, then the sum of the inradii of the three right triangles ( $\triangle A B C, \triangle A C D$, $\triangle B C D$ ) is equal to ( ). (A) $C D$ (B) $A C$ (C) $B C$ (D) $A B$	A
Freddy the frog is jumping around the coordinate plane searching for a river, which lies on the horizontal line $y = 24$. A fence is located at the horizontal line $y = 0$. On each jump Freddy randomly chooses a direction parallel to one of the coordinate axes and moves one unit in that direction. When he is at a point where $y=0$, with equal likelihoods he chooses one of three directions where he either jumps parallel to the fence or jumps away from the fence, but he never chooses the direction that would have him cross over the fence to where $y < 0$. Freddy starts his search at the point $(0, 21)$ and will stop once he reaches a point on the river. Find the expected number of jumps it will take Freddy to reach the river.	273
The figure below represents a bookshelf with two shelves and five stacks of books, three of them with two books and two of them with only one book. Alice and Luiz invented a game in which each of them, alternately, removes one or two books from one of the stacks of books. The one who takes the last book wins. Alice starts the challenge. Which of them has a winning strategy, regardless of the opponent's moves? ![](https://cdn.mathpix.com/cropped/2024_05_01_a50b5476db779c3986d5g-09.jpg?height=486&width=793&top_left_y=1962&top_left_x=711) #	Alice
13. 6 different points are given on the plane, no three of which are collinear. Each pair of points is to be joined by a red line or a blue line subject to the following restriction: if the lines joining $A B$ and $A C$ (where $A, B, C$ denote the given points) are both red, then the line joining $B C$ is also red. How many different colourings of the lines are possible? (2 marks) 在平面上給定 6 個不同的點, 當中沒有三點共線。現要把任意兩點均以一條紅線或監線連起, 並須符合以下規定: 若 $A B$ 和 $A C$ (這裡 $A 、 B 、 C$ 代表給定的點）均以紅線連起, 則 $B C$ 亦必須以紅線連起。那麼, 連線的顏色有多少個不同的組合？	203
Let $\triangle ABC$ be a triangle with $AB = 7$, $AC = 8$, and $BC = 3$. Let $P_1$ and $P_2$ be two distinct points on line $AC$ ($A, P_1, C, P_2$ appear in that order on the line) and $Q_1$ and $Q_2$ be two distinct points on line $AB$ ($A, Q_1, B, Q_2$ appear in that order on the line) such that $BQ_1 = P_1Q_1 = P_1C$ and $BQ_2 = P_2Q_2 = P_2C$. Find the distance between the circumcenters of $BP_1P_2$ and $CQ_1Q_2$.	3
A $4\times4\times4$ cube is composed of $64$ unit cubes. The faces of $16$ unit cubes are to be coloured red. A colouring is called interesting if there is exactly $1$ red unit cube in every $1\times1\times 4$ rectangular box composed of $4$ unit cubes. Determine the number of interesting colourings.	576
12. After removing one element from the set $\{1!, 2!, \cdots, 24!\}$, the product of the remaining elements is exactly a perfect square.	12!
3. In astronomy, "parsec" is commonly used as a unit of distance. If in a right triangle $\triangle ABC$, $$ \angle ACB=90^{\circ}, CB=1.496 \times 10^{8} \text{ km}, $$ i.e., the length of side $CB$ is equal to the average distance between the Moon $(C)$ and the Earth $(B)$, then, when the size of $\angle BAC$ is 1 second (1 degree equals 60 minutes, 1 minute equals 60 seconds), the length of the hypotenuse $AB$ is 1 parsec. Therefore, 1 parsec = $\qquad$ km (expressed in scientific notation, retaining four significant figures).	3.086 \times 10^{13}
For any two coprime positive integers $p, q$, define $f(i)$ to be the remainder of $p\cdot i$ divided by $q$ for $i = 1, 2,\ldots,q -1$. The number $i$ is called a[b] large [/b]number (resp. [b]small[/b] number) when $f(i)$ is the maximum (resp. the minimum) among the numbers $f(1), f(2),\ldots,f(i)$. Note that $1$ is both large and small. Let $a, b$ be two fixed positive integers. Given that there are exactly $a$ large numbers and $b$ small numbers among $1, 2,\ldots , q - 1$, find the least possible number for $q$. [i] Proposed by usjl[/i]	ab + 1
Let $(a_n)_{n\geq 1}$ be a sequence such that $a_1=1$ and $3a_{n+1}-3a_n=1$ for all $n\geq 1$. Find $a_{2002}$. $\textbf{(A) }666\qquad\textbf{(B) }667\qquad\textbf{(C) }668\qquad\textbf{(D) }669\qquad\textbf{(E) }670$	668
4. Let $a$ be a real number such that the graph of the function $y=f(x)=a \sin 2x + \cos 2x + \sin x + \cos x$ is symmetric about the line $x=-\pi$. Let the set of all such $a$ be denoted by $S$. Then $S$ is ( ). (A) empty set (B) singleton set (contains only one element) (C) finite set with more than one element (D) infinite set (contains infinitely many elements)	A
Let $S_n$ be the sum of the reciprocals of the non-zero digits of the integers from 1 to $10^n$ inclusive. Find the smallest positive integer $n$ for which $S_n$ is an integer.	63
The graph shows the fuel used per $100 \mathrm{~km}$ of driving for five different vehicles. Which vehicle would travel the farthest using 50 litres of fuel? (A) $U$ (B) $V$ (C) $W$ (D) $X$ (E) $Y$ ![](https://cdn.mathpix.com/cropped/2024_04_20_6027bc27089ed4fc493cg-059.jpg?height=431&width=380&top_left_y=1297&top_left_x=1271)	D
If $x=\frac{1}{4}$, which of the following has the largest value? (A) $x$ (B) $x^{2}$ (C) $\frac{1}{2} x$ (D) $\frac{1}{x}$ (E) $\sqrt{x}$	D
Let $a, b, c,$ and $d$ be positive integers such that $\gcd(a, b)=24$, $\gcd(b, c)=36$, $\gcd(c, d)=54$, and $70<\gcd(d, a)<100$. Which of the following must be a divisor of $a$? $\textbf{(A)} \text{ 5} \qquad \textbf{(B)} \text{ 7} \qquad \textbf{(C)} \text{ 11} \qquad \textbf{(D)} \text{ 13} \qquad \textbf{(E)} \text{ 17}$	13
Exercise 4. We want to color the three-element subsets of $\{1,2,3,4,5,6,7\}$ such that if two of these subsets have no element in common, then they must be of different colors. What is the minimum number of colors needed to achieve this goal?	3
4. Find all real numbers $p$ such that the cubic equation $5 x^{3}-5(p+1) x^{2}+(71 p-1) x+1=66 p$ has three positive integer roots. untranslated text remains the same as requested.	76
Let $ABC$ be a right triangle with a right angle at $C.$ Two lines, one parallel to $AC$ and the other parallel to $BC,$ intersect on the hypotenuse $AB.$ The lines split the triangle into two triangles and a rectangle. The two triangles have areas $512$ and $32.$ What is the area of the rectangle? [i]Author: Ray Li[/i]	256
4. Given the curve $y=x^{3}-x$, draw a tangent line to the curve from a point $A(t, 0)$ on the $x$-axis, then the maximum number of tangent lines is $\qquad$.	3
7. Given a quartic polynomial $f(x)$ whose four real roots form an arithmetic sequence with a common difference of 2. Then the difference between the largest and smallest roots of $f^{\prime}(x)$ is $\qquad$	2 \sqrt{5}
6. $\frac{2 \cos 10^{\circ}-\sin 20^{\circ}}{\cos 20^{\circ}}$ The value is $\qquad$	\sqrt{3}
3. If point $P\left(x_{0}, y_{0}\right)$ is such that the chord of contact of the ellipse $E: \frac{x}{4}+y^{2}=1$ and the hyperbola $H: x^{2}-\frac{y^{2}}{4}=1$ are perpendicular to each other, then $\frac{y_{0}}{x_{0}}=$ $\qquad$	\pm 1
6. Determine the largest natural number $n \geqq 10$ such that for any 10 different numbers $z$ from the set $\{1,2, \ldots, n\}$, the following statement holds: If none of these 10 numbers is a prime number, then the sum of some two of them is a prime number. (Ján Mazák)	21
Long) and 1 public F: foundation code. A customer wants to buy two kilograms of candy. The salesperson places a 1-kilogram weight on the left pan and puts the candy on the right pan to balance it, then gives the candy to the customer. Then, the salesperson places the 1-kilogram weight on the right pan and puts more candy on the left pan to balance it, and gives this to the customer as well. The two kilograms of candy given to the customer (). (A) is fair (B) the customer gains (C) the store loses (D) if the long arm is more than twice the length of the short arm, the store loses; if less than twice, the customer gains	C
Given that $\sum_{k=1}^{35}\sin 5k=\tan \frac mn,$ where angles are measured in degrees, and $m_{}$ and $n_{}$ are relatively prime positive integers that satisfy $\frac mn<90,$ find $m+n.$	177
3A. Given the set of quadratic functions $f(x)=a x^{2}+b x+c$, for which $a<b$ and $f(x) \geq 0$ for every $x \in \mathbb{R}$. Determine the smallest possible value of the expression $A=\frac{a+b+c}{b-a}$.	3
[b]p1.[/b] You can do either of two operations to a number written on a blackboard: you can double it, or you can erase the last digit. Can you get the number $14$ starting from the number $458$ by using these two operations? [b]p2.[/b] Show that the first $2011$ digits after the point in the infinite decimal fraction representing $(7 + \sqrt50)^{2011}$ are all $9$’s. [b]p3.[/b] Inside of a trapezoid find a point such that segments which join this point with the midpoints of the edges divide the trapezoid into four pieces of equal area. [b]p4.[/b] New chess figure called “lion” is placed on the $8 \times 8$ chess board. It can move one square horizontally right, or one square vertically down or one square diagonally left up. Can lion visit every square of the chess board exactly once and return in his last move to the initial position? You can choose the initial position arbitrarily. PS. You should use hide for answers.	14
A rectangular cuboid has the sum of the three edges starting from one vertex as $p$, and the length of the space diagonal is $q$. What is the surface area of the body? - Can its volume be calculated?	p^2-q^2

