Olympiad problems

1998 IMO, 5

Let $I$ be the incenter of triangle $ABC$. Let $K,L$ and $M$ be the points of tangency of the incircle of $ABC$ with $AB,BC$ and $CA$, respectively. The line $t$ passes through $B$ and is parallel to $KL$. The lines $MK$ and $ML$ intersect $t$ at the points $R$ and $S$. Prove that $\angle RIS$ is acute.

2023 Caucasus Mathematical Olympiad, 7

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Numbers $1, 2,\ldots, n$ are written on the board. By one move, we replace some two numbers $ a, b$ with the number $a^2-b{}$. Find all $n{}$ such that after $n-1$ moves it is possible to obtain $0$.

2010 Gheorghe Vranceanu, 2

Tags: inequalities , linear algebra , matrix , determinant , real analysis

Let be a natural number $ n, $ a number $ t\in (0,1) $ and $ n+1 $ numbers $ a_0\ge a_1\ge a_2\ge\cdots\ge a_n\ge 0. $ Prove the following matrix inequality: $$ \begin{vmatrix}\frac{(1+t\sqrt{-1})^2}{1+t^2} & -1 & 0& 0 & \cdots & 0 & 0 \\ 0 & \frac{(1+t\sqrt{-1})^2}{1+t^2} & -1 & 0 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & \cdots & \frac{(1+t\sqrt{-1})^2}{1+t^2} & -1 \\ a_0 & a_1 & a_2 & a_3 & \cdots & a_{n-1} & a_n \end{vmatrix}^2\le a_0^2\left( 1+\frac{1}{t^2} \right) $$

2021-IMOC, N6

Tags: number theory , diaphontine

Show that there do not exist positive integers $x,y,z$ such that $$x^x + y^y = 9^z$$ [i]usjl[/i]

2011 Tuymaada Olympiad, 4

Tags: 3d geometry , number theory , geometry

In a set of consecutive positive integers, there are exactly $100$ perfect cubes and $10$ perfect fourth powers. Prove that there are at least $2000$ perfect squares in the set.

2009 Sharygin Geometry Olympiad, 3

Tags: geometry , circumcircle , incircle , concurrency

Quadrilateral $ABCD$ is circumscribed, rays $BA$ and $CD$ intersect in point $E$, rays $BC$ and $AD$ intersect in point $F$. The incircle of the triangle formed by lines $AB, CD$ and the bisector of angle $B$, touches $AB$ in point $K$, and the incircle of the triangle formed by lines $AD, BC$ and the bisector of angle $B$, touches $BC$ in point $L$. Prove that lines $KL, AC$ and $EF$ concur. (I.Bogdanov)

2019 239 Open Mathematical Olympiad, 1

Tags: combinatorics

On the island of knights and liars, a tennis tournament was held, in which $100$ people participated in. Each two of them played exactly $1$ time with the other one. After the tournament, each of the participants declared: “I have beaten as many knights as liars,” while all the knights told the truth, and all the liars lied. What is the largest number of knights that could participate in the tournament?

2020 JBMO Shortlist, 2

Tags: number theory

Find all positive integers $a$, $b$, $c$, and $p$, where $p$ is a prime number, such that $73p^2 + 6 = 9a^2 + 17b^2 + 17c^2$.

1986 Spain Mathematical Olympiad, 4

Tags: function , mean , algebra

Denote by $m(a,b)$ the arithmetic mean of positive real numbers $a,b$. Given a positive real function $g$ having positive derivatives of the first and second order, define $\mu (a,b)$ the mean value of $a$ and $b$ with respect to $g$ by $2g(\mu (a,b)) = g(a)+g(b)$. Decide which of the two mean values $m$ and $\mu$ is larger.

2009 Harvard-MIT Mathematics Tournament, 6

Tags: calculus , algebra , polynomial , function

Let $p_0(x),p_1(x),p_2(x),\ldots$ be polynomials such that $p_0(x)=x$ and for all positive integers $n$, $\dfrac{d}{dx}p_n(x)=p_{n-1}(x)$. Define the function $p(x):[0,\infty)\to\mathbb{R}$ by $p(x)=p_n(x)$ for all $x\in [n,n+1)$. Given that $p(x)$ is continuous on $[0,\infty)$, compute \[\sum_{n=0}^\infty p_n(2009).\]

2007 QEDMO 5th, 7

Tags: combinatorics

In a group of $20$ people, each person sends a letter to $10$ of the others. Prove that there are two persons who send a letter to each other.

2007 Harvard-MIT Mathematics Tournament, 4

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A sequence consists of the digits $122333444455555\ldots$ such that the each positive integer $n$ is repeated $n$ times, in increasing order. Find the sum of the $4501$st and $4052$nd digits of this sequence.

