Olympiad problems

2006 Tournament of Towns, 4

A circle of radius $R$ is inscribed into an acute triangle. Three tangents to the circle split the triangle into three right angle triangles and a hexagon that has perimeter $Q$. Find the sum of diameters of circles inscribed into the three right triangles. (6)

2022 Latvia Baltic Way TST, P9

Tags: geometry , cyclic quadrilateral , circumcircle

Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\Omega$. Let the lines $AB$ and $CD$ intersect at $P$, and the lines $AD$ and $BC$ intersect at $Q$. Let then the circumcircle of the triangle $\triangle APQ$ intersect $\Omega$ at $R \neq A$. Prove that the line $CR$ goes through the midpoint of the segment $PQ$.

1997 Denmark MO - Mohr Contest, 2

Tags: square , area , geometry

Two squares, both with side length $1$, are arranged so that one has one vertex in the center of the other. Determine the area of the gray area. [img]https://1.bp.blogspot.com/-xt3pe0rp1SI/XzcGLgEw1EI/AAAAAAAAMYM/vFKxvvVuLvAJ5FO_yX315X3Fg_iFaK2fACLcBGAsYHQ/s0/1997%2BMohr%2Bp2.png[/img]

2002 India Regional Mathematical Olympiad, 4

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Suppose the integers $1,2,\ldots 10$ are split into two disjoint collections $a_1,a_2, \ldots a_5$ and $b_1 , \ldots b_5$ such that $a_1 <a _2 < a_3 <a_4 <a _5 , b_1 < b_2 < b_3 < b_4 < b_5$ (i) Show that the larger number in any pair $\{ a_j, b_j \}$ , $1 \leq j \leq 5$ is at least $6$. (ii) Show that $\sum_{i=1} ^{5} | a_i - b_i|$ = 25 for every such partition.

2013 Dutch IMO TST, 4

Tags: combinatorics , combinatorial geometry , rectangle , board

Let $n \ge 3$ be an integer, and consider a $n \times n$-board, divided into $n^2$ unit squares. For all $m \ge 1$, arbitrarily many $1\times m$-rectangles (type I) and arbitrarily many $m\times 1$-rectangles (type II) are available. We cover the board with $N$ such rectangles, without overlaps, and such that every rectangle lies entirely inside the board. We require that the number of type I rectangles used is equal to the number of type II rectangles used.(Note that a $1 \times 1$-rectangle has both types.) What is the minimal value of $N$ for which this is possible?

PEN H Problems, 76

Tags: diophantine equation

Find all pairs $(m,n)$ of integers that satisfy the equation \[(m-n)^{2}=\frac{4mn}{m+n-1}.\]

2023 Bulgarian Spring Mathematical Competition, 9.3

Tags: number theory

Given a prime $p$, find $\gcd(\binom{2^pp}{1},\binom{2^pp}{3},\ldots, \binom{2^pp}{2^pp-1}) $.

2008 All-Russian Olympiad, 7

Tags: number theory

For which integers $ n>1$ do there exist natural numbers $ b_1,b_2,\ldots,b_n$ not all equal such that the number $ (b_1\plus{}k)(b_2\plus{}k)\cdots(b_n\plus{}k)$ is a power of an integer for each natural number $ k$? (The exponents may depend on $ k$, but must be greater than $ 1$)

2013 Sharygin Geometry Olympiad, 21

Tags: power of a point , reflection , angle chasing , similar triangles , geometry

Chords $BC$ and $DE$ of circle $\omega$ meet at point $A$. The line through $D$ parallel to $BC$ meets $\omega$ again at $F$, and $FA$ meets $\omega$ again at $T$. Let $M = ET \cap BC$ and let $N$ be the reflection of $A$ over $M$. Show that $(DEN)$ passes through the midpoint of $BC$.

2022 USEMO, 1

Tags: perimeter , combinatorics , grid

A [i]stick[/i] is defined as a $1 \times k$ or $k\times 1$ rectangle for any integer $k \ge 1$. We wish to partition the cells of a $2022 \times 2022$ chessboard into $m$ non-overlapping sticks, such that any two of these $m$ sticks share at most one unit of perimeter. Determine the smallest $m$ for which this is possible. [i]Holden Mui[/i]

2017 International Zhautykov Olympiad, 3

Tags: 3d geometry , tetrahedron , inequalities , geometry

Let $ABCD$ be the regular tetrahedron, and $M, N$ points in space. Prove that: $AM \cdot AN + BM \cdot BN + CM \cdot CN \geq DM \cdot DN$

2010 Today's Calculation Of Integral, 555

Tags: calculus , integration , derivative , function , calculus computations , logarithm

For $ \frac {1}{e} < t < 1$, find the minimum value of $ \int_0^1 |xe^{ \minus{} x} \minus{} tx|dx$.

