Olympiad problems

2005 Poland - Second Round, 1

Tags: algebra , polynomial , modular arithmetic , quadratic , number theory

The polynomial $W(x)=x^2+ax+b$ with integer coefficients has the following property: for every prime number $p$ there is an integer $k$ such that both $W(k)$ and $W(k+1)$ are divisible by $p$. Show that there is an integer $m$ such that $W(m)=W(m+1)=0$.

2014 Harvard-MIT Mathematics Tournament, 23

Tags: probability , expected value , absolute value

Let $S=\{-100,-99,-98,\ldots,99,100\}$. Choose a $50$-element subset $T$ of $S$ at random. Find the expected number of elements of the set $\{|x|:x\in T\}$.

2010 F = Ma, 15

Tags:

A small block moving with initial speed $v_\text{0}$ moves smoothly onto a sloped big block of mass $M$. After the small block reaches the height $h$ on the slope, it slides down. Find the height $h$. (A) $h=\frac{v_\text{0}^2}{2g}$ (B) $h=\frac{1}{g}\frac{Mv_\text{0}^2}{m+M}$ (C) $h=\frac{1}{2g}\frac{Mv_\text{0}^2}{m+M}$ (D) $h=\frac{1}{2g}\frac{mv_\text{0}^2}{m+M}$ (e) $h=\frac{v_\text{0}^2}{g}$

2018 Moscow Mathematical Olympiad, 3

Tags: geometry , circumcircle

$O$ is circumcircle and $AH$ is the altitude of $\triangle ABC$. $P$ is the point on line $OC$ such that $AP \perp OC$. Prove, that midpoint of $AB$ lies on the line $HP$.

2020 Saint Petersburg Mathematical Olympiad, 4.

Tags: inequalities , number theory

Let $m$ be a given positive integer. Prove that there exists a positive integer $k$ such that it holds $$1\leq \frac{1^m+2^m+3^m+\ldots +(k-1)^m}{k^m}<2.$$

1999 Slovenia National Olympiad, Problem 3

Tags: geometry

A semicircle with diameter $AB$ is given. Two non-intersecting circles $k_1$ and $k_2$ with different radii touch the diameter $AB$ and touch the semicircle internally at $C$ and $D$, respectively. An interior common tangent $t$ of $k_1$ and $k_2$ touches $k_1$ at $E$ and $k_2$ at $F$. Prove that the lines $CE$ and $DF$ intersect on the semicircle.

2018 Taiwan TST Round 1, 2

Tags: geometry , circumcircle

In a plane, we are given $ 100 $ circles with radius $ 1 $ so that the area of any triangle whose vertices are circumcenters of those circles is at most $ 100 $. Prove that one may find a line that intersects at least $ 10 $ circles.

2010 Contests, 523

Tags: calculus , integration , calculus computations , logarithm

Prove the following inequality. \[ \ln \frac {\sqrt {2009} \plus{} \sqrt {2010}}{\sqrt {2008} \plus{} \sqrt {2009}} < \int_{\sqrt {2008}}^{\sqrt {2009}} \frac {\sqrt {1 \minus{} e^{ \minus{} x^2}}}{x}\ dx < \sqrt {2009} \minus{} \sqrt {2008}\]

2019 Regional Olympiad of Mexico Southeast, 1

Tags: number theory

Found the smaller multiple of $2019$ of the form $abcabc\dots abc$, where $a,b$ and $c$ are digits.

1962 Czech and Slovak Olympiad III A, 1

Tags: perfect square , prime , algebra , quadratic polynomial

Determine all integers $x$ such that $2x^2-x-36$ is a perfect square of a prime.

1997 All-Russian Olympiad Regional Round, 8.3

Tags: geometry , equilateral , angle

On sides $AB$ and $BC$ of an equilateral triangle $ABC$ are taken points $D$ and $K$, and on the side $AC$ , points $E$ and $M$ so that $DA + AE = KC +CM = AB$. Prove that the angle between lines $DM$ and $KE$ is equal to $60^o$.

2011 Tournament of Towns, 5

Tags: geometry

$AD$ and $BE$ are altitudes of an acute triangle $ABC$. From $D$, perpendiculars are dropped to $AB$ at $G$ and $AC$ at $K$. From $E$, perpendiculars are dropped to $AB$ at $F$ and $BC$ at $H$. Prove that $FG$ is parallel to $HK$ and $FK = GH$.

2016 Regional Olympiad of Mexico Southeast, 5

Tags: combinatorics

Martin and Chayo have an bag with $2016$ chocolates each one. Both empty his bag on a table making a pile of chocolates. They decide to make a competence to see who gets the chocolates, as follows: A movement consist that a player take two chocolates of his pile, keep a chocolate in his bag and put the other chocolate in the pile of the other player, in his turn the player needs to make at least one movement and he can repeat as many times as he wish before passing his turn. Lost the player that can not make at least one movement in his turn. If Martin starts the game, who can ensure the victory and keep all the chocolates?

2014 Sharygin Geometry Olympiad, 9

Tags: geometry

Two circles $\omega_1$ and $\omega_2$ touching externally at point $L$ are inscribed into angle $BAC$. Circle $\omega_1$ touches ray $AB$ at point $E$, and circle $\omega_2$ touches ray $AC$ at point $M$. Line $EL$ meets $\omega_2$ for the second time at point $Q$. Prove that $MQ\parallel AL$.

