Olympiad problems

2015 Dutch IMO TST, 4

Each of the numbers $1$ up to and including $2014$ has to be coloured; half of them have to be coloured red the other half blue. Then you consider the number $k$ of positive integers that are expressible as the sum of a red and a blue number. Determine the maximum value of $k$ that can be obtained.

2024 Bangladesh Mathematical Olympiad, P5

Tags: angle chasing , parallelogram , geometry , incenter

Consider $\triangle XPQ$ and $\triangle YPQ$ such that $X$ and $Y$ are on the opposite sides of $PQ$ and the circumradius of $\triangle XPQ$ and the circumradius of $\triangle YPQ$ are the same. $I$ and $J$ are the incenters of $\triangle XPQ$ and $\triangle YPQ$ respectively. Let $M$ be the midpoint of $PQ$. Suppose $I, M, J$ are collinear. Prove that $XPYQ$ is a parallelogram.

1966 Bulgaria National Olympiad, Problem 1

Tags: diophantine equation , number theory

Prove that the equation $$3x(x-3y)=y^2+z^2$$doesn't have any integer solutions except $x=0,y=0,z=0$.

2007 AMC 10, 20

Tags:

A set of $ 25$ square blocks is arranged into a $ 5\times 5$ square. How many different combinations of $ 3$ blocks can be selected from that set so that no two are in the same row or column? $ \textbf{(A)}\ 100\qquad \textbf{(B)}\ 125\qquad \textbf{(C)}\ 600\qquad \textbf{(D)}\ 2300\qquad \textbf{(E)}\ 3600$

1999 Israel Grosman Mathematical Olympiad, 3

Tags: equilateral , geometry , incircle

For every triangle $ABC$, denote by $D(ABC)$ the triangle whose vertices are the tangency points of the incircle of $\vartriangle ABC$ with the sides. Assume that $\vartriangle ABC$ is not equilateral. (a) Prove that $D(ABC)$ is also not equilateral. (b) Find in the sequence $T_1 = \vartriangle ABC, T_{k+1} = D(T_k)$ for $k \in N$ a triangle whose largest angle $\alpha$ satisfies $0 < \alpha -60^o < 0.0001^o$

2009 China Western Mathematical Olympiad, 4

Tags: inequalities

The real numbers $a_{1},a_{2},\ldots ,a_{n}$ where $n\ge 3$ are such that $\sum_{i=1}^{n}a_{i}=0$ and $2a_{k}\le\ a_{k-1}+a_{k+1}$ for all $k=2,3,\ldots ,n-1$. Find the least $f(n)$ such that, for all $k\in\left\{1,2,\ldots ,n\right\}$, we have $|a_{k}|\le f(n)\max\left\{|a_{1}|,|a_{n}|\right\}$.

2013 Denmark MO - Mohr Contest, 2

Tags: geometry , area , semicircle , circumcircle , rectangle

The figure shows a rectangle, its circumscribed circle and four semicircles, which have the rectangle’s sides as diameters. Prove that the combined area of the four dark gray crescentshaped regions is equal to the area of the light gray rectangle. [img]https://1.bp.blogspot.com/-gojv6KfBC9I/XzT9ZMKrIeI/AAAAAAAAMVU/NB-vUldjULI7jvqiFWmBC_Sd8QFtwrc7wCLcBGAsYHQ/s0/2013%2BMohr%2Bp3.png[/img]

2022 Romania Team Selection Test, 2

Tags: sequence , number theory

Fix a nonnegative integer $a_0$ to define a sequence of integers $a_0,a_1,\ldots$ by letting $a_k,k\geq 1$ be the smallest integer (strictly) greater than $a_{k-1}$ making $a_{k-1}+a_k{}$ into a perfect square. Let $S{}$ be the set of positive integers not expressible as the difference of two terms of the sequence $(a_k)_{k\geq 0}.$ Prove that $S$ is finite and determine its size in terms of $a_0.$

2015 Irish Math Olympiad, 3

Tags: perfect cube , number theory

Find all positive integers $n$ for which both $837 + n$ and $837 - n$ are cubes of positive integers.

