Olympiad problems

2009 Jozsef Wildt International Math Competition, W. 6

Prove that$$p (n)= 2+ \left (p (1) + \cdots + p\left ( \left [\frac {n}{2} \right ] + \chi_1 (n)\right ) + \left (p'_2(n) + \cdots + p' _{ \left [\frac {n}{2} \right ] - 1}(n)\right )\right )$$for every $n \in \mathbb {N}$ with $n>2$ where $\chi $ denotes the principal character Dirichlet modulo 2, i.e.$$ \chi _1 (n) = \begin{cases} 1 & \text{if } (n,2)=1 \\ 0 &\text{if } (n,2)>1 \end{cases} $$with $p (n) $ we denote number of possible partitions of $n $ and $p' _m(n) $ we denote the number of partitions of $n$ in exactly $m$ sumands.

2021 Ecuador NMO (OMEC), 4

Tags: combinatorics

In a board $8$x$8$, the unit squares have numbers $1-64$, as shown. The unit square with a multiple of $3$ on it are red. Find the minimum number of chess' bishops that we need to put on the board such that any red unit square either has a bishop on it or is attacked by at least one bishop. Note: A bishops moves diagonally. [img]https://i.imgur.com/03baBwp.jpeg[/img]

2010 Regional Competition For Advanced Students, 1

Tags: trigonometry , inequalities

Let $0 \le a$, $b \le 1$ be real numbers. Prove the following inequality: \[\sqrt{a^3b^3}+ \sqrt{(1-a^2)(1-ab)(1-b^2)} \le 1.\] [i](41th Austrian Mathematical Olympiad, regional competition, problem 1)[/i]

2003 National High School Mathematics League, 3

Tags: graph theory

A space figure is consisted of $n$ vertexes and $l$ lines connecting these vertices, where $n=q^2+q+1, l\geq\frac{1}{2}q(q+1)^2+1,q\geq2,q\in\mathbb{Z}_+$. The figure satisfies: every four vertices are not coplane, every vertex is connected by at least one line, and there is a vertex connected by at least $q+2$ lines. Prove that there exists a space quadrilateral in the figure. Note: a space quadrilateral is figure with four vertices $A, B, C, D$ and four lines $ AB, BC, CD, DA$.

2025 Bulgarian Spring Mathematical Competition, 9.4

Tags: functional equation , number theory , divisibility , sophie germain identity , function , prime

Determine all functions $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ such that $f(a) + 2ab + 2f(b)$ divides $f(a)^2 + 4f(b)^2$ for any positive integers $a$ and $b$.

2001 All-Russian Olympiad Regional Round, 8.5

Tags: algebra

Let $a, b, c, d, e$ and $f$ be some numbers, and $ a \cdot c \cdot e \ne 0$.It is known that the values of the expressions $|ax+b|+|cx+d| $and $|ex+f|$ equal at all values of $x$. Prove that $ad = bc$.

2008 Alexandru Myller, 2

Tags: binomial coefficient , complex number , number theory , trigonometry , binomial theorem

There are no integers $ a,b,c $ that satisfy $ \left( a+b\sqrt{-3}\right)^{17}=c+\sqrt{-3} . $ [i]Dorin Andrica, Mihai Piticari[/i]

2024 Francophone Mathematical Olympiad, 4

Tags: prime number , number theory , divisor

Let $p$ be a fixed prime number. Find all integers $n \ge 1$ with the following property: One can partition the positive divisors of $n$ in pairs $(d,d')$ satisfying $d<d'$ and $p \mid \left\lfloor \frac{d'}{d}\right\rfloor$.

2019 PUMaC Combinatorics B, 5

Tags: combinatorics , expected value

Marko lives on the origin of the Cartesian plane. Every second, Marko moves $1$ unit up with probability $\tfrac{2}{9}$, $1$ unit right with probability $\tfrac{2}{9}$, $1$ unit up and $1$ unit right with probability $\tfrac{4}{9}$, and he doesn’t move with probability $\tfrac{1}{9}$. After $2019$ seconds, Marko ends up on the point $(A, B)$. What is the expected value of $A\cdot B$?

