Olympiad problems

2018 Peru IMO TST, 1

A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even. [i]Proposed by Jeck Lim, Singapore[/i]

2010 Today's Calculation Of Integral, 574

Tags: logarithm , calculus , integration , calculus computations

Let $ n$ be a positive integer. Prove that $ x^ne^{1\minus{}x}\leq n!$ for $ x\geq 0$,

2007 AMC 8, 22

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A lemming sits at a corner of a square with side length $10$ meters. The lemming runs $6.2$ meters along a diagonal toward the opposite corner. It stops, makes a $90^{\circ}$ right turn and runs $2$ more meters. A scientist measures the shortest distance between the lemming and each side of the square. What is the average of these four distances in meters? $\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4.5 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6.2 \qquad \textbf{(E)}\ 7$

1994 Bundeswettbewerb Mathematik, 2

Tags: sequence , divisibility , number theory

Let $k$ be an integer and define a sequence $a_0 , a_1 ,a_2 ,\ldots$ by $$ a_0 =0 , \;\; a_1 =k \;\;\text{and} \;\; a_{n+2} =k^{2}a_{n+1}-a_n \; \text{for} \; n\geq 0.$$ Prove that $a_{n+1} a_n +1$ divides $a_{n+1}^{2} +a_{n}^{2}$ for all $n$.

2021 LMT Spring, A13

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In a round-robin tournament, where any two players play each other exactly once, the fact holds that among every three students $A$, $B$, and $C$, one of the students beats the other two. Given that there are six players in the tournament and Aidan beats Zach but loses to Andrew, find how many ways there are for the tournament to play out. Note: The order in which the matches take place does not matter. [i]Proposed by Kevin Zhao[/i]

2017 NIMO Problems, 3

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In rectangle $ABCD$ with center $O$, $AB=10$ and $BC=8$. Circle $\gamma$ has center $O$ and lies tangent to $\overline{AB}$ and $\overline{CD}$. Points $M$ and $N$ are chosen on $\overline{AD}$ and $\overline{BC}$, respectively; segment $MN$ intersects $\gamma$ at two distinct points $P$ and $Q$, with $P$ between $M$ and $Q$. If $MP : PQ : QN = 3 : 5 : 2$, then the length $MN$ can be expressed in the form $\sqrt{a} - \sqrt{b}$, where $a$, $b$ are positive integers. Find $100a + b$. [i]Proposed by Michael Tang[/i]

2019 China Team Selection Test, 6

Tags: number theory

Given coprime positive integers $p,q>1$, call all positive integers that cannot be written as $px+qy$(where $x,y$ are non-negative integers) [i]bad[/i], and define $S(p,q)$ to be the sum of all bad numbers raised to the power of $2019$. Prove that there exists a positive integer $n$, such that for any $p,q$ as described, $(p-1)(q-1)$ divides $nS(p,q)$.

2004 Gheorghe Vranceanu, 2

Tags: inequalities , algebra

[b]a)[/b] Let be an even number $ n\ge 4 $ and $ n $ positive real numbers $ x_1,x_2,\ldots ,x_n. $ Prove that: $$ \min_{1\le i\le n/2} \frac{x_i}{x_{i+n/2}}\le \frac{x_1+x_2+\cdots +x_{n/2}}{x_{1+n/2}+ x_{2+n/2} +\cdots + x_n}\le \max_{1\le i\le n/2} \frac{x_i}{x_{i+n/2}}$$ [b]b)[/b] Let be $ m\ge 1 $ pairwise distinct natural numbers $ a,b,\ldots ,c. $ Show that: $$ \frac{ab\cdots c}{a+b+\cdots +c}\ge (m-1)!\cdot\frac{2}{m+1} $$ [i]M. Tetiva[/i]

2017 Online Math Open Problems, 17

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For a positive integer $n$, define $f(n)=\sum_{i=0}^{\infty}\frac{\gcd(i,n)}{2^i}$ and let $g:\mathbb N\rightarrow \mathbb Q$ be a function such that $\sum_{d\mid n}g(d)=f(n)$ for all positive integers $n$. Given that $g(12321)=\frac{p}{q}$ for relatively prime integers $p$ and $q$, find $v_2(p)$. [i]Proposed by Michael Ren[/i]

2010 Contests, 1

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A student is asked to measure $30.0 {\text{g}}$ of methanol $(d=0.7914 \text{g/mL at 25}^{\circ}\text{C})$ but has only a graduated cylinder with which to measure it. What volume of methanol should the student use to obtain the required ${30.0 \text{g}}$? ${ \textbf{(A)}\ 23.7 \text{mL} \qquad\textbf{(B)}\ 30.0 \text{mL} \qquad\textbf{(C)}\ 32.4 \text{mL} \qquad\textbf{(D)}\ 37.9 \text{mL} }$

PEN Q Problems, 10

Tags: polynomial , algebra

Suppose that the integers $a_{1}$, $a_{2}$, $\cdots$, $a_{n}$ are distinct. Show that \[(x-a_{1})(x-a_{2}) \cdots (x-a_{n})-1\] cannot be expressed as the product of two nonconstant polynomials with integer coefficients.

