Olympiad problems

1978 Czech and Slovak Olympiad III A, 6

Show that the number \[p_n=\left(\frac{3+\sqrt5}{2}\right)^n+\left(\frac{3-\sqrt5}{2}\right)^n-2\] is a positive integer for any positive integer $n.$ Furthermore, show that the numbers $p_{2n-1}$ and $p_{2n}/5$ are perfect squares $($for any positive integer $n).$

1993 Miklós Schweitzer, 1

Tags: combinatorics , geometry

There are n points in the plane with the property that the distance between any two points is at least 1. Prove that for sufficiently large n , the number of pairs of points whose distance is in $[ t_1 , t_1 + 1] \cup [ t_2 , t_2 + 1]$ for some $t_1, t_2$ , is at most $[\frac{n^2}{3}]$ , and the bound is sharp.

2013 AIME Problems, 14

Tags: induction , modular arithmetic , number theory , pattern finding

For positive integers $n$ and $k$, let $f(n,k)$ be the remainder when $n$ is divided by $k$, and for $n>1$ let $F(n) = \displaystyle\max_{1 \le k \le \frac{n}{2}} f(n,k)$. Find the remainder when $\displaystyle\sum_{n=20}^{100} F(n)$ is divided by $1000$.

2004 Denmark MO - Mohr Contest, 2

Tags: number theory , divide , divisible

Show that if $a$ and $b$ are integer numbers, and $a^2 + b^2 + 9ab$ is divisible by $11$, then $a^2-b^2$ divisible by $11$.

2018 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: number theory , coprime , coprime numbers , sum of powers , divide

Determine all the triples $\{a, b, c \}$ of positive integers coprime (not necessarily pairwise prime) such that $a + b + c$ simultaneously divides the three numbers $a^{12} + b^{12}+ c^{12}$, $ a^{23} + b^{23} + c^{23} $ and $ a^{11004} + b^{11004} + c^{11004}$

2007 Postal Coaching, 6

Tags: digit , number theory , divide , divisible

Consider all the $7$-digit numbers formed by the digits $1,2 , 3,...,7$ each digit being used exactly once in all the $7! $ numbers. Prove that no two of them have the property that one divides the other.

2019 BMT Spring, 6

Tags: probability , combinatorics

At a party, $2019$ people decide to form teams of three. To do so, each turn, every person not on a team points to one other person at random. If three people point to each other (that is, $A$ points to $B$, $B$ points to $C$, and $C$ points to $A$), then they form a team. What is the probability that after $65, 536$ turns, exactly one person is not on a team

1990 IMO Longlists, 89

Tags: function , algebra , domain , combinatorics

Let $n$ be a positive integer. $S_1, S_2, \ldots, S_n$ are pairwise non-intersecting sets, and $S_k $ has exactly $k$ elements $(k = 1, 2, \ldots, n)$. Define $S = S_1\cup S_2\cup\cdots \cup S_n$. The function $f: S \to S $ maps all elements in $S_k$ to a fixed element of $S_k$, $k = 1, 2, \ldots, n$. Find the number of functions $g: S \to S$ satisfying $f(g(f(x))) = f(x).$

2009 Indonesia TST, 2

Tags: combinatorics

Prove that there exists two different permutations $ (a_1,a_2,\dots,a_{2009})$ and $ (b_1,b_2,\dots,b_{2009})$ of $ (1,2,\dots,2009)$ such that \[ \sum_{i\equal{}1}^{2009}i^i a_i \minus{} \sum_{i\equal{}1}^{2009} i^i b_i\] is divisible by $ 2009!$.

2008 ITest, 74

Tags: analytic geometry

Points $C$ and $D$ lie on opposite sides of line $\overline{AB}$. Let $M$ and $N$ be the centroids of $\triangle ABC$ and $\triangle ABD$ respectively. If $AB=841$, $BC=840$, $AC=41$, $AD=609$, and $BD=580$, find the sum of the numerator and denominator of the value of $MN$ when expressed as a fraction in lowest terms.

2005 IMO Shortlist, 1

Tags: coefficient , algebra , polynomial , root , quadratic

Find all pairs of integers $a,b$ for which there exists a polynomial $P(x) \in \mathbb{Z}[X]$ such that product $(x^2+ax+b)\cdot P(x)$ is a polynomial of a form \[ x^n+c_{n-1}x^{n-1}+\cdots+c_1x+c_0 \] where each of $c_0,c_1,\ldots,c_{n-1}$ is equal to $1$ or $-1$.

2017 China Northern MO, 1

Tags: sequence , inequalities

A sequence $\{a_n\}$ is defined as follows: $a_1 = 1$, $a_2 = \frac{1}{3}$, and for all $n \geq 1,$ $\frac{(1+a_n)(1+a_{n+2})}{(1+a_n+1)^2} = \frac{a_na_{n+2}}{a_{n+1}^2}$. Prove that, for all $n \geq 1$, $a_1 + a_2 + ... + a_n < \frac{34}{21}$.