2002 Junior Balkan Team Selection Tests - Moldova, 3

Tags: altitude , geometry , perpendicular , orthocenter

Let $ABC$ be a an acute triangle. Points $A_1, B_1$ and $C_1$ are respectively the projections of the vertices $A, B$ and $C$ on the opposite sides of the triangle, the point $H$ is the orthocenter of the triangle, and the point $P$ is the middle of the segment $[AH]$. The lines $BH$ and $A_1C_1$, $P B_1$ and $AB$ intersect respectively at the points $M$ and $N$. Prove that the lines $MN$ and $BC$ are perpendicular.

2001 Saint Petersburg Mathematical Olympiad, 9.4

Tags: gcd , number theory , divisibility , greatest common divisor

Let $a,b,c\in\mathbb{Z^{+}}$ such that $$(a^2-1, b^2-1, c^2-1)=1$$ Prove that $$(ab+c, bc+a, ca+b)=(a,b,c)$$ (As usual, $(x,y,z)$ means the greatest common divisor of numbers $x,y,z$) [I]Proposed by A. Golovanov[/i]

2000 Harvard-MIT Mathematics Tournament, 3

Tags:

Five students take a test on which any integer score from $0$ to $100$ inclusive is possible. What is the largest possible difference between the median and the mean of the scores?

1998 Romania National Olympiad, 1

Tags: integer , algebra , combinatorics

Let $n \ge 2$ be an integer and $M= \{1,2,\ldots,n\}.$ For each $k \in \{1,2,\ldots,n-1\}$ we define $$x_k= \frac{1}{n+1} \sum_{\substack{A \subset M \\ |A|=k}} (\min A + \max A).$$ Prove that the numbers $x_k$ are integers and not all of them are divisible by $4.$ [hide=Notations]$|A|$ is the cardinal of $A$ $\min A$ is the smallest element in $A$ $\max A$ is the largest element in $A$[/hide]

1990 AMC 8, 17

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A straight concrete sidewalk is to be $3$ feet wide, $60$ feet long, and $3$ inches thick. How many cubic yards of concrete must a contractor order for the sidewalk if concrete must be ordered in a whole number of cubic yards? $ \text{(A)}\ 2\qquad\text{(B)}\ 5\qquad\text{(C)}\ 12\qquad\text{(D)}\ 20\qquad\text{(E)}\ \text{more than 20} $

2018 Korea - Final Round, 4

Tags: perpendicular bisector , geometry

Triangle $ABC$ satisfies $\angle C=90^{\circ}$. A circle passing $A,B$ meets segment $AC$ at $G(\neq A,C)$ and it meets segment $BC$ at point $D(\neq B)$. Segment $AD$ cuts segment $BG$ at $H$, and let $l$, the perpendicular bisector of segment $AD$, cuts the perpendicular bisector of segment $AB$ at point $E$. A line passing $D$ is perpendicular to $DE$ and cuts $l$ at point $F$. If the circumcircle of triangle $CFH$ cuts $AC$, $BC$ at $P(\neq C),Q(\neq C)$ respectively, then prove that $PQ$ is perpendicular to $FH$.

1992 IMO, 3

Tags: combinatorics , ramsey theory , graph theory , coloring , extremal combinatorics

Consider $9$ points in space, no four of which are coplanar. Each pair of points is joined by an edge (that is, a line segment) and each edge is either colored blue or red or left uncolored. Find the smallest value of $\,n\,$ such that whenever exactly $\,n\,$ edges are colored, the set of colored edges necessarily contains a triangle all of whose edges have the same color.

2019 USA EGMO Team Selection Test, 3

Tags: number theory

Let $n$ be a positive integer such that the number \[\frac{1^k + 2^k + \dots + n^k}{n}\] is an integer for any $k \in \{1, 2, \dots, 99\}$. Prove that $n$ has no divisors between 2 and 100, inclusive.