2023 Turkey Olympic Revenge, 5

Tags: combinatorics , game theory , combinatorial game theory

There are $10$ cups, each having $10$ pebbles in them. Two players $A$ and $B$ play a game, repeating the following in order each move: $\bullet$ $B$ takes one pebble from each cup and redistributes them as $A$ wishes. $\bullet$ After $B$ distributes the pebbles, he tells how many pebbles are in each cup to $A$. Then $B$ destroys all the cups having no pebbles. $\bullet$ $B$ switches the places of two cups without telling $A$. After finitely many moves, $A$ can guarantee that $n$ cups are destroyed. Find the maximum possible value of $n$. (Note that $A$ doesn't see the cups while playing.) [i]Proposed by Emre Osman[/i]

2020 Korean MO winter camp, #7

Tags: algebra

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $2f(x^2+y^2)=(x+f(y))^2+f(x-f(y))^2$ for all $x,y\in\mathbb{R}$.

2004 South africa National Olympiad, 1

Tags: greatest common divisor , algorithm , number theory

Let $a=1111\dots1111$ and $b=1111\dots1111$ where $a$ has forty ones and $b$ has twelve ones. Determine the greatest common divisor of $a$ and $b$.

2021 Romania EGMO TST, P2

Tags: geometry

Two circles intersect at points $A\neq B$. A line passing through $A{}$ intersects the circles again at $C$ and $D$. Let $E$ and $F$ be the midpoints of the arcs $\overarc{BC}$ and $\overarc{BD}$ which do not contain $A{}$ and let $M$ be the midpoint of the segment $CD$. Prove that $ME$ and $MF$ are perpendicular.

2023 Harvard-MIT Mathematics Tournament, 4

Tags: hmmt

Suppose $P (x)$ is a polynomial with real coefficients such that $P (t) = P (1)t^2 + P (P (1))t + P (P (P (1)))$ for all real numbers $t$. Compute the largest possible value of $P(P(P(P(1))))$.

MOAA Individual Speed General Rounds, 2023.5

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Let $P(x)$ be a nonzero quadratic polynomial such that $P(1) = P(2) = 0$. Given that $P(3)^2 = P(4)+P(5)$, find $P(6)$. [i]Proposed by Andy Xu[/i]

2016 Korea National Olympiad, 4

Tags: combinatorics

For a positive integer $n$, $S_n$ is the set of positive integer $n$-tuples $(a_1,a_2, \cdots ,a_n)$ which satisfies the following. (i). $a_1=1$. (ii). $a_{i+1} \le a_i+1$. For $k \le n$, define $N_k$ as the number of $n$-tuples $(a_1, a_2, \cdots a_n) \in S_n$ such that $a_k=1, a_{k+1}=2$. Find the sum $N_1 + N_2+ \cdots N_{k-1}$.

2009 Croatia Team Selection Test, 4

Tags: quadratic , diophantine equation , number theory

Prove that there are infinite many positive integers $ n$ such that $ n^2\plus{}1\mid n!$, and infinite many of those for which $ n^2\plus{}1 \nmid n!$.

1986 Traian Lălescu, 2.3

Tags: integral , function , calculus , derivative , ftc , rolle theorem , inequalities

Let $ f:[0,2]\longrightarrow \mathbb{R} $ a differentiable function having a continuous derivative and satisfying $ f(0)=f(2)=1 $ and $ |f’|\le 1. $ Show that $$ \left| \int_0^2 f(t) dt\right| >1. $$

2017 Regional Olympiad of Mexico Southeast, 4

Tags: number theory , diophantine equation , factorial

Find all couples of positive integers $m$ and $n$ such that $$n!+5=m^3$$

1995 Putnam, 4

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Suppose we have a necklace of $n$ beads. Each bead is labelled with an integer and the sum of all these labels is $n-1$. Prove that we can cut the necklace to form a string whose consecutive labels $x_1, x_2,\cdots , x_n$ satisfy \[ \sum_{i=1}^{k}x_i\le k-1\quad \forall \;\;1\le k\le n \]

2004 USAMTS Problems, 4

Tags: algebra , system of equations

Find, with proof, all integers $n$ such that there is a solution in nonnegative real numbers $(x,y,z)$ to the system of equations \[2x^2+3y^2+6z^2=n\text{ and }3x+4y+5z=23.\]

2000 JBMO ShortLists, 1

Tags: number theory

Prove that there are at least $666$ positive composite numbers with $2006$ digits, having a digit equal to $7$ and all the rest equal to $1$.