Olympiad problems

1998 Mexico National Olympiad, 4

Find all integers that can be written in the form $\frac{1}{a_1}+\frac{2}{a_2}+...+\frac{9}{a_9}$ where $a_1,a_2, ...,a_9$ are nonzero digits, not necessarily different.

2011 QEDMO 10th, 4

Tags: combinatorics

In year $2525$ the QED has $3n + 1$ members, of which $n$ are identical robots and $2n + 1$ (uncloned and therefore distinguishable) people. For the $263^{th}$ board election in Wurzburg there will be exactly $n$ members. Find out how many distinguishable compositions are conceivable for this.

2024 AMC 10, 10

Tags: 2024 AMC 10b , AMC , 2024 AMC , AMC 10 , geometry , parallelogram

Quadrilateral $ABCD$ is a parallelogram, and $E$ is the midpoint of the side $\overline{AD}$. Let $F$ be the intersection of lines $EB$ and $AC$. What is the ratio of the area of quadrilateral $CDEF$ to the area of triangle $CFB$? $\textbf{(A) } 5 : 4 \qquad \textbf{(B) } 4 : 3 \qquad \textbf{(C) } 3 : 2 \qquad \textbf{(D) } 5 : 3 \qquad \textbf{(E) } 2 : 1$

2021 Adygea Teachers' Geometry Olympiad, 3

Tags: geometry , excircle

In a triangle, one excircle touches side $AB$ at point $C_1$ and the other touches side $BC$ at point $A_1$. Prove that on the straight line $A_1C_1$ the constructed excircles cut out equal segments.

1972 AMC 12/AHSME, 4

Tags: AMC

The number of solutions to $\{1,~2\}\subseteq~X~\subseteq~\{1,~2,~3,~4,~5\}$, where $X$ is a subset of $\{1,~2,~3,~4,~5\}$ is $\textbf{(A) }2\qquad\textbf{(B) }4\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad \textbf{(E) }\text{None of these}$

1991 Brazil National Olympiad, 3

Tags: algebra , polynomial , function , rational function , algebra proposed

Given $k > 0$, the sequence $a_n$ is defined by its first two members and \[ a_{n+2} = a_{n+1} + \frac{k}{n}a_n \] a)For which $k$ can we write $a_n$ as a polynomial in $n$? b) For which $k$ can we write $\frac{a_{n+1}}{a_n} = \frac{p(n)}{q(n)}$? ($p,q$ are polynomials in $\mathbb R[X]$).

1999 Dutch Mathematical Olympiad, 5

Tags: algorithm , greatest common divisor , number theory , Euclidean algorithm

Let $c$ be a nonnegative integer, and define $a_n = n^2 + c$ (for $n \geq 1)$. Define $d_n$ as the greatest common divisor of $a_n$ and $a_{n + 1}$. (a) Suppose that $c = 0$. Show that $d_n = 1,\ \forall n \geq 1$. (b) Suppose that $c = 1$. Show that $d_n \in \{1,5\},\ \forall n \geq 1$. (c) Show that $d_n \leq 4c + 1,\ \forall n \geq 1$.

1972 Miklós Schweitzer, 4

Tags: abstract algebra , group theory , search , superior algebra , superior algebra unsolved

Let $ G$ be a solvable torsion group in which every Abelian subgroup is finitely generated. Prove that $ G$ is finite. [i]J. Pelikan[/i]

2022 Francophone Mathematical Olympiad, 3

Tags: geometry , circles , parallel

Let $ABC$ be a triangle and $\Gamma$ its circumcircle. Denote $\Delta$ the tangent at $A$ to the circle $\Gamma$. $\Gamma_1$ is a circle tangent to the lines $\Delta$, $(AB)$ and $(BC)$, and $E$ its touchpoint with the line $(AB)$. Let $\Gamma_2$ be a circle tangent to the lines $\Delta$, $(AC)$ and $(BC)$, and $F$ its touchpoint with the line $(AC)$. We suppose that $E$ and $F$ belong respectively to the segments $[AB]$ and $[AC]$, and that the two circles $\Gamma_1$ and $\Gamma_2$ lie outside triangle $ABC$. Show that the lines $(BC)$ and $(EF)$ are parallel.

2015 Cono Sur Olympiad, 6

Tags: combinatorics , Sets , arithmetic

Let $S = \{1, 2, 3, \ldots , 2046, 2047, 2048\}$. Two subsets $A$ and $B$ of $S$ are said to be [i]friends[/i] if the following conditions are true: [list] [*] They do not share any elements. [*] They both have the same number of elements. [*] The product of all elements from $A$ equals the product of all elements from $B$. [/list] Prove that there are two subsets of $S$ that are [i]friends[/i] such that each one of them contains at least $738$ elements.

2005 Germany Team Selection Test, 3

Tags: geometry , modular arithmetic , combinatorics , circles , Intersection , IMO Shortlist , double counting

Let ${n}$ and $k$ be positive integers. There are given ${n}$ circles in the plane. Every two of them intersect at two distinct points, and all points of intersection they determine are pairwise distinct (i. e. no three circles have a common point). No three circles have a point in common. Each intersection point must be colored with one of $n$ distinct colors so that each color is used at least once and exactly $k$ distinct colors occur on each circle. Find all values of $n\geq 2$ and $k$ for which such a coloring is possible. [i]Proposed by Horst Sewerin, Germany[/i]

1966 Kurschak Competition, 2

Tags: number theory , Digits

Show that the $n$ digits after the decimal point in $(5 +\sqrt{26})^n$ are all equal.