Prove that number $1$ can be represented as a sum of a finite number $n$ of real numbers, less than $1,$ not necessarily distinct, which contain in their decimal representation only the digits $0$ and/or $7.$ Which is the least possible number $n$?

8

A right-angled triangle has side lengths that are integers. What could be the last digit of the area's measure, if the length of the hypotenuse is not divisible by 5?

0

Let $a_n\ (n\geq 1)$ be the value for which $\int_x^{2x} e^{-t^n}dt\ (x\geq 0)$ is maximal. Find $\lim_{n\to\infty} \ln a_n.$

-\ln 2

6. As shown in Figure 2, let $P$ be a point inside the equilateral $\triangle ABC$ with side length 12. Draw perpendiculars from $P$ to the sides $BC$, $CA$, and $AB$, with the feet of the perpendiculars being $D$, $E$, and $F$ respectively. Given that $PD: PE: PF = 1: 2: 3$. Then, the area of quadrilateral $BDPF$ is

11 \sqrt{3}

Problem 6. (8 points) In the plane, there is a non-closed, non-self-intersecting broken line consisting of 31 segments (adjacent segments do not lie on the same straight line). For each segment, the line defined by it is constructed. It is possible for some of the 31 constructed lines to coincide. What is the minimum number of different lines that can be obtained? Answer. 9.