2021 China Team Selection Test, 3

Tags: algebra , number theory , diophantine equation

Given positive integers $a,b,c$ which are pairwise coprime. Let $f(n)$ denotes the number of the non-negative integer solution $(x,y,z)$ to the equation $$ax+by+cz=n.$$ Prove that there exists constants $\alpha, \beta, \gamma \in \mathbb{R}$ such that for any non-negative integer $n$, $$|f(n)- \left( \alpha n^2+ \beta n + \gamma \right) | < \frac{1}{12} \left( a+b+c \right).$$

2009 Hong Kong TST, 1

Tags: inequalities , trigonometry

Let $ \theta_1, \theta_2,\ldots , \theta_{2008}$ be real numbers. Find the maximum value of $ \sin\theta_1\cos\theta_2 \plus{} \sin\theta_2\cos\theta_3 \plus{} \ldots \plus{} \sin\theta_{2007}\cos\theta_{2008} \plus{} \sin\theta_{2008}\cos\theta_1$

2012 India Regional Mathematical Olympiad, 5

Tags: geometry , trigonometry , area of a triangle , ratio

Let $ABC$ be a triangle. Let $D,E$ be points on the segment $BC$ such that $BD=DE=EC$. Let $F$ be the mid-point of $AC$. Let $BF$ intersect $AD$ in $P$ and $AE$ in $Q$ respectively. Determine the ratio of the area of the triangle $APQ$ to that of the quadrilateral $PDEQ$.

2016 Baltic Way, 12

Tags: tiling , combinatorics

Does there exist a hexagon (not necessarily convex) with side lengths $1, 2, 3, 4, 5, 6$ (not necessarily in this order) that can be tiled with a) $31$ b) $32$ equilateral triangles with side length $1?$

2016 Iran Team Selection Test, 4

Tags: algebra

Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.

2016 Grand Duchy of Lithuania, 3

Tags: equal segments , isosceles , geometry

Let $ABC$ be an isosceles triangle with $AB = AC$. Let $D, E$ and $F$ be points on line segments $BC, CA$ and $AB$, respectively, such that $BF = BE$ and such that $ED$ is the angle bisector of $\angle BEC$. Prove that $BD = EF$ if and only if $AF = EC$.

1987 AIME Problems, 5

Tags: special factorizations , sfft

Find $3x^2 y^2$ if $x$ and $y$ are integers such that $y^2 + 3x^2 y^2 = 30x^2 + 517$.

2020 Brazil Team Selection Test, 8

Tags: inequalities

Let $a_1, a_2,\dots$ be an infinite sequence of positive real numbers such that for each positive integer $n$ we have \[\frac{a_1+a_2+\cdots+a_n}n\geq\sqrt{\frac{a_1^2+a_2^2+\cdots+a_{n+1}^2}{n+1}}.\] Prove that the sequence $a_1,a_2,\dots$ is constant. [i]Proposed by Alex Zhai[/i]

2005 Serbia Team Selection Test, 4

Tags: inequalities , trigonometry , geometry

Let $T$ be the centroid of triangle $ABC$. Prove that \[ \frac 1{\sin \angle TAC} + \frac 1{\sin \angle TBC} \geq 4 \]

2013 Online Math Open Problems, 2

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The figure below consists of several unit squares, $M$ of which are white and $N$ of which are green. Compute $100M+N$. [asy] size(4cm); int N = 4; path square; for (int x=-N; x<=N; ++x) { for (int y=-N+abs(x); y<=N-abs(x); ++y) { square = rotate(9)*((x+0.5,y+0.5)--(x+0.5,y-0.5)--(x-0.5,y-0.5)--(x-0.5,y+0.5)--cycle); if ((x+y) % 2 == 0) { filldraw(square, green, black); } else { filldraw(square, white, black); } } } [/asy] [i]Proposed by Evan Chen[/i]

2022-2023 OMMC FINAL ROUND, 10

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Initially, one cow is located on every negative integer on the number line. Each day, Farmer John chooses an integer $k$ where the interval $[k, k+5000]$ has cows. First, he moves each cow in $[k, k+5000]$ to another integer in the interval, so that no two cows move to the same integer. Then, he chooses a cow in the interval and removes it. Can Farmer John get a cow on $100,000,000$ after some time?

2015 AMC 10, 4

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Pablo, Sofia, and Mia got some candy eggs at a party. Pablo had three times as many eggs as Sofia, and Sofia had twice as many eggs as Mia. Pablo decides to give some of his eggs to Sofia and Mia so that all three will have the same number of eggs. What fraction of his eggs should Pablo give to Sofia? $ \textbf{(A) }\dfrac{1}{12}\qquad\textbf{(B) }\dfrac{1}{6}\qquad\textbf{(C) }\dfrac{1}{4}\qquad\textbf{(D) }\dfrac{1}{3}\qquad\textbf{(E) }\dfrac{1}{2} $

1988 Czech And Slovak Olympiad IIIA, 4

Tags: coloring , combinatorial number theory , number theory , combinatorics

Prove that each of the numbers $1, 2, 3, ..., 2^n$ can be written in one of two colors (red and blue) such that no non-constant $2n$-term arithmetic sequence chosen from these numbers is monochromatic .

2021-IMOC qualification, N3

Tags: number theory

Prove: There exists a positive integer $n$ with $2021$ prime divisors, satisfying $n|2^n+1$.