Olympiad problems

2010 Turkey MO (2nd round), 2

For integers $a$ and $b$ with $0 \leq a,b < {2010}^{18}$ let $S$ be the set of all polynomials in the form of $P(x)=ax^2+bx.$ For a polynomial $P$ in $S,$ if for all integers n with $0 \leq n <{2010}^{18}$ there exists a polynomial $Q$ in $S$ satisfying $Q(P(n)) \equiv n \pmod {2010^{18}},$ then we call $P$ as a [i]good polynomial.[/i] Find the number of [i]good polynomials.[/i]

2001 Estonia National Olympiad, 1

Tags: convex , polygon , geometry , angle

The angles of a convex $n$-gon are $a,2a, ... ,na$. Find all possible values of $n$ and the corresponding values of $a$.

2012 CHMMC Spring, 5

Tags: combinatorics

Suppose $S$ is a subset of $\{1, 2, 3, 4, 5, 6, 7\}$. How many different possible values are there for the product of the elements in $S$?

2024 Saint Petersburg Mathematical Olympiad, 3

Tags: geometry , circumcircle

On the side $BC$ of acute triangle $ABC$ point $P$ was chosen. Point $E$ is symmetric to point $B$ onto line $AP$. Segment $PE$ meets circumcircle of triangle $ABP$ in point $D$. $M$ is midpoint of side $AC$. Prove that $DE+AC>2BM$.

2009 AMC 8, 6

Tags:

Steve's empty swimming pool will hold $ 24,000$ gallons of water when full. It will be filled by $ 4$ hoses, each of which supplies $ 2.5$ gallons of water per minute. How many hours will it take to fill Steve's pool? $ \textbf{(A)}\ 40 \qquad \textbf{(B)}\ 42 \qquad \textbf{(C)}\ 44 \qquad \textbf{(D)}\ 46 \qquad \textbf{(E)}\ 48$

1989 Iran MO (2nd round), 1

Tags: floor function , function , limit , inequalities

[b](a)[/b] Let $n$ be a positive integer, prove that \[ \sqrt{n+1} - \sqrt{n} < \frac{1}{2 \sqrt n}\] [b](b)[/b] Find a positive integer $n$ for which \[ \bigg\lfloor 1 +\frac{1}{\sqrt 2} +\frac{1}{\sqrt 3} +\frac{1}{\sqrt 4} + \cdots +\frac{1}{\sqrt n} \bigg\rfloor =12\]

BIMO 2022, 5

Tags: number theory

Find all functions $f : \mathbb{Z}\rightarrow \mathbb{Z}$ such that for all prime $p$ the following condition holds: $$p \mid ab + bc + ca \iff p \mid f(a)f(b) + f(b)f(c) + f(c)f(a)$$ [i]Proposed by Anzo Teh Zhao Yang[/i]

2014 Baltic Way, 10

Tags: combinatorics

In a country there are $100$ airports. Super-Air operates direct flights between some pairs of airports (in both directions). The [i]traffic[/i] of an airport is the number of airports it has a direct Super-Air connection with. A new company, Concur-Air, establishes a direct flight between two airports if and only if the sum of their traffics is at least $100.$ It turns out that there exists a round-trip of Concur-Air flights that lands in every airport exactly once. Show that then there also exists a round-trip of Super-Air flights that lands in every airport exactly once.

2022 Federal Competition For Advanced Students, P1, 2

Tags: geometry , cyclic quadrilateral , equal segments

The points $A, B, C, D$ lie in this order on a circle with center $O$. Furthermore, the straight lines $AC$ and $BD$ should be perpendicular to each other. The base of the perpendicular from $O$ on $AB$ is $F$. Prove $CD = 2 OF$. [i](Karl Czakler)[/i]

2018 Slovenia Team Selection Test, 4

Tags: geometry , circumcircle

Let $\mathcal{K}$ be a circle centered in $A$. Let $p$ be a line tangent to $\mathcal{K}$ in $B$ and let a line parallel to $p$ intersect $\mathcal{K}$ in $C$ and $D$. Let the line $AD$ intersect $p$ in $E$ and let $F$ be the intersection of the lines $CE$ and $AB$. Prove that the line through $D$, parallel to the tangent through $A$ to the circumcircle of $AFD$ intersects the line $CF$ on $\mathcal{K}$.

