Olympiad problems

2019 IMO Shortlist, N4

Find all functions $f:\mathbb Z_{>0}\to \mathbb Z_{>0}$ such that $a+f(b)$ divides $a^2+bf(a)$ for all positive integers $a$ and $b$ with $a+b>2019$.

2021 CHKMO, 2

Tags: number theory , prime factorization , prime

For each positive integer $n$ larger than $1$ with prime factorization $p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}$, its [i]signature[/i] is defined as the sum $\alpha_1+\alpha_2+\cdots+\alpha_k$. Does there exist $2020$ consecutive positive integers such that among them, there are exactly $1812$ integers whose signatures are strictly smaller than $11$?

1997 AMC 12/AHSME, 25

Tags: geometry , parallelogram

Let $ ABCD$ be a parallelogram and let $ \overrightarrow{AA^\prime}$, $ \overrightarrow{BB^\prime}$, $ \overrightarrow{CC^\prime}$, and $ \overrightarrow{DD^\prime}$ be parallel rays in space on the same side of the plane determined by $ ABCD$. If $ AA^\prime \equal{} 10$, $ BB^\prime \equal{} 8$, $ CC^\prime \equal{} 18$, $ DD^\prime \equal{} 22$, and $ M$ and $ N$ are the midpoints of $ \overline{A^{\prime}C^{\prime}}$ and $ \overline{B^{\prime}D^{\prime}}$, respectively, then $ MN \equal{}$ $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ 3\qquad \textbf{(E)}\ 4$

Kvant 2020, M1387

Tags: combinatorics , polyhedron

An ant crawls clockwise along the contour of each face of a convex polyhedron. It is known that their speeds at any given time are not less than 1 mm/h. Prove that sooner or later two ants will collide. [i]Proposed by A. Klyachko[/i]

2012 Korea Junior Math Olympiad, 2

Tags: geometry , pentagon , concyclic

A pentagon $ABCDE$ is inscribed in a circle $O$, and satisfies $\angle A = 90^o, AB = CD$. Let $F$ be a point on segment $AE$. Let $BF$ hit $O$ again at $J(\ne B)$, $CE \cap DJ = K$, $BD\cap FK = L$. Prove that $B,L,E,F$ are cyclic.

2020 AMC 12/AHSME, 13

Tags: logarithm

Which of the following is the value of $\sqrt{\log_2{6}+\log_3{6}}?$ $\textbf{(A) } 1 \qquad\textbf{(B) } \sqrt{\log_5{6}} \qquad\textbf{(C) } 2 \qquad\textbf{(D) } \sqrt{\log_2{3}}+\sqrt{\log_3{2}} \qquad\textbf{(E) } \sqrt{\log_2{6}}+\sqrt{\log_3{6}}$

2013 Dutch Mathematical Olympiad, 1

Tags: maximum , combinatorics , table , coloring

In a table consisting of $n$ by $n$ small squares some squares are coloured black and the other squares are coloured white. For each pair of columns and each pair of rows the four squares on the intersections of these rows and columns must not all be of the same colour. What is the largest possible value of $n$?

2022 Taiwan TST Round 2, C

Tags: combinatorics , lattice points

There are $2022$ distinct integer points on the plane. Let $I$ be the number of pairs among these points with exactly $1$ unit apart. Find the maximum possible value of $I$. ([i]Note. An integer point is a point with integer coordinates.[/i]) [i]Proposed by CSJL.[/i]

Gheorghe Țițeica 2024, P3

Tags: algebra

Let $a,b,c,d\in\mathbb{R}$ such that for all $x\in(-1,1)$ we have $$(x^2+ax+b)\cdot\lfloor x^2+cx+d\rfloor = \lfloor x^2+ax+b\rfloor \cdot (x^2 + cx + d).$$ Prove that $a=c$ and $b=d$. [i]Cristi Săvescu[/i]

2024 Euler Olympiad, Round 2, 5

Tags: euler , geometry

Consider a circle with an arc $AB$ and a point $C$ on this arc. Let $D$ be the midpoint of arc $BC$ and $M$ the midpoint of chord $AD$. Suppose the tangent lines to the circle at point $D$ intersect the ray $AC$ at point $K$. Prove that the areas of triangle $MBD$ and quadrilateral $MCKD$ are equal if and only if the measure of arc $AB$ is $180^\circ$. [i]Proposed by Irakli Shalibashvili, Georgia [/i]

2008 Bulgarian Autumn Math Competition, Problem 12.2

Tags: geometry , incenter , excircle , collinear points

Let $ABC$ be a triangle, such that the midpoint of $AB$, the incenter and the touchpoint of the excircle opposite $A$ with $\overline{AC}$ are collinear. Find $AB$ and $BC$ if $AC=3$ and $\angle ABC=60^{\circ}$.