2021 Middle European Mathematical Olympiad, 4

Tags: number theory , meme

Let $n \ge 3$ be an integer. Zagi the squirrel sits at a vertex of a regular $n$-gon. Zagi plans to make a journey of $n-1$ jumps such that in the $i$-th jump, it jumps by $i$ edges clockwise, for $i \in \{1, \ldots,n-1 \}$. Prove that if after $\lceil \tfrac{n}{2} \rceil$ jumps Zagi has visited $\lceil \tfrac{n}{2} \rceil+1$ distinct vertices, then after $n-1$ jumps Zagi will have visited all of the vertices. ([i]Remark.[/i] For a real number $x$, we denote by $\lceil x \rceil$ the smallest integer larger or equal to $x$.)

2010 Peru MO (ONEM), 4

Tags: geometry , 3d geometry , combinatorics

A parallelepiped is said to be [i]integer [/i] when at least one of its edges measures a integer number of units. We have a group of integer parallelepipeds with which a larger parallelepiped is assembled, which has no holes inside or on its edge. Prove that the assembled parallelepiped is also integer. Example. The following figure shows an assembled parallelepiped with a certain group of integer parallelepipeds. [img]https://cdn.artofproblemsolving.com/attachments/3/7/f88954d6fe3a59fd2db6dcee9dddb120012826.png[/img]

2023 Olympic Revenge, 6

Tags: group theory , random , algebra

We say that $H$ permeates $G$ if $G$ and $H$ are finite groups and for all subgroup $F$ of $G$ there is $H'\cong H$ with $H'\le F$ or $F\le H'\le G$. Suppose that a non-abelian group $H$ permeates $G$ and let $S=\langle H'\le G | H'\cong H\rangle$. Show that $$|\bigcap_{H'\in S} H'|>1$$

2017 Miklós Schweitzer, 8

Tags: algebra , binary representation , number base , function , inequalities

Let the base $2$ representation of $x\in[0;1)$ be $x=\sum_{i=0}^\infty \frac{x_i}{2^{i+1}}$. (If $x$ is dyadically rational, i.e. $x\in\left\{\frac{k}{2^n}\,:\, k,n\in\mathbb{Z}\right\}$, then we choose the finite representation.) Define function $f_n:[0;1)\to\mathbb{Z}$ by $$f_n(x)=\sum_{j=0}^{n-1}(-1)^{\sum_{i=0}^j x_i}.$$Does there exist a function $\varphi:[0;\infty)\to[0;\infty)$ such that $\lim_{x\to\infty} \varphi(x)=\infty$ and $$\sup_{n\in\mathbb{N}}\int_0^1 \varphi(|f_n(x)|)\mathrm{d}x<\infty\, ?$$

1997 National High School Mathematics League, 2

Tags: complex number

For real numbers $x_0,x_1,\cdots,x_n$, there exists real numbers $y_0,y_1,\cdots,y_n$, satisfying that $z_0^2=z_1^2+z_2^2+\cdots+z_n^2$, where $z_k=x_k+\text{i}y_{k}(k=0,1,\cdots,n)$. Find all such $(x_0,x_1,\cdots,x_n)$.

1989 Dutch Mathematical Olympiad, 1

Tags: algebra , sequence , recurrence relation

For a sequence of integers $a_1,a_2,a_3,...$ with $0<a_1<a_2<a_3<...$ applies: $$a_n=4a_{n-1}-a_{n-2} \,\,\, for \,\,\, n > 2$$ It is further given that $a_4 = 194$. Calculate $a_5$.

2024 Malaysian APMO Camp Selection Test, 1

Tags: number theory

Let $a_1<a_2< \cdots$ be a strictly increasing sequence of positive integers. Suppose there exist $N$ such that for all $n>N$, $$a_{n+1}\mid a_1+a_2+\cdots+a_n$$ Prove that there exist $M$ such that $a_{m+1}=2a_m$ for all $m>M$. [i]Proposed by Ivan Chan Kai Chin[/i]

2024 LMT Fall, 22

Tags: guts

Find the number of real numbers $0 \leq \alpha < 50$ such that $\alpha^2 + 2\{\alpha\}$ is an integer. (Here $\{\alpha\}$ denotes the fractional part of $\alpha$.)

CIME II 2018, 6

Tags:

Define $f(x)=-\frac{2x}{4x+3}$ and $g(x)=\frac{x+2}{2x+1}$. Moreover, let $h^{n+1} (x)=g(f(h^n(x)))$, where $h^1(x)=g(f(x))$. If the value of $\sum_{k=1}^{100} (-1)^k\cdot h^{100}(k)$ can be written in the form $ab^c$, for some integers $a,b,c$ where $c$ is as maximal as possible and $b\ne 1$, find $a+b+c$. [i]Proposed by [b]AOPS12142015[/b][/i]

2018 Purple Comet Problems, 3

Tags: algebra

The fraction $$\left(\frac{\frac13+1}{3} +\frac{1+ \frac13}{3} \right) / \left(\frac{3}{\frac{1}{3+1}+\frac{ 1}{1+3}}\right)$$ can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

1999 Ukraine Team Selection Test, 2

Tags: integer , algebra

Show that there exist integers $j,k,l,m,n$ greater than $100$ such that $j^2 +k^2 +l^2 +m^2 +n^2 = jklmn-12$.