2019 Iran Team Selection Test, 2

Tags:

$a, a_1,a_2,\dots ,a_n$ are natural numbers. We know that for any natural number $k$ which $ak+1$ is square, at least one of $a_1k+1,\dots ,a_n k+1$ is also square. Prove $a$ is one of $a_1,\dots ,a_n$ [i]Proposed by Mohsen Jamali[/i]

2022 CMIMC, 2.5

Tags: combinatorics

Daniel, Ethan, and Zack are playing a multi-round game of Tetris. Whoever wins $11$ rounds first is crowned the champion. However Zack is trying to pull off a "reverse-sweep", where (at-least) one of the other two players first hits $10$ wins while Zack is still at $0$, but Zack still ends up being the first to reach $11$. How many possible sequences of round wins can lead to Zack pulling off a reverse sweep? [i]Proposed by Dilhan Salgado[/i]

1991 AMC 12/AHSME, 24

Tags: geometric transformation , rotation , logarithm , geometry

The graph, $G$ of $y = \log_{10}x$ is rotated $90^{\circ}$ counter-clockwise about the origin to obtain a new graph $G'$. Which of the following is an equation for $G'$? $ \textbf{(A)}\ y = \log_{10}\left(\frac{x + 90}{9}\right)\qquad\textbf{(B)}\ y = \log_{x}10\qquad\textbf{(C)}\ y = \frac{1}{x + 1}\qquad\textbf{(D)}\ y = 10^{-x}\qquad\textbf{(E)}\ y = 10^{x} $

1974 AMC 12/AHSME, 20

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Let \[ T \equal{} \frac1{3\minus{}\sqrt8} \minus{} \frac1{\sqrt8 \minus{} \sqrt7} \plus{} \frac1{\sqrt7\minus{}\sqrt6} \minus{} \frac1{\sqrt6\minus{}\sqrt5} \plus{} \frac1{\sqrt5\minus{}2}\] then $ \textbf{(A)}\ T<1 \qquad \textbf{(B)}\ T\equal{}1 \qquad \textbf{(C)}\ 1<T<2 \qquad \textbf{(D)}\ T>2 \qquad$ $ \textbf{(E)}\ T \equal{} \frac1{(3\minus{}\sqrt8)(\sqrt8\minus{}\sqrt7)(\sqrt7\minus{}\sqrt6)(\sqrt6\minus{}\sqrt5)(\sqrt5\minus{}2)}$

1957 Moscow Mathematical Olympiad, 352

Tags: geometry , parallelogram , minimum , diagonal , area

Of all parallelograms of a given area find the one with the shortest possible longer diagonal.

2004 Cono Sur Olympiad, 6

Tags: combinatorial geometry , combinatorics

Let $m$, $n$ be positive integers. On an $m\times{n}$ checkerboard, divided into $1\times1$ squares, we consider all paths that go from upper right vertex to the lower left vertex, travelling exclusively on the grid lines by going down or to the left. We define the area of a path as the number of squares on the checkerboard that are below this path. Let $p$ be a prime such that $r_{p}(m)+r_{p}(n)\geq{p}$, where $r_{p}(m)$ denotes the remainder when $m$ is divided by $p$ and $r_{p}(n)$ denotes the remainder when $n$ is divided by $p$. How many paths have an area that is a multiple of $p$?

2009 Mediterranean Mathematics Olympiad, 4

Tags: inequalities

Let $x,y,z$ be positive real numbers. Prove that \[ \sum_{cyclic} \frac{xy}{xy+x^2+y^2} ~\le~ \sum_{cyclic} \frac{x}{2x+z} \] [i](Proposed by Šefket Arslanagić, Bosnia and Herzegovina)[/i]

2022 Tuymaada Olympiad, 5

Tags: permutation , combinatorics

Each row of a $24 \times 8$ table contains some permutation of the numbers $1, 2, \cdots , 8.$ In each column the numbers are multiplied. What is the minimum possible sum of all the products? [i](C. Wu)[/i]