Olympiad problems

2001 Bundeswettbewerb Mathematik, 4

Tags: domain , function , algebra , perimeter , geometry , rotation

A square $ R$ of sidelength $ 250$ lies inside a square $ Q$ of sidelength $ 500$. Prove that: One can always find two points $ A$ and $ B$ on the perimeter of $ Q$ such that the segment $ AB$ has no common point with the square $ R$, and the length of this segment $ AB$ is greater than $ 521$.

KoMaL A Problems 2023/2024, A. 872

Tags: sequence , algebra , combinatorics

For every positive integer $k$ let $a_{k,1},a_{k,2},\ldots$ be a sequence of positive integers. For every positive integer $k$ let sequence $\{a_{k+1,i}\}$ be the difference sequence of $\{a_{k,i}\}$, i.e. for all positive integers $k$ and $i$ the following holds: $a_{k,i+1}-a_{k,i}=a_{k+1,i}$. Is it possible that every positive integer appears exactly once among numbers $a_{k,i}$? [i]Proposed by Dávid Matolcsi, Berkeley[/i]

ICMC 4, 5

Tags: powers of 2 , elementary-number-theory , number theory

Find all composite positive integers $m$ such that, whenever the product of two positive integers $a$ and $b$ is $m$, their sum is a power of $2$. [i]Proposed by Harun Khan[/i]

2024 India Regional Mathematical Olympiad, 4

Tags: real , algebra , inequalities

Let $a_1,a_2,a_3,a_4$ be real numbers such that $a_1^2 + a_2^2 + a_3^2 + a_4^2 = 1$. Show that there exist $i,j$ with $ 1 \leq i < j \leq 4$, such that $(a_i - a_j)^2 \leq \frac{1}{5}$.

2006 Hungary-Israel Binational, 3

Tags: geometry

Let $ \mathcal{H} \equal{} A_1A_2\ldots A_n$ be a convex $ n$-gon. For $ i \equal{} 1, 2, \ldots, n$, let $ A'_{i}$ be the point symmetric to $ A_i$ with respect to the midpoint of $ A_{i \minus{} 1}A_{i \plus{} 1}$ (where $ A_{n \plus{} 1} \equal{} A_1$). We say that the vertex $ A_i$ is [i]good[/i] if $ A'_{i}$ lies inside $ \mathcal{H}$. Show that at least $ n \minus{} 3$ vertices of $ \mathcal{H}$ are [i]good[/i].

2007 AMC 10, 6

Tags:

At Euclid High School, the number of students taking the AMC10 was 60 in 2002, 66 in 2003, 70 in 2004, 76 in 2005, 78 in 2006, and is 85 in 2007. Between what two consecutive years was there the largest percentage increase? $ \textbf{(A)}\ 2002\ \text{and}\ 2003 \qquad \textbf{(B)}\ 2003\ \text{and}\ 2004 \qquad \textbf{(C)}\ 2004\ \text{and}\ 2005 \qquad \textbf{(D)}\ 2005\ \text{and}\ 2006 \qquad \textbf{(E)}\ 2006\ \text{and}\ 2007$

2009 Brazil Team Selection Test, 1

Tags: right triangle , inradius , geometry

Let $r$ be a positive real number. Prove that the number of right triangles with prime positive integer sides that have an inradius equal to $r$ are zero or a power of $2$. [hide=original wording]Seja r um numero real positivo. Prove que o numero de triangulos retangulos com lados inteiros positivos primos entre si que possuem inraio igual a r e zero ou uma potencia de 2.[/hide]

1973 Chisinau City MO, 63

Tags: point , coloring , combinatorics

Each point in space is colored in one of four different colors. Prove that there is a segment $1$ cm long with endpoints of the same color.

1986 Austrian-Polish Competition, 3

Tags: coloring , combinatorics , square

Each point in space is colored either blue or red. Show that there exists a unit square having exactly $0, 1$ or $4$ blue vertices.

2010 Iran MO (3rd Round), 2

Tags: combinatorics

suppose that $\mathcal F\subseteq \bigcup_{j=k+1}^{n}X^{(j)}$ and $|X|=n$. we know that $\mathcal F$ is a sperner family and it's also $H_k$. prove that: $\sum_{B\in \mathcal F}\frac{1}{\dbinom{n-1}{|B|-1}}\le 1$ (15 points)

2018 USAMO, 1

Tags: inequality , homogenous

Let $a,b,c$ be positive real numbers such that $a+b+c=4\sqrt[3]{abc}$. Prove that \[2(ab+bc+ca)+4\min(a^2,b^2,c^2)\ge a^2+b^2+c^2.\]

2024 Indonesia TST, G

Tags: incenter , geometry

Given an acute triangle $ABC$. The incircle with center $I$ touches $BC,CA,AB$ at $D,E,F$ respectively. Let $M,N$ be the midpoint of the minor arc of $AB$ and $AC$ respectively. Prove that $M,F,E,N$ are collinear if and only if $\angle BAC =90$$^{\circ}$

2009 Stanford Mathematics Tournament, 5

Tags: derivative , calculus

Find the minimum possible value of $2x^2+2xy+4y+5y^2-x$ for real numbers $x$ and $y$.