2007 Purple Comet Problems, 8

Tags: probability , number theory , relatively prime

You know that the Jones family has five children, and the Smith family has three children. Of the eight children you know that there are five girls and three boys. Let $\dfrac{m}{n}$ be the probability that at least one of the families has only girls for children. Given that $m$ and $n$ are relatively prime positive integers, find $m+ n$.

2024 CMIMC Algebra and Number Theory, 5

Tags: algebra

Let \[f(x)=(x+1)^{6}+(x-1)^{5}+(x+1)^{4}+(x-1)^3+(x+1)^2+(x-1)^1+1.\] Find the remainder when $\sum_{j=-126}^{126}jf(j)$ is divided by 1000. [i]Proposed by Hari Desikan[/i]

2017 Romania EGMO TST, P3

Tags: algebra , function , functional equation

Determine all functions $f:\mathbb R\to\mathbb R$ such that \[f(xy-1)+f(x)f(y)=2xy-1,\]for any real numbers $x{}$ and $y{}.$

2023 Bangladesh Mathematical Olympiad, P1

Tags: factorial , number theory , diophantine equation

Find all possible non-negative integer solution ($x,$ $y$) of the following equation- $$x!+2^y=z!$$ Note: $x!=x\cdot(x-1)!$ and $0!=1$. For example, $5!=5\times4\times3\times2\times1=120$.

Gheorghe Țițeica 2025, P3

Tags: perimeter , geometry

Out of all the nondegenerate triangles with positive integer sides and perimeter $100$, find the one with the smallest area.

2014 Stars Of Mathematics, 2

Tags: number theory

Let $N$ be an arbitrary positive integer. Prove that if, from among any $n$ consecutive integers larger than $N$, one may select $7$ of them, pairwise co-prime, then $n\geq 22$. ([i]Dan Schwarz[/i])

2017-IMOC, A3

Tags: algebra , system of equations

Solve the following system of equations: $$\begin{cases} x^3+y+z=1\\ x+y^3+z=1\\ x+y+z^3=1\end{cases}$$

2021 Israel TST, 3

Tags: geometry , incircle , orthocenter , common tangent

Let $ABC$ be an acute triangle with orthocenter $H$. Prove that there is a line $l$ which is parallel to $BC$ and tangent to the incircles of $ABH$ and $ACH$.

1999 All-Russian Olympiad, 2

Tags: induction , number theory

Find all bounded sequences $(a_n)_{n=1}^\infty$ of natural numbers such that for all $n \ge 3$, \[ a_n = \frac{a_{n-1} + a_{n-2}}{\gcd(a_{n-1}, a_{n-2})}. \]

2000 Korea Junior Math Olympiad, 3

Tags: geometry

Acute triangle $ABC$ is inscribed in circle $O$. $P$ is the foot of altitude from $A$ to $BC$, and $D$ is the intersection of $O$ and line $AP$. $M, N$ are midpoint of $AB, AC$ respectively. $MP$ and $CD$ intersects at $Q$, and $NP$ and $BD$ intersects at $R$. Show that $AD, BQ, CR$ meet at one point if and only if $AB=AC$.

1998 National Olympiad First Round, 3

Tags:

How many ways are there to divide a set with 6 elements into 3 disjoint subsets? $\textbf{(A)}\ 90 \qquad\textbf{(B)}\ 105 \qquad\textbf{(C)}\ 120 \qquad\textbf{(D)}\ 180 \qquad\textbf{(E)}\ 243$

1989 IMO Shortlist, 24

Tags: geometry , 3d geometry , sphere , combinatorics , maximization , geometric inequality

For points $ A_1, \ldots ,A_5$ on the sphere of radius 1, what is the maximum value that $ min_{1 \leq i,j \leq 5} A_iA_j$ can take? Determine all configurations for which this maximum is attained. (Or: determine the diameter of any set $ \{A_1, \ldots ,A_5\}$ for which this maximum is attained.)

2013 Moldova Team Selection Test, 3

Tags: geometry , parallelogram , reflection , moving points , ptolemy sinus lemma

Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ meet at $E$. The extensions of the sides $AD$ and $BC$ beyond $A$ and $B$ meet at $F$. Let $G$ be the point such that $ECGD$ is a parallelogram, and let $H$ be the image of $E$ under reflection in $AD$. Prove that $D,H,F,G$ are concyclic.

2021 Girls in Math at Yale, 9

Tags: college

Ali defines a [i]pronunciation[/i] of any sequence of English letters to be a partition of those letters into substrings such that each substring contains at least one vowel. For example, $\text{A } \vert \text{ THEN } \vert \text{ A}$, $\text{ATH } \vert \text{ E } \vert \text{ NA}$, $\text{ATHENA}$, and $\text{AT } \vert \text{ HEN } \vert \text{ A}$ are all pronunciations of the sequence $\text{ATHENA}$. How many distinct pronunciations does $\text{YALEMATHCOMP}$ have? (Y is not a vowel.) [i]Proposed by Andrew Wu, with significant inspiration from ali cy[/i]