2006 Kurschak Competition, 2

Tags: combinatorics

Let $a,t,n$ be positive integers such that $a\le n$. Consider the subsets of $\{1,2,\dots,n\}$ whose any two elements differ by at least $t$. Prove that the number of such subsets not containing $a$ is at most $t^2$ times the number of those that do contain $a$.

1996 IMC, 8

Tags: trigonometry

Let $\theta$ be a positive real number. Show that if $k\in \mathbb{N}$ and both $\cosh k \theta$ and $\cosh(k+1) \theta$ are rational, then so is $\cosh \theta$.

1990 Bulgaria National Olympiad, Problem 5

Tags: triangle , geometry , arc , circles

Given a circular arc, find a triangle of the smallest possible area which covers the arc so that the endpoints of the arc lie on the same side of the triangle.

2012 Today's Calculation Of Integral, 819

Tags: limit , calculus computations , calculus , trigonometry , integration

For real numbers $a,\ b$ with $0\leq a\leq \pi,\ a<b$, let $I(a,\ b)=\int_{a}^{b} e^{-x} \sin x\ dx.$ Determine the value of $a$ such that $\lim_{b\rightarrow \infty} I(a,\ b)=0.$

2010 Today's Calculation Of Integral, 654

Tags: function , limit , calculus , calculus computations , integration

A function $f(x)$ defined in $x\geq 0$ satisfies $\lim_{x\to\infty} \frac{f(x)}{x}=1$. Find $\int_0^{\infty} \{f(x)-f'(x)\}e^{-x}dx$. [i]1997 Hokkaido University entrance exam/Science[/i]

2017 AIME Problems, 2

Tags:

When each of 702, 787, and 855 is divided by the positive integer $m$, the remainder is always the positive integer $r$. When each of 412, 722, and 815 is divided by the positive integer $n$, the remainder is always the positive integer $s \neq r$. Fine $m+n+r+s$.

2019 Puerto Rico Team Selection Test, 2

Tags: perpendicular , geometry , square

Let $ABCD$ be a square. Let $M$ and $K$ be points on segments $BC$ and $CD$ respectively, such that $MC = KD$. Let $ P$ be the intersection of the segments $MD$ and $BK$. Prove that $AP$ is perpendicular to $MK$.

2003 Kazakhstan National Olympiad, 2

Tags: inequalities , algebra

For positive real numbers $ x, y, z $, prove the inequality: $$ \displaylines {\frac {x ^ 3} {x + y} + \frac {y ^ 3} {y + z} + \frac {z ^ 3} {z + x} \geq \frac {xy + yz + zx} {2}.} $$

2017 Junior Regional Olympiad - FBH, 5

Tags: rational , sqaure roots

Find all positive integers $a$ and $b$ such that number $p=\frac{\sqrt{2}+\sqrt{a}}{\sqrt{3}+\sqrt{b}}$ is rational number

2011 Macedonia National Olympiad, 4

Tags: function , search , algebra

Find all functions $~$ $f: \mathbb{R} \to \mathbb{R}$ $~$ which satisfy the equation \[ f(x+yf(x))\, =\, f(f(x)) + xf(y)\, . \]

2004 Greece National Olympiad, 2

Tags: induction , algebra

If $m\geq 2$ show that there does not exist positive integers $x_1, x_2, ..., x_m,$ such that \[x_1< x_2<...< x_m \ \ \text{and} \ \ \frac{1}{x_1^3}+\frac{1}{x_2^3}+...+\frac{1}{x_m^3}=1.\]

1980 IMO, 9

Tags: algebra

Prove that is $x,y$ are non negative integers then $5x\ge 7y$ if and only if there exist non-negative integers $(a,b,c,d)$ such that \[\left\{\begin{array}{l}x=a+2b+3c+7d\qquad\\ y=b+2c+5d\qquad\\ \end{array}\right.\]

2023 China Team Selection Test, P21