2023 Thailand TST, 2

Tags: number theory

For each $1\leq i\leq 9$ and $T\in\mathbb N$, define $d_i(T)$ to be the total number of times the digit $i$ appears when all the multiples of $1829$ between $1$ and $T$ inclusive are written out in base $10$. Show that there are infinitely many $T\in\mathbb N$ such that there are precisely two distinct values among $d_1(T)$, $d_2(T)$, $\dots$, $d_9(T)$.

2024 Thailand October Camp, 4

Tags: prime number , number theory

The sequence $(a_n)_{n\in\mathbb{N}}$ is defined by $a_1=3$ and $$a_n=a_1a_2\cdots a_{n-1}-1$$ Show that there exist infinitely many prime number that divide at least one number in this sequences

2018 Sharygin Geometry Olympiad, 5

Tags: geometry

The vertex $C$ of equilateral triangles $ABC$ and $CDE$ lies on the segment $AE$, and the vertices $B$ and $D$ lie on the same side with respect to this segment. The circumcircles of these triangles centered at $O_1$ and $O_2$ meet for the second time at point $F$. The lines $O_1O_2$ and $AD$ meet at point $K$. Prove that $AK = BF$.

2018 CCA Math Bonanza, T2

Tags:

Arnold has plates weighing $5$, $15$, $25$, $35$, or $45$ pounds. He lifts a barbell, which consists of a $45$-pound bar and any number of plates that he has. Vlad looks at Arnold's bar and is impressed to see him bench-press $600$ pounds. Unfortunately, Vlad mistook each plate on Arnold's bar for the plate one size heavier, and Arnold was actually lifting $470$ pounds. How many plates did Arnold have on the bar? [i]2018 CCA Math Bonanza Team Round #2[/i]

2014 Balkan MO, 4

Tags: perimeter , binomial coefficient , combinatorics , geometry

Let $n$ be a positive integer. A regular hexagon with side length $n$ is divided into equilateral triangles with side length $1$ by lines parallel to its sides. Find the number of regular hexagons all of whose vertices are among the vertices of those equilateral triangles. [i]UK - Sahl Khan[/i]

2018 CHMMC (Fall), 2

Tags: algebra

Compute the sum $\sum^{200}_{n=1}\frac{1}{n(n+1)(n+2)}$ .

2015 Math Prize for Girls Problems, 9

Tags:

Say that a rational number is [i]special[/i] if its decimal expansion is of the form $0.\overline{abcdef}$, where $a$, $b$, $c$, $d$, $e$, and $f$ are digits (possibly equal) that include each of the digits $2$, $0$, $1$, and $5$ at least once (in some order). How many special rational numbers are there?

2019 Saudi Arabia Pre-TST + Training Tests, 1.1

Tags: combinatorics

In a school there are $40$ different clubs, each of them contains exactly $30$ children. For every $i$ from $1$ to $30$ define $n_i$ as a number of children who attend exactly $i$ clubs. Prove that it is possible to organize $40$ new clubs with $30$ children in each of them such, that the analogical numbers $n_1, n_2,..., n_{30}$ will be the same for them.

1998 South africa National Olympiad, 2

Tags: euler , trig identities , law of sines , perimeter , trigonometry , geometry , inequalities

Find the maximum value of \[ \sin{2\alpha} + \sin{2\beta} + \sin{2\gamma} \] where $\alpha,\beta$ and $\gamma$ are positive and $\alpha + \beta + \gamma = 180^{\circ}$.

1999 Israel Grosman Mathematical Olympiad, 6

Tags: midpoint , 3d geometry , coplanar , parallelogram , geometry

Let $A,B,C,D,E,F$ be points in space such that the quadrilaterals $ABDE,BCEF, CDFA$ are parallelograms. Prove that the six midpoints of the sides $AB,BC,CD,DE,EF,FA$ are coplanar

2015 May Olympiad, 5

Tags: combinatorics

Twenty-six people gather in a house. Alicia is friends with only one person, Bruno is friends with two people, Carlos is a friend of three, Daniel is four, Elías is five, and so following each person is friend of a person more than the previous person, until reaching Yvonne, the person number twenty-five, who is a friend to everyone. How many people is Zoila a friend of, person number twenty-six? Clarification: If $A$ is a friend of $B$ then $B$ is a friend of $A$.

1947 Moscow Mathematical Olympiad, 138

Tags: points in space , combinatorial geometry , 3d geometry , geometry

In space, $n$ wire triangles are situated so that any two of them have a common vertex and each vertex is the vertex of $k$ triangles. Find all $n$ and $k$ for which this is possible.