Olympiad problems

2025 Sharygin Geometry Olympiad, 18

Tags: geometry

Let $ABCD$ be a quadrilateral such that the excircles $\omega_{1}$ and $\omega_{2}$ of triangles $ABC$ and $BCD$ touching their sides $AB$ and $BD$ respectively touch the extension of $BC$ at the same point $P$. The segment $AD$ meets $\omega_{2}$ at point $Q$, and the line $AD$ meets $\omega_{1}$ at $R$ and $S$. Prove that one of angles $RPQ$ and $SPQ$ is right Proposed by: I.Kukharchuk

2020 USA EGMO Team Selection Test, 6

Tags: algebra , polynomial , integer polynomial , periodic sequence , modular arithmetic

Find the largest integer $N \in \{1, 2, \ldots , 2019 \}$ such that there exists a polynomial $P(x)$ with integer coefficients satisfying the following property: for each positive integer $k$, $P^k(0)$ is divisible by $2020$ if and only if $k$ is divisible by $N$. Here $P^k$ means $P$ applied $k$ times, so $P^1(0)=P(0), P^2(0)=P(P(0)),$ etc.

2020 MBMT, 3

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Square $ABCD$ has a side length of 1. Point $E$ lies on the interior of $ABCD$, and is on the line $\overleftrightarrow{AC}$ such that the length of $\overline{AE}$ is 1. Find the shortest distance from point $E$ to a side of square $ABCD$. [i]Proposed by Chris Tong[/i]

2022 Moldova Team Selection Test, 7

Tags: number theory

Let $f:\mathbb{N} \rightarrow \mathbb{N},$ $f(n)=n^2-69n+2250$ be a function. Find the prime number $p$, for which the sum of the digits of the number $f(p^2+32)$ is as small as possible.

1990 Romania Team Selection Test, 9

Tags: distance , combinatorial geometry , geometry

The distance between any two of six given points in the plane is at least $1$. Prove that the distance between some two points is at least $\sqrt{\frac{5+\sqrt5}{2}}$

2016 CMIMC, 9

Tags: computer science

Ryan has three distinct eggs, one of which is made of rubber and thus cannot break; unfortunately, he doesn't know which egg is the rubber one. Further, in some 100-story building there exists a floor such that all normal eggs dropped from below that floor will not break, while those dropped from at or above that floor will break and cannot be dropped again. What is the minimum number of times Ryan must drop an egg to determine the floor satisfying this property?

2008 Portugal MO, 6

Tags: combinatorics

Let $n$ be a natural number larger than $2$. Vanessa has $n$ piles of jade stones, and all the piles have a different number of stones. Vanessa can distribute the stones from any pile by the other piles and stay with $n-1$ piles with the same number of stones. She also can distribute the stones from any two piles by the other piles and stay with $n-2$ piles with the same number of stones. Find the smallest possible number of jade's stones that the pile with the largest number of stones can have.

2009 Philippine MO, 2

Tags: diophantine , number theory

[b](a)[/b] Find all pairs $(n,x)$ of positive integers that satisfy the equation $2^n + 1 = x^2$. [b](b)[/b] Find all pairs $(n,x)$ of positive integers that satisfy the equation $2^n = x^2 + 1$.

2016 Costa Rica - Final Round, N3

Tags: diophantine equation , diophantine , number theory

Find all natural values of $n$ and $m$, such that $(n -1)2^{n - 1} + 5 = m^2 + 4m$.

2017 BAMO, 5

Tags:

Call a number $T$ [i]persistent[/i] if the following holds: Whenever $a,b,c,d$ are real numbers different from $0$ and $1$ such that $$a+b+c+d = T$$ and $$\frac{1}{a}+\frac{1}{b} +\frac{1}{c}+\frac{1}{d} = T,$$ we also have $$\frac{1}{1 - a}+\frac{1}{1-b}+\frac{1}{1-c}+\frac{1}{1-d}= T.$$ (a) If $T$ is persistent, prove that $T$ must be equal to $2$. (b) Prove that $2$ is persistent. Note: alternatively, we can just ask “Show that there exists a unique persistent number, and determine its value”.

2006 Hong Kong TST., 4

Tags: inequalities

Let x,y,z be positive real numbers such that $x+y+z=1$. For positive integer n, define $S_n = x^n+y^n+z^n$ Furthermore, let $P=S_2 S_{2005}$ and $Q=S_3 S_{2004}$. (a) Find the smallest possible value of Q. (b) If $x,y,z$ are pairwise distinct, determine whether P or Q is larger.

2018 Brazil EGMO TST, 2

Tags: inequalities , min , algebra

(a) Let $x$ be a real number with $x \ge 1$. Prove that $x^3 - 5x^2 + 8x - 4 \ge 0$. (b) Let $a, b \ge 1$ real numbers. Find the minimum value of the expression $ab(a + b - 10) + 8(a + b)$. Determine also the real number pairs $(a, b)$ that make this expression equal to this minimum value.