9

Four, (50 points) In an $n \times n$ grid, fill each cell with one of the numbers 1 to $n^{2}$. If no matter how you fill it, there must be two adjacent cells where the difference between the two numbers is at least 1011, find the minimum value of $n$. --- The translation preserves the original text's formatting and structure.

2020

On a board, the numbers from 1 to 2009 are written. A couple of them are erased and instead of them, on the board is written the remainder of the sum of the erased numbers divided by 13. After a couple of repetition of this erasing, only 3 numbers are left, of which two are 9 and 999. Find the third number.

8

4. The number of real solutions to the equation $\left|x^{2}-3 x+2\right|+\left|x^{2}+2 x-3\right|=11$ is ( ). (A) 0 (B) 1 (C) 2 (D) 4

C

## Problem Statement Calculate the limit of the numerical sequence: $\lim _{n \rightarrow \infty} \frac{(n+1)^{4}-(n-1)^{4}}{(n+1)^{3}+(n-1)^{3}}$

4

6. Let $[x]$ denote the greatest integer not exceeding the real number $x$, $$ \begin{array}{c} S=\left[\frac{1}{1}\right]+\left[\frac{2}{1}\right]+\left[\frac{1}{2}\right]+\left[\frac{2}{2}\right]+\left[\frac{3}{2}\right]+ \\ {\left[\frac{4}{2}\right]+\left[\frac{1}{3}\right]+\left[\frac{2}{3}\right]+\left[\frac{3}{3}\right]+\left[\frac{4}{3}\right]+} \\ {\left[\frac{5}{3}\right]+\left[\frac{6}{3}\right]+\cdots} \end{array} $$ up to 2016 terms, where, for a segment with denominator $k$, there are $2 k$ terms $\left[\frac{1}{k}\right],\left[\frac{2}{k}\right], \cdots,\left[\frac{2 k}{k}\right]$, and only the last segment may have fewer than $2 k$ terms. Then the value of $S$ is

1078

19. Given $m \in\{11,13,15,17,19\}$, $n \in\{1999,2000, \cdots, 2018\}$. Then the probability that the unit digit of $m^{n}$ is 1 is ( ). (A) $\frac{1}{5}$ (B) $\frac{1}{4}$ (C) $\frac{3}{10}$ (D) $\frac{7}{20}$ (E) $\frac{2}{5}$

E

1. A line $l$ intersects a hyperbola $c$, then the maximum number of intersection points is ( ). A. 1 B. 2 C. 3 D. 4

B

4. As shown in Figure 1, in the right triangular prism $A B C-A_{1} B_{1} C_{1}$, $A A_{1}=A B=A C$, and $M$ and $Q$ are the midpoints of $C C_{1}$ and $B C$ respectively. If for any point $P$ on the line segment $A_{1} B_{1}$, $P Q \perp A M$, then $\angle B A C$ equals ( ). (A) $30^{\circ}$ (B) $45^{\circ}$ (C) $60^{\circ}$ (D) $90^{\circ}$

D

The equation of line $\ell_1$ is $24x-7y = 319$ and the equation of line $\ell_2$ is $12x-5y = 125$. Let $a$ be the number of positive integer values $n$ less than $2023$ such that for both $\ell_1$ and $\ell_2$ there exists a lattice point on that line that is a distance of $n$ from the point $(20,23)$. Determine $a$. [i]Proposed by Christopher Cheng[/i] [hide=Solution][i]Solution. [/i] $\boxed{6}$ Note that $(20,23)$ is the intersection of the lines $\ell_1$ and $\ell_2$. Thus, we only care about lattice points on the the two lines that are an integer distance away from $(20,23)$. Notice that $7$ and $24$ are part of the Pythagorean triple $(7,24,25)$ and $5$ and $12$ are part of the Pythagorean triple $(5,12,13)$. Thus, points on $\ell_1$ only satisfy the conditions when $n$ is divisible by $25$ and points on $\ell_2$ only satisfy the conditions when $n$ is divisible by $13$. Therefore, $a$ is just the number of positive integers less than $2023$ that are divisible by both $25$ and $13$. The LCM of $25$ and $13$ is $325$, so the answer is $\boxed{6}$.[/hide]

6

1B. If for the non-zero real numbers $a, b$ and $c$ the equalities $a^{2}+a=b^{2}, b^{2}+b=c^{2}$ and $c^{2}+c=a^{2}$ hold, determine the value of the expression $(a-b)(b-c)(c-a)$.

1

2. As shown in Figure 1, the side length of rhombus $A B C D$ is $a$, and $O$ is a point on the diagonal $A C$, with $O A=a, O B=$ $O C=O D=1$. Then $a$ equals ( ). (A) $\frac{\sqrt{5}+1}{2}$ (B) $\frac{\sqrt{5}-1}{2}$ (C) 1 (D) 2

A

1. Given $a, b>0, a \neq 1$, and $a^{b}=\log _{a} b$, then the value of $a^{a^{b}}-\log _{a} \log _{a} b^{a}$ is

-1

There is a regular $17$-gon $\mathcal{P}$ and its circumcircle $\mathcal{Y}$ on the plane. The vertices of $\mathcal{P}$ are coloured in such a way that $A,B \in \mathcal{P}$ are of different colour, if the shorter arc connecting $A$ and $B$ on $\mathcal{Y}$ has $2^k+1$ vertices, for some $k \in \mathbb{N},$ including $A$ and $B.$ What is the least number of colours which suffices?