2012 Czech And Slovak Olympiad IIIA, 2

Tags: geometry , area of a triangle , maximum , median

Find out the maximum possible area of the triangle $ABC$ whose medians have lengths satisfying inequalities $m_a \le 2, m_b \le 3, m_c \le 4$.

2023 Israel TST, P2

Tags: combinatorics , modular arithmetic , abstract algebra

Let $n>3$ be an integer. Integers $a_1, \dots, a_n$ are given so that $a_k\in \{k, -k\}$ for all $1\leq k\leq n$. Prove that there is a sequence of indices $1\leq k_1, k_2, \dots, k_n\leq n$, not necessarily distinct, for which the sums \[a_{k_1}\] \[a_{k_1}+a_{k_2}\] \[a_{k_1}+a_{k_2}+a_{k_3}\] \[\vdots\] \[a_{k_1}+a_{k_2}+\cdots+a_{k_n}\] have distinct residues modulo $2n+1$, and so that the last one is divisible by $2n+1$.

1980 IMO, 20

Tags: inequalities , inradius , circumcircle , geometry

The radii of the circumscribed circle and the inscribed circle of a regular $n$-gon, $n\ge 3$ are denoted by $R_n$ and $r_n$, respectively. Prove that \[\frac{r_n}{R_n}\ge\left(\frac{r_{n+1}}{R_{n+1}}\right)^2.\]

2021 MOAA, 3

Tags:

What is the last digit of $2021^{2021}$? [i]Proposed by Yifan Kang[/i]

2022 Olympic Revenge, Problem 4

Tags: number theory

Let $\{a_n\}_{n=1}^{\infty}$ be a sequence of positive integers such that $a_1=1$. For each $n \geq 1$, $a_{n+1}$ is the smallest positive integer, distinct from $a_1,a_2,...,a_n$, such that $\gcd(a_{n+1}a_n+1,a_i)=1$ for each $i=1,2,...,n$. Prove that every positive integer appears in $\{a_n\}_{n=1}^{\infty}$.

1992 Austrian-Polish Competition, 1

Tags: sum , positive , number theory , product , even , divisor

For a natural number $n$, denote by $s(n)$ the sum of all positive divisors of n. Prove that for every $n > 1$ the product $s(n - 1)s(n)s(n + 1)$ is even.

1991 Arnold's Trivium, 86

Tags: geometry , 3d geometry , tetrahedron , icosahedron

Through the centre of a cube (tetrahedron, icosahedron) draw a straight line in such a way that the sum of the squares of its distances from the vertices is a) minimal, b) maximal.

2019 IMO Shortlist, A2

Tags: inequalities , algebra

Let $u_1, u_2, \dots, u_{2019}$ be real numbers satisfying \[u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1.\] Let $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ and $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that \[ a b \leqslant-\frac{1}{2019}. \]

2014 USA Team Selection Test, 2

Tags: quadratic residue , quadratic , induction , modular arithmetic , number theory

Let $a_1,a_2,a_3,\ldots$ be a sequence of integers, with the property that every consecutive group of $a_i$'s averages to a perfect square. More precisely, for every positive integers $n$ and $k$, the quantity \[\frac{a_n+a_{n+1}+\cdots+a_{n+k-1}}{k}\] is always the square of an integer. Prove that the sequence must be constant (all $a_i$ are equal to the same perfect square). [i]Evan O'Dorney and Victor Wang[/i]

2014 Miklós Schweitzer, 10

Tags: 3d geometry , sphere , geometry

To each vertex of a given triangulation of the two-dimensional sphere, we assign a convex subset of the plane. Assume that the three convex sets corresponding to the three vertices of any two-dimensional face of the triangulation have at least one point in common. Show that there exist four vertices such that the corresponding convex sets have at least one point in common.

2010 Today's Calculation Of Integral, 614

Tags: calculus , integration , ratio , trigonometry , calculus computations

Evaluate $\int_0^1 \{x(1-x)\}^{\frac 32}dx.$ [i]2010 Hirosaki University School of Medicine entrance exam[/i]

2023 Romania Team Selection Test, P4

Tags: geometry , combinatorics , combinatorial geometry

Fix a positive integer $n.{}$ Consider an $n{}$-point set $S{}$ in the plane. An [i]eligible[/i] set is a non-empty set of the form $S\cap D,{}$ where $D$ is a closed disk in the plane. In terms of $n,$ determine the smallest possible number of eligible subsets $S{}$ may contain. [i]Proposed by Cristi Săvescu[/i]

2006 Iran MO (3rd Round), 1

Tags: incenter , geometric transformation , homothety , ratio , radical axis , geometry , angle bisector

Prove that in triangle $ABC$, radical center of its excircles lies on line $GI$, which $G$ is Centroid of triangle $ABC$, and $I$ is the incenter.