2016 IMO Shortlist, C2

Tags: combinatorics , IMO Shortlist , AZE IMO TST , TST

Find all positive integers $n$ for which all positive divisors of $n$ can be put into the cells of a rectangular table under the following constraints: [list] [*]each cell contains a distinct divisor; [*]the sums of all rows are equal; and [*]the sums of all columns are equal. [/list]

2018 Israel National Olympiad, 6

Tags: circles , tangent circles , inradius , geometry

In the corners of triangle $ABC$ there are three circles with the same radius. Each of them is tangent to two of the triangle's sides. The vertices of triangle $MNK$ lie on different sides of triangle $ABC$, and each edge of $MNK$ is also tangent to one of the three circles. Likewise, the vertices of triangle $PQR$ lie on different sides of triangle $ABC$, and each edge of $PQR$ is also tangent to one of the three circles (see picture below). Prove that triangles $MNK,PQR$ have the same inradius. [img]https://i.imgur.com/bYuBabS.png[/img]

2000 Korea Junior Math Olympiad, 8

Tags: counting , derangement , combinatorics , KJMO

$n$ men and one woman are in the meeting room with $n+1$ chairs, each of them having their own seat. Show that the following two number of cases are equal. (1) Number of cases to choose one man to get out of the room, and make the left $n-1$ men to sit to each other's chair. (2) Number of cases to make $n+1$ people to sit to each other's chair.

2016 Romania Team Selection Tests, 1

Tags: algebra , number theory

Given a positive integer $n$, determine all functions $f$ from the first $n$ positive integers to the positive integers, satisfying the following two conditions: [b](1)[/b] $\sum_{k=1}^{n}{f(k)}=2n$; and [b](2)[/b] $\sum_{k\in K}{f(k)}=n$ for no subset $K$ of the first $n$ positive integers.

2019 CCA Math Bonanza, I14

Tags:

Call an odd prime $p$ [i]adjective[/i] if there exists an infinite sequence $a_0,a_1,a_2,\ldots$ of positive integers such that \[a_0\equiv1+\frac{1}{a_1}\equiv1+\frac{1}{1+\frac{1}{a_2}}\equiv1+\frac{1}{1+\frac{1}{1+\frac{1}{a_3}}}\equiv\ldots\pmod p.\] What is the sum of the first three odd primes that are [i]not[/i] adjective? Note: For two common fractions $\frac{a}{b}$ and $\frac{c}{d}$, we say that $\frac{a}{b}\equiv\frac{c}{d}\pmod p$ if $p$ divides $ad-bc$ and $p$ does not divide $bd$. [i]2019 CCA Math Bonanza Individual Round #14[/i]

2011-2012 SDML (High School), 6

Tags: 2011-12 SDML Round 2 , SDML High School

A positive integer is equal to the sum of the squares of its four smallest positive divisors. What is the largest prime that divides this positive integer?

2023 Francophone Mathematical Olympiad, 3

Tags: geometry , circumcircle , tangent circles

Let $\Gamma$ and $\Gamma'$ be two circles with centres $O$ and $O'$, such that $O$ belongs to $\Gamma'$. Let $M$ be a point on $\Gamma'$, outside of $\Gamma$. The tangents to $\Gamma$ that go through $M$ touch $\Gamma$ in two points $A$ and $B$, and cross $\Gamma'$ again in two points $C$ and $D$. Finally, let $E$ be the crossing point of the lines $AB$ and $CD$. Prove that the circumcircles of the triangles $CEO'$ and $DEO'$ are tangent to $\Gamma'$.

2015 German National Olympiad, 5

Tags: geometry , geometry proposed , tangent circles , parallel , Germany

Let $ABCD$ be a convex quadrilateral such that the circle with diameter $AB$ touches the line $CD$. Prove that that the circle with diameter $CD$ touches the line $AB$ if and only if $BC$ and $AD$ are parallel.

2008 Harvard-MIT Mathematics Tournament, 8

Tags: trigonometry

Compute $ \arctan\left(\tan65^\circ \minus{} 2\tan40^\circ\right)$. (Express your answer in degrees.)

2018 Taiwan TST Round 3, 2

Tags: IMO Shortlist , function , algebra

Let $S$ be a finite set, and let $\mathcal{A}$ be the set of all functions from $S$ to $S$. Let $f$ be an element of $\mathcal{A}$, and let $T=f(S)$ be the image of $S$ under $f$. Suppose that $f\circ g\circ f\ne g\circ f\circ g$ for every $g$ in $\mathcal{A}$ with $g\ne f$. Show that $f(T)=T$.

2021 Korea Winter Program Practice Test, 4

Tags: combinatorics , geometry , combinatorial geometry

A positive integer $m(\ge 2$) is given. From circle $C_1$ with a radius 1, construct $C_2, C_3, C_4, ... $ through following acts: In the $i$th act, select a circle $P_i$ inside $C_i$ with a area $\frac{1}{m}$ of $C_i$. If such circle dosen't exist, the act ends. If not, let $C_{i+1}$ a difference of sets $C_i -P_i$. Prove that this act ends within a finite number of times.

2003 Junior Macedonian Mathematical Olympiad, Problem 5

Tags: combinatorics

Is it possible to cover a $2003 \times 2003$ chessboard (without overlap) using only horizontal $1 \times 2$ dominoes and only vertical $3 \times 1$ trominoes?

2014 Bulgaria National Olympiad, 2

Tags: function , induction , limit , geometric series , algebra solved , algebra

Find all functions $f: \mathbb{Q}^+ \to \mathbb{R}^+ $ with the property: \[f(xy)=f(x+y)(f(x)+f(y)) \,,\, \forall x,y \in \mathbb{Q}^+\] [i]Proposed by Nikolay Nikolov[/i]