4

Find all triples $(m,p,q)$ where $ m $ is a positive integer and $ p , q $ are primes. \[ 2^m p^2 + 1 = q^5 \]

(1, 11, 3)

2. In the complex plane, there are 7 points corresponding to the 7 roots of the equation $x^{7}=$ $-1+\sqrt{3} i$. Among the four quadrants where these 7 points are located, only 1 point is in ( ). (A) the I quadrant (B) the II quadrant (C) the III quadrant (D) the IV quadrant

C

Of the 36 students in Richelle's class, 12 prefer chocolate pie, 8 prefer apple, and 6 prefer blueberry. Half of the remaining students prefer cherry pie and half prefer lemon. For Richelle's pie graph showing this data, how many degrees should she use for cherry pie? $ \text{(A)}\ 10\qquad\text{(B)}\ 20\qquad\text{(C)}\ 30\qquad\text{(D)}\ 50\qquad\text{(E)}\ 72 $

50

Father played chess with uncle. For a won game, the winner received 8 crowns from the opponent, and for a draw, nobody got anything. Uncle won four times, there were five draws, and in the end, father earned 24 crowns. How many games did father play with uncle? (M. Volfová)

16

## Problem 4 Given the numbers $1,2,3, \ldots, 1000$. Find the largest number $m$ with the property that by removing any $m$ numbers from these 1000 numbers, among the $1000-m$ remaining numbers, there exist two such that one divides the other. Selected problems by Prof. Cicortas Marius Note: a) The actual working time is 3 hours. b) All problems are mandatory. c) Each problem is graded from 0 to 7. ## NATIONAL MATHEMATICS OLYMPIAD Local stage - 15.02.2014 ## Grade IX ## Grading Rubric

499

4. Let $A$ and $B$ be $n$-digit numbers, where $n$ is odd, which give the same remainder $r \neq 0$ when divided by $k$. Find at least one number $k$, which does not depend on $n$, such that the number $C$, obtained by appending the digits of $A$ and $B$, is divisible by $k$.

11

Example: Given the radii of the upper and lower bases of a frustum are 3 and 6, respectively, and the height is $3 \sqrt{3}$, the radii $O A$ and $O B$ of the lower base are perpendicular, and $C$ is a point on the generatrix $B B^{\prime}$ such that $B^{\prime} C: C B$ $=1: 2$. Find the shortest distance between points $A$ and $C$ on the lateral surface of the frustum.

4 \sqrt{13-6 \sqrt{2}}

6. Let set $A=\{1,2,3,4,5,6\}$, and a one-to-one mapping $f: A \rightarrow A$ satisfies that for any $x \in A$, $f(f(f(x)))$ $=x$. Then the number of mappings $f$ that satisfy the above condition is ( ). (A) 40 (B) 41 (C) 80 (D) 81

D

5. For a convex $n$-sided polygon, if circles are constructed with each side as the diameter, the convex $n$-sided polygon must be covered by these $n$ circles. Then the maximum value of $n$ is: ( ). (A) 3 . (B) 4 . (C) 5 . (D) Greater than 5 .

B

1. As shown in Figure 1, in the Cartesian coordinate system, the graph of the quadratic function $y=a x^{2}+m c(a \neq$ $0)$ passes through three vertices $A$, $B$, and $C$ of the square $A B O C$, and $a c=-2$. Then the value of $m$ is ( ). (A) 1 (B) -1 (C) 2 (D) -2

A

Let $a_1,a_2,a_3,a_4,a_5$ be distinct real numbers. Consider all sums of the form $a_i + a_j$ where $i,j \in \{1,2,3,4,5\}$ and $i \neq j$. Let $m$ be the number of distinct numbers among these sums. What is the smallest possible value of $m$?

7

7. A circle with a radius of 1 has six points, these six points divide the circle into six equal parts. Take three of these points as vertices to form a triangle. If the triangle is neither equilateral nor isosceles, then the area of this triangle is ( ). (A) $\frac{\sqrt{3}}{3}$ (B) $\frac{\sqrt{3}}{2}$ (C) 1 (D) $\sqrt{2}$ (E) 2

B

15.15 A paper punch can be placed at any point in the plane, and when it operates, it can punch out points at an irrational distance from it. What is the minimum number of paper punches needed to punch out all points in the plane? (51st Putnam Mathematical Competition, 1990)

3

1. A three-digit number is 29 times the sum of its digits. Then this three-digit number is $\qquad$

261

The area of the region in the $xy$-plane satisfying the inequality \[\min_{1 \le n \le 10} \max\left(\frac{x^2+y^2}{4n^2}, \, 2 - \frac{x^2+y^2}{4n^2-4n+1}\right) \le 1\] is $k\pi$, for some integer $k$. Find $k$. [i]Proposed by Michael Tang[/i]

210

2. On a line, several points were marked, including points $A$ and $B$. All possible segments with endpoints at the marked points are considered. Vasya calculated that point $A$ is inside 50 of these segments, and point $B$ is inside 56 segments. How many points were marked? (The endpoints of a segment are not considered its internal points.)

16

Lines $l_1^{}$ and $l_2^{}$ both pass through the origin and make first-quadrant angles of $\frac{\pi}{70}$ and $\frac{\pi}{54}$ radians, respectively, with the positive x-axis. For any line $l^{}_{}$, the transformation $R(l)^{}_{}$ produces another line as follows: $l^{}_{}$ is reflected in $l_1^{}$, and the resulting line is reflected in $l_2^{}$. Let $R^{(1)}(l)=R(l)^{}_{}$ and $R^{(n)}(l)^{}_{}=R\left(R^{(n-1)}(l)\right)$. Given that $l^{}_{}$ is the line $y=\frac{19}{92}x^{}_{}$, find the smallest positive integer $m^{}_{}$ for which $R^{(m)}(l)=l^{}_{}$.