2020 European Mathematical Cup, 4

Tags: algebra , functional equation , function

Let $\mathbb{R^+}$ denote the set of all positive real numbers. Find all functions $f: \mathbb{R^+}\rightarrow \mathbb{R^+}$ such that $$xf(x + y) + f(xf(y) + 1) = f(xf(x))$$ for all $x, y \in\mathbb{R^+}.$ [i]Proposed by Amadej Kristjan Kocbek, Jakob Jurij Snoj[/i]

2006 Germany Team Selection Test, 3

Tags: modular arithmetic , number theory

Is the following statement true? For each positive integer $n$, we can find eight nonnegative integers $a$, $b$, $c$, $d$, $e$, $f$, $g$, $h$ such that $n=\frac{2^a-2^b}{2^c-2^d}\cdot\frac{2^e-2^f}{2^g-2^h}$.

2016 NIMO Summer Contest, 7

Tags:

Suppose that $a$ and $b$ are real numbers such that $\sin(a)+\sin(b)=1$ and $\cos(a)+\cos(b)=\frac{3}{2}$. If the value of $\cos(a-b)$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, determine $100m+n$. [i]Proposed by Michael Ren[/i]

2022 District Olympiad, P1

Tags: algebra , logarithm

Determine all $x\in(0,3/4)$ which satisfy \[\log_x(1-x)+\log_2\frac{1-x}{x}=\frac{1}{(\log_2x)^2}.\]

2015 Geolympiad Spring, 2

Tags:

Let $ABC$ be a triangle and $w$ its incircle. $w$ touches $BC,CA$ at $A_1,B_1$ respectively. The second intersection of $AA_1$ and $w$ is $A_2$, similarly define $B_2$. Then $AB,A_1B_1,A_2B_2$ concur at a point $C_3$.

2019 Purple Comet Problems, 8

Tags: geometry

The diagram below shows a $12$ by $20$ rectangle split into four strips of equal widths all surrounding an isosceles triangle. Find the area of the shaded region. [img]https://cdn.artofproblemsolving.com/attachments/9/e/ed6be5110d923965c64887a2ca8e858c977700.png[/img]

2009 Poland - Second Round, 2

Tags: combinatorics

Find all integer numbers $n\ge 4$ which satisfy the following condition: from every $n$ different $3$-element subsets of $n$-element set it is possible to choose $2$ subsets, which have exactly one element in common.

1982 All Soviet Union Mathematical Olympiad, 327

Tags: geometry , circles , area

Given two points $M$ and $K$ on the circumference with radius $r_1$ and centre $O_1$. The circumference with radius $r_2$ and centre $O_2$ is inscribed in $\angle MO_1K$ . Find the area of quadrangle $MO_1KO_2$ .

2010 Purple Comet Problems, 15

Tags: number theory , relatively prime

Find the smallest possible sum $a + b + c + d + e$ where $a, b, c, d,$ and $e$ are positive integers satisfying the conditions $\star$ each of the pairs of integers $(a, b), (b, c), (c, d),$ and $(d, e)$ are [b]not[/b] relatively prime $\star$ all other pairs of the five integers [b]are[/b] relatively prime.

1955 Miklós Schweitzer, 8

Tags: 3d geometry , tetrahedron , geometry

[b]8.[/b] Show that on any tetrahedron there can be found three acute bihedral angles such that the faces including these angles count among them all faces of tetrahedron. [b](G. 10)[/b]

2012 Albania National Olympiad, 3

Tags: invariant , arithmetic sequence , algebra

Let $S_i$ be the sum of the first $i$ terms of the arithmetic sequence $a_1,a_2,a_3\ldots $. Show that the value of the expression \[\frac{S_i}{i}(j-k) + \frac{S_j}{j}(k-i) +\ \frac{S_k}{k}(i-j)\] does not depend on the numbers $i,j,k$ nor on the choice of the arithmetic sequence $a_1,a_2,a_3,\ldots$.

2009 ITAMO, 2

Tags: circumcircle , projective geometry , geometry

Let $ABC$ be an acute-angled scalene triangle and $\Gamma$ be its circumcircle. $K$ is the foot of the internal bisector of $\angle BAC$ on $BC$. Let $M$ be the midpoint of the arc $BC$ containing $A$. $MK$ intersect $\Gamma$ again at $A'$. $T$ is the intersection of the tangents at $A$ and $A'$. $R$ is the intersection of the perpendicular to $AK$ at $A$ and perpendicular to $A'K$ at $A'$. Show that $T, R$ and $K$ are collinear.

MOAA Gunga Bowls, 2021.19

Tags:

Let $S$ be the set of triples $(a,b,c)$ of non-negative integers with $a+b+c$ even. The value of the sum \[\sum_{(a,b,c)\in S}\frac{1}{2^a3^b5^c}\] can be expressed as $\frac{m}{n}$ for relative prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Nathan Xiong[/i]