2008 German National Olympiad, 3

Tags: function , algebra

Find all functions $ f$ defined on non-negative real numbers having the following properties: (i) For all non-negative $ x$ it is $ f(x) \geq 0$. (ii) It is $ f\left(1\right)\equal{}\frac 12$. (iii) For all non-negative numbers $ x,y$ it is $ f\left( y \cdot f(x) \right) \cdot f(x) \equal{} f(x\plus{}y)$.

2004 Olympic Revenge, 4

Tags: functional equation , functional , algebra

Find all functions $f:R \rightarrow R$ such that for any reals $x,y$, $f(x^2+y)=f(x)f(x+1)+f(y)+2x^2y$.

2005 China Team Selection Test, 2

Tags: algebra , logarithm

Determine whether $\sqrt{1001^2+1}+\sqrt{1002^2+1}+ \cdots + \sqrt{2000^2+1}$ be a rational number or not?

2011 Postal Coaching, 2

Tags: combinatorics

For which $n \ge 1$ is it possible to place the numbers $1, 2, \ldots, n$ in some order $(a)$ on a line segment, or $(b)$ on a circle so that for every $s$ from $1$ to $\frac{n(n+1)}{2}$, there is a connected subset of the segement or circle such that the sum of the numbers in that subset is $s$?

2022 Moldova EGMO TST, 11

Tags: geometry , trapezoid

Let there be a trapezoid $ABCD$ with bases $AD$ and $BC$. Points $M$ and $P$ are on sides $AB$ and $CD$ such that $CM$ and $BP$ intersect in $N$ and the pentagon $AMNPD$ is cyclic. Prove that the triangle $ADN$ is isosceles.

2019 India IMO Training Camp, P2

Tags: number theory

Show that there do not exist natural numbers $a_1, a_2, \dots, a_{2018}$ such that the numbers \[ (a_1)^{2018}+a_2, (a_2)^{2018}+a_3, \dots, (a_{2018})^{2018}+a_1 \] are all powers of $5$ [i]Proposed by Tejaswi Navilarekallu[/i]

2013 IMO Shortlist, N3

Tags: number theory , prime divisor , polynomial

Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.

1998 Slovenia Team Selection Test, 6

Tags: sequence , recurrence relation , algebra

Let $a_0 = 1998$ and $a_{n+1} =\frac{a_n^2}{a_n +1}$ for each nonnegative integer $n$. Prove that $[a_n] = 1994- n$ for $0 \le n \le 1000$

2002 Bundeswettbewerb Mathematik, 4

Tags: geometry , circumcircle , search , incenter , analytic geometry , ratio , trigonometry

In an acute-angled triangle $ABC$, we consider the feet $H_a$ and $H_b$ of the altitudes from $A$ and $B$, and the intersections $W_a$ and $W_b$ of the angle bisectors from $A$ and $B$ with the opposite sides $BC$ and $CA$ respectively. Show that the centre of the incircle $I$ of triangle $ABC$ lies on the segment $H_aH_b$ if and only if the centre of the circumcircle $O$ of triangle $ABC$ lies on the segment $W_aW_b$.

2017 China Team Selection Test, 1

Tags: combinatorics , equation , generating functions

Prove that :$$\sum_{k=0}^{58}C_{2017+k}^{58-k}C_{2075-k}^{k}=\sum_{p=0}^{29}C_{4091-2p}^{58-2p}$$

2012 Kazakhstan National Olympiad, 3

Tags: function , induction , combinatorics

There are $n$ balls numbered from $1$ to $n$, and $2n-1$ boxes numbered from $1$ to $2n-1$. For each $i$, ball number $i$ can only be put in the boxes with numbers from $1$ to $2i-1$. Let $k$ be an integer from $1$ to $n$. In how many ways we can choose $k$ balls, $k$ boxes and put these balls in the selected boxes so that each box has exactly one ball?

2016 CMIMC, 4

Tags: function , number theory , prime factorization

For some positive integer $n$, consider the usual prime factorization \[n = \displaystyle \prod_{i=1}^{k} p_{i}^{e_{i}}=p_1^{e_1}p_2^{e_2}\ldots p_k^{e_k},\] where $k$ is the number of primes factors of $n$ and $p_{i}$ are the prime factors of $n$. Define $Q(n), R(n)$ by \[ Q(n) = \prod_{i=1}^{k} p_{i}^{p_{i}} \text{ and } R(n) = \prod_{i=1}^{k} e_{i}^{e_{i}}. \] For how many $1 \leq n \leq 70$ does $R(n)$ divide $Q(n)$?

2019 Greece JBMO TST, 1

Tags: geometry , concyclic , circumcircle , perpendicular

Consider an acute triangle $ABC$ with $AB>AC$ inscribed in a circle of center $O$. From the midpoint $D$ of side $BC$ we draw line $(\ell)$ perpendicular to side $AB$ that intersects it at point $E$. If line $AO$ intersects line $(\ell)$ at point $Z$, prove that points $A,Z,D,C$ are concyclic.