945

2、D Teacher has five vases, these five vases are arranged in a row from shortest to tallest, the height difference between adjacent vases is 2 centimeters, and the tallest vase is exactly equal to the sum of the heights of the two shortest vases, then the total height of the five vases is _ centimeters

50

5. Let the base of the right quadrilateral prism $A^{\prime} B^{\prime} C^{\prime} D^{\prime}-A B C D$ be a rhombus, with an area of $2 \sqrt{3} \mathrm{~cm}^{2}, \angle A B C=60^{\circ}, E$ and $F$ be points on the edges $C C^{\prime}$ and $B B^{\prime}$, respectively, such that $E C=B C=$ $2 F B$. Then the volume of the quadrilateral pyramid $A-B C E F$ is ( ). (A) $\sqrt{3} \mathrm{~cm}^{3}$ (B) $\sqrt{5} \mathrm{~cm}^{3}$ (C) $6 \mathrm{~cm}^{3}$ (D) $\dot{9} \mathrm{~cm}^{3}$

A

3. In circle $\odot O$, the radius $r=5 \mathrm{~cm}$, $A B$ and $C D$ are two parallel chords, and $A B=8 \mathrm{~cm}, C D=6 \mathrm{~cm}$. Then the length of $A C$ has ( ). (A) 1 solution (B) 2 solutions (C) 3 solutions (D) 4 solutions

C

4. Zhao, Qian, Sun, and Li each prepared a gift to give to one of the other three classmates when the new semester started. It is known that Zhao's gift was not given to Qian, Sun did not receive Li's gift, Sun does not know who Zhao gave the gift to, and Li does not know who Qian received the gift from. So, Qian gave the gift to ( ). (A) Zhao (B) Sun (C) Li (D) None of the above

B

3. As shown in Figure 1, $EF$ is the midline of $\triangle ABC$, $O$ is a point on $EF$, and satisfies $OE=2OF$. Then the ratio of the area of $\triangle ABC$ to the area of $\triangle AOC$ is ( ). (A) 2 (B) $\frac{3}{2}$ (C) $\frac{5}{3}$ (D) 3

D

2. Let $x$ be a positive integer, and $y$ is obtained from $x$ when the first digit of $x$ is moved to the last place. Determine the smallest number $x$ for which $3 x=y$.

142857

18. Let $a_{k}$ be the coefficient of $x^{k}$ in the expansion of $$ (x+1)+(x+1)^{2}+(x+1)^{3}+(x+1)^{4}+\cdots+(x+1)^{99} \text {. } $$ Determine the value of $\left\lfloor a_{4} / a_{3}\right\rfloor$.

19

Henry starts with a list of the first 1000 positive integers, and performs a series of steps on the list. At each step, he erases any nonpositive integers or any integers that have a repeated digit, and then decreases everything in the list by 1. How many steps does it take for Henry's list to be empty? [i]Proposed by Michael Ren[/i]

11

\section*{Problem $6-330916=331016$} It is known that $2^{10}=1024$. Formulate a computer program that can determine the smallest natural exponent $p>10$ for which the number $2^p$ also ends in the digits ...024! Explain why the program you have formulated solves this problem! Hint: Note that for the numbers involved in the calculations, there are limitations on the number of digits when using typical computer usage.

110

rainbow is the name of a bird. this bird has $n$ colors and it's colors in two consecutive days are not equal. there doesn't exist $4$ days in this bird's life like $i,j,k,l$ such that $i<j<k<l$ and the bird has the same color in days $i$ and $k$ and the same color in days $j$ and $l$ different from the colors it has in days $i$ and $k$. what is the maximum number of days rainbow can live in terms of $n$?

2n - 1

【Question 6】Using 2 unit squares (unit squares) can form a 2-connected square, which is commonly known as a domino. Obviously, dominoes that can coincide after translation, rotation, or symmetry transformation should be considered as the same one, so there is only one domino. Similarly, the different 3-connected squares formed by 3 unit squares are only 2. Using 4 unit squares to form different 4-connected squares, there are 5. Then, using 5 unit squares to form different 5-connected squares, there are $\qquad$. (Note: The blank space at the end is left as in the original text for the answer to be filled in.)

12

3. Given a convex quadrilateral $ABCD$ with area $P$. We extend side $AB$ beyond $B$ to $A_1$ such that $\overline{AB}=\overline{BA_1}$, then $BC$ beyond $C$ to $B_1$ such that $\overline{BC}=\overline{CB_1}$, then $CD$ beyond $D$ to $C_1$ so that $\overline{CD}=\overline{DC_1}$, and $DA$ beyond $A$ to $D_1$ such that $\overline{DA}=\overline{AD_1}$. What is the area of quadrilateral $A_1B_1C_1D_1$?

5P

B. As shown in Figure 2, in the square $A B C D$ with side length 1, $E$ and $F$ are points on $B C$ and $C D$ respectively, and $\triangle A E F$ is an equilateral triangle. Then the area of $\triangle A E F$ is

2\sqrt{3}-3

1. If $x-y=12$, find the value of $x^{3}-y^{3}-36 x y$. (1 mark) If $x-y=12$, find the value of $x^{3}-y^{3}-36 x y$. (1 mark)

1728

1. If $\log _{4}(x+2 y)+\log _{4}(x-2 y)=1$, then the minimum value of $|x|-|y|$ is $\qquad$

\sqrt{3}

5. Given a sphere with a radius of 6. Then the maximum volume of a regular tetrahedron inscribed in the sphere is ( ). (A) $32 \sqrt{3}$ (B) $54 \sqrt{3}$ (C) $64 \sqrt{3}$ (D) $72 \sqrt{3}$

C

Let $n$ be a nonzero natural number, and $x_1, x_2,..., x_n$ positive real numbers that $ \frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}= n$. Find the minimum value of the expression $x_1 +\frac{x_2^2}{2}++\frac{x_3^3}{3}+...++\frac{x_n^n}{n}$.

1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}

8. Let $a>0, b>0$. Then the inequality that does not always hold is ( ). (A) $\frac{2}{\frac{1}{a}+\frac{1}{b}} \geqslant \sqrt{a b}$ (B) $\frac{1}{a}+\frac{1}{b} \geqslant \frac{4}{a+b}$ (C) $\sqrt{|a-b|} \geqslant \sqrt{a}-\sqrt{b}$ (D) $a^{2}+b^{2}+1>a+b$

A

A cuboid has an integer volume. Three of the faces have different areas, namely $7, 27$, and $L$. What is the smallest possible integer value for $L$?

21

11. The solution set of the equation $x^{2}|x|+|x|^{2}-x^{2}-|x|=0$ in the complex number set corresponds to a figure in the complex plane which is ( ). A. Several points and a line B. Unit circle and a line C. Several lines D. Origin and unit circle

D

\section*{Problem 1 - 101211} In a parent-teacher meeting, exactly 18 fathers and exactly 24 mothers were present, with at least one parent of each student in the class attending. Of exactly 10 boys and exactly 8 girls, both parents were present for each. For exactly 4 boys and exactly 3 girls, only the mother was present, while for exactly 1 boy and exactly 1 girl, only the father was present. Determine the number of all those children in this class who have siblings in the same class! (There are no children in this class who have step-parents or step-siblings.)

4

Let $k$ and $a$ are positive constants. Denote by $V_1$ the volume of the solid generated by a rotation of the figure enclosed by the curve $C: y=\frac{x}{x+k}\ (x\geq 0)$, the line $x=a$ and the $x$-axis around the $x$-axis, and denote by $V_2$ that of the solid by a rotation of the figure enclosed by the curve $C$, the line $y=\frac{a}{a+k}$ and the $y$-axis around the $y$-axis. Find the ratio $\frac{V_2}{V_1}.$

k

Problem 5.4. At the end-of-the-year school dance, there were twice as many boys as girls. Masha counted that there were 8 fewer girls, besides herself, than boys. How many boys came to the dance?

14

Suppose the side lengths of triangle $ABC$ are the roots of polynomial $x^3 - 27x^2 + 222x - 540$. What is the product of its inradius and circumradius?

10

5. The number of $n$ that makes each interior angle of a regular $n$-sided polygon an even number of degrees is ( ). (A) 15 (B) 16 (C) 17 (D) 18

B

Let $A$, $B$, $C$, $D$ be four points on a line in this order. Suppose that $AC = 25$, $BD = 40$, and $AD = 57$. Compute $AB \cdot CD + AD \cdot BC$. [i]Proposed by Evan Chen[/i]

1000

\section*{Problem 4 - 261014} Jürgen claims that there is a positional system with base $m$ in which the following calculation is correct: \begin{tabular}{lllllll} & 7 & 0 & 1 &. & 3 & 4 \\ \hline 2 & 5 & 0 & 3 & & & \\ & 3 & 4 & 0 & 4 & & \\ \hline 3 & 0 & 4 & 3 & 4 & & \end{tabular} Determine all natural numbers $m$ for which this is true! Hint: In a positional system with base $m$, there are exactly the digits $0,1, \ldots, m-2, m-1$. Each natural number is represented as a sum of products of a power of $m$ with one of the digits; the powers are ordered by decreasing exponents. The sequence of digits is then written as it is known for $m=10$ in the decimal notation of natural numbers.

8

22. Consider a list of six numbers. When the largest number is removed from the list, the average is decreased by 1 . When the smallest number is removed, the average is increased by 1 . When both the largest and the smallest numbers are removed, the average of the remaining four numbers is 20 . Find the product of the largest and the smallest numbers.

375

$9 \cdot 23$ The largest of the following four numbers is (A) $\operatorname{tg} 48^{\circ}+\operatorname{ctg} 48^{\circ}$. (B) $\sin 48^{\circ}+\cos 48^{\circ}$. (C) $\operatorname{tg} 48^{\circ}+\cos 48^{\circ}$. (D) $\operatorname{ctg} 48^{\circ}+\sin 48^{\circ}$. (Chinese Junior High School Mathematics League, 1988)

A

10. On a plane, 2011 points are marked. We will call a pair of marked points $A$ and $B$ isolated if all other points are strictly outside the circle constructed on $A B$ as its diameter. What is the smallest number of isolated pairs that can exist?

2010

From the identity $$ \int_{0}^{\pi \slash 2} \log \sin 2x \, dx = \int_{0}^{\pi \slash 2} \log \sin x \, dx + \int_{0}^{\pi \slash 2} \log \cos x \, dx +\int_{0}^{\pi \slash 2} \log 2 \, dx, $$ deduce the value of $\int_{0}^{\pi \slash 2} \log \sin x \, dx.$

-\frac{\pi}{2} \log 2

Example 1 (An Ancient Chinese Mathematical Problem) Emperor Taizong of Tang ordered the counting of soldiers: if 1,001 soldiers make up one battalion, then one person remains; if 1,002 soldiers make up one battalion, then four people remain. This time, the counting of soldiers has at least $\qquad$ people.

1000000

1. Let $\mathbb{N}$ be the set of all natural numbers and $S=\left\{(a, b, c, d) \in \mathbb{N}^{4}: a^{2}+b^{2}+c^{2}=d^{2}\right\}$. Find the largest positive integer $m$ such that $m$ divides $a b c d$ for all $(a, b, c, d) \in S$.

12

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

Florián was thinking about what bouquet he would have tied for his mom for Mother's Day. In the florist's, according to the price list, he calculated that whether he buys 5 classic gerberas or 7 mini gerberas, the bouquet, when supplemented with a decorative ribbon, would cost the same, which is 295 crowns. However, if he bought only 2 mini gerberas and 1 classic gerbera without any additional items, he would pay 102 crowns. How much does one ribbon cost? (L. Šimůnek)

85

6. Given the function $f(x)=|| x-1|-1|$, if the equation $f(x)=m(m \in \mathbf{R})$ has exactly 4 distinct real roots $x_{1}, x_{2}, x_{3}, x_{4}$, then the range of $x_{1} x_{2} x_{3} x_{4}$ is $\qquad$ .

(-3,0)

Let's find those prime numbers $p$ for which the number $p^{2}+11$ has exactly 6 positive divisors.

3

4. In Rt $\triangle A B C$, $C D$ is the altitude to the hypotenuse $A B$, then the sum of the inradii of the three right triangles ( $\triangle A B C, \triangle A C D$, $\triangle B C D$ ) is equal to ( ). (A) $C D$ (B) $A C$ (C) $B C$ (D) $A B$

A

Freddy the frog is jumping around the coordinate plane searching for a river, which lies on the horizontal line $y = 24$. A fence is located at the horizontal line $y = 0$. On each jump Freddy randomly chooses a direction parallel to one of the coordinate axes and moves one unit in that direction. When he is at a point where $y=0$, with equal likelihoods he chooses one of three directions where he either jumps parallel to the fence or jumps away from the fence, but he never chooses the direction that would have him cross over the fence to where $y < 0$. Freddy starts his search at the point $(0, 21)$ and will stop once he reaches a point on the river. Find the expected number of jumps it will take Freddy to reach the river.

273

The figure below represents a bookshelf with two shelves and five stacks of books, three of them with two books and two of them with only one book. Alice and Luiz invented a game in which each of them, alternately, removes one or two books from one of the stacks of books. The one who takes the last book wins. Alice starts the challenge. Which of them has a winning strategy, regardless of the opponent's moves? ![](https://cdn.mathpix.com/cropped/2024_05_01_a50b5476db779c3986d5g-09.jpg?height=486&width=793&top_left_y=1962&top_left_x=711) #

Alice

13. 6 different points are given on the plane, no three of which are collinear. Each pair of points is to be joined by a red line or a blue line subject to the following restriction: if the lines joining $A B$ and $A C$ (where $A, B, C$ denote the given points) are both red, then the line joining $B C$ is also red. How many different colourings of the lines are possible? (2 marks) 在平面上給定 6 個不同的點, 當中沒有三點共線。現要把任意兩點均以一條紅線或監線連起, 並須符合以下規定: 若 $A B$ 和 $A C$ (這裡 $A 、 B 、 C$ 代表給定的點）均以紅線連起, 則 $B C$ 亦必須以紅線連起。那麼, 連線的顏色有多少個不同的組合？

203

Let $\triangle ABC$ be a triangle with $AB = 7$, $AC = 8$, and $BC = 3$. Let $P_1$ and $P_2$ be two distinct points on line $AC$ ($A, P_1, C, P_2$ appear in that order on the line) and $Q_1$ and $Q_2$ be two distinct points on line $AB$ ($A, Q_1, B, Q_2$ appear in that order on the line) such that $BQ_1 = P_1Q_1 = P_1C$ and $BQ_2 = P_2Q_2 = P_2C$. Find the distance between the circumcenters of $BP_1P_2$ and $CQ_1Q_2$.

3

A $4\times4\times4$ cube is composed of $64$ unit cubes. The faces of $16$ unit cubes are to be coloured red. A colouring is called interesting if there is exactly $1$ red unit cube in every $1\times1\times 4$ rectangular box composed of $4$ unit cubes. Determine the number of interesting colourings.

576

12. After removing one element from the set $\{1!, 2!, \cdots, 24!\}$, the product of the remaining elements is exactly a perfect square.

12!

3. In astronomy, "parsec" is commonly used as a unit of distance. If in a right triangle $\triangle ABC$, $$ \angle ACB=90^{\circ}, CB=1.496 \times 10^{8} \text{ km}, $$ i.e., the length of side $CB$ is equal to the average distance between the Moon $(C)$ and the Earth $(B)$, then, when the size of $\angle BAC$ is 1 second (1 degree equals 60 minutes, 1 minute equals 60 seconds), the length of the hypotenuse $AB$ is 1 parsec. Therefore, 1 parsec = $\qquad$ km (expressed in scientific notation, retaining four significant figures).

3.086 \times 10^{13}

For any two coprime positive integers $p, q$, define $f(i)$ to be the remainder of $p\cdot i$ divided by $q$ for $i = 1, 2,\ldots,q -1$. The number $i$ is called a[b] large [/b]number (resp. [b]small[/b] number) when $f(i)$ is the maximum (resp. the minimum) among the numbers $f(1), f(2),\ldots,f(i)$. Note that $1$ is both large and small. Let $a, b$ be two fixed positive integers. Given that there are exactly $a$ large numbers and $b$ small numbers among $1, 2,\ldots , q - 1$, find the least possible number for $q$. [i] Proposed by usjl[/i]

ab + 1

Let $(a_n)_{n\geq 1}$ be a sequence such that $a_1=1$ and $3a_{n+1}-3a_n=1$ for all $n\geq 1$. Find $a_{2002}$. $\textbf{(A) }666\qquad\textbf{(B) }667\qquad\textbf{(C) }668\qquad\textbf{(D) }669\qquad\textbf{(E) }670$

668

4. Let $a$ be a real number such that the graph of the function $y=f(x)=a \sin 2x + \cos 2x + \sin x + \cos x$ is symmetric about the line $x=-\pi$. Let the set of all such $a$ be denoted by $S$. Then $S$ is ( ). (A) empty set (B) singleton set (contains only one element) (C) finite set with more than one element (D) infinite set (contains infinitely many elements)

A

Let $S_n$ be the sum of the reciprocals of the non-zero digits of the integers from 1 to $10^n$ inclusive. Find the smallest positive integer $n$ for which $S_n$ is an integer.

63

The graph shows the fuel used per $100 \mathrm{~km}$ of driving for five different vehicles. Which vehicle would travel the farthest using 50 litres of fuel? (A) $U$ (B) $V$ (C) $W$ (D) $X$ (E) $Y$ ![](https://cdn.mathpix.com/cropped/2024_04_20_6027bc27089ed4fc493cg-059.jpg?height=431&width=380&top_left_y=1297&top_left_x=1271)

D

If $x=\frac{1}{4}$, which of the following has the largest value? (A) $x$ (B) $x^{2}$ (C) $\frac{1}{2} x$ (D) $\frac{1}{x}$ (E) $\sqrt{x}$

D

Let $a, b, c,$ and $d$ be positive integers such that $\gcd(a, b)=24$, $\gcd(b, c)=36$, $\gcd(c, d)=54$, and $70<\gcd(d, a)<100$. Which of the following must be a divisor of $a$? $\textbf{(A)} \text{ 5} \qquad \textbf{(B)} \text{ 7} \qquad \textbf{(C)} \text{ 11} \qquad \textbf{(D)} \text{ 13} \qquad \textbf{(E)} \text{ 17}$

13

Exercise 4. We want to color the three-element subsets of $\{1,2,3,4,5,6,7\}$ such that if two of these subsets have no element in common, then they must be of different colors. What is the minimum number of colors needed to achieve this goal?

3

4. Find all real numbers $p$ such that the cubic equation $5 x^{3}-5(p+1) x^{2}+(71 p-1) x+1=66 p$ has three positive integer roots. untranslated text remains the same as requested.

76

Let $ABC$ be a right triangle with a right angle at $C.$ Two lines, one parallel to $AC$ and the other parallel to $BC,$ intersect on the hypotenuse $AB.$ The lines split the triangle into two triangles and a rectangle. The two triangles have areas $512$ and $32.$ What is the area of the rectangle? [i]Author: Ray Li[/i]

256

4. Given the curve $y=x^{3}-x$, draw a tangent line to the curve from a point $A(t, 0)$ on the $x$-axis, then the maximum number of tangent lines is $\qquad$.

3

7. Given a quartic polynomial $f(x)$ whose four real roots form an arithmetic sequence with a common difference of 2. Then the difference between the largest and smallest roots of $f^{\prime}(x)$ is $\qquad$

2 \sqrt{5}

6. $\frac{2 \cos 10^{\circ}-\sin 20^{\circ}}{\cos 20^{\circ}}$ The value is $\qquad$

\sqrt{3}

3. If point $P\left(x_{0}, y_{0}\right)$ is such that the chord of contact of the ellipse $E: \frac{x}{4}+y^{2}=1$ and the hyperbola $H: x^{2}-\frac{y^{2}}{4}=1$ are perpendicular to each other, then $\frac{y_{0}}{x_{0}}=$ $\qquad$

\pm 1

6. Determine the largest natural number $n \geqq 10$ such that for any 10 different numbers $z$ from the set $\{1,2, \ldots, n\}$, the following statement holds: If none of these 10 numbers is a prime number, then the sum of some two of them is a prime number. (Ján Mazák)

21

Long) and 1 public F: foundation code. A customer wants to buy two kilograms of candy. The salesperson places a 1-kilogram weight on the left pan and puts the candy on the right pan to balance it, then gives the candy to the customer. Then, the salesperson places the 1-kilogram weight on the right pan and puts more candy on the left pan to balance it, and gives this to the customer as well. The two kilograms of candy given to the customer (). (A) is fair (B) the customer gains (C) the store loses (D) if the long arm is more than twice the length of the short arm, the store loses; if less than twice, the customer gains

C

Given that $\sum_{k=1}^{35}\sin 5k=\tan \frac mn,$ where angles are measured in degrees, and $m_{}$ and $n_{}$ are relatively prime positive integers that satisfy $\frac mn<90,$ find $m+n.$

177

3A. Given the set of quadratic functions $f(x)=a x^{2}+b x+c$, for which $a<b$ and $f(x) \geq 0$ for every $x \in \mathbb{R}$. Determine the smallest possible value of the expression $A=\frac{a+b+c}{b-a}$.

3

[b]p1.[/b] You can do either of two operations to a number written on a blackboard: you can double it, or you can erase the last digit. Can you get the number $14$ starting from the number $458$ by using these two operations? [b]p2.[/b] Show that the first $2011$ digits after the point in the infinite decimal fraction representing $(7 + \sqrt50)^{2011}$ are all $9$’s. [b]p3.[/b] Inside of a trapezoid find a point such that segments which join this point with the midpoints of the edges divide the trapezoid into four pieces of equal area. [b]p4.[/b] New chess figure called “lion” is placed on the $8 \times 8$ chess board. It can move one square horizontally right, or one square vertically down or one square diagonally left up. Can lion visit every square of the chess board exactly once and return in his last move to the initial position? You can choose the initial position arbitrarily. PS. You should use hide for answers.

14

A rectangular cuboid has the sum of the three edges starting from one vertex as $p$, and the length of the space diagonal is $q$. What is the surface area of the body? - Can its volume be calculated?

p^2-q^2