2010 National Chemistry Olympiad, 10

Tags:

Magnesium chloride dissolves in water to form: $ \textbf{(A)}\hspace{.05in}\text{hydrated MgCl}_2 \text{molecules}\qquad$ $\textbf{(B)}\hspace{.05in}\text{hydrated Mg}^{2+} \text{ions and hydrated Cl}^- \text{ions} \qquad$ $\textbf{(C)}\hspace{.05in}\text{hydrated Mg}^{2+} \text{ions and hydrated Cl}_2 ^{2-} \text{ions}\qquad$ $\textbf{(D)}\hspace{.05in}\text{hydrated Mg atoms and hydrated Cl}_2 \text{molecules}\qquad$

2015 Abels Math Contest (Norwegian MO) Final, 1a

Tags: system of equations , algebra

Find all triples $(x, y, z) \in R^3$ satisfying the equations $\begin{cases} x^2 + 4y^2 = 4zx \\ y^2 + 4z^2 = 4xy \\ z^2 + 4x^2 = 4yz \end{cases}$

PEN A Problems, 40

Tags: number theory , greatest common divisor , divisibility theory

Determine the greatest common divisor of the elements of the set \[\{n^{13}-n \; \vert \; n \in \mathbb{Z}\}.\]

2012 USAMO, 4

Tags: function , induction , factorial , modular arithmetic , algebra , number theory

Find all functions $f:\mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ (where $\mathbb{Z}^+$ is the set of positive integers) such that $f(n!) = f(n)!$ for all positive integers $n$ and such that $m-n$ divides $f(m) - f(n)$ for all distinct positive integers $m, n$.

1984 Tournament Of Towns, (078) 3

Tags: combinatorial geometry , combinatorics , decagon

We are given a regular decagon with all diagonals drawn. The number "$+ 1$ " is attached to each vertex and to each point where diagonals intersect (we consider only internal points of intersection). We can decide at any time to simultaneously change the sign of all such numbers along a given side or a given diagonal . Is it possible after a certain number of such operations to have changed all the signs to negative?

1985 Balkan MO, 1

Tags: circumcircle , analytic geometry , graphing lines , complex number , geometry

In a given triangle $ABC$, $O$ is its circumcenter, $D$ is the midpoint of $AB$ and $E$ is the centroid of the triangle $ACD$. Show that the lines $CD$ and $OE$ are perpendicular if and only if $AB=AC$.

2008 All-Russian Olympiad, 1

Tags: number theory

Do there exist $ 14$ positive integers, upon increasing each of them by $ 1$,their product increases exactly $ 2008$ times?

2003 JBMO Shortlist, 3

Tags: geometry

Let $G$ be the centroid of triangle $ABC$, and $A'$ the symmetric of $A$ wrt $C$. Show that $G, B, C, A'$ are concyclic if and only if $GA \perp GC$.

LMT Speed Rounds, 2010.10

Tags:

How many integers less than $2502$ are equal to the square of a prime number?

2011 Sharygin Geometry Olympiad, 17

Tags: geometry , median , bisector , compare , geometric inequality

a) Does there exist a triangle in which the shortest median is longer that the longest bisectrix? b) Does there exist a triangle in which the shortest bisectrix is longer that the longest altitude?

2012 AIME Problems, 9

Tags: ratio , trigonometry , function , quadratic , number theory , relatively prime

Let $x$ and $y$ be real numbers such that $\frac{\sin{x}}{\sin{y}} = 3$ and $\frac{\cos{x}}{\cos{y}} = \frac{1}{2}$. The value of $\frac{\sin{2x}}{\sin{2y}} + \frac{\cos{2x}}{\cos{2y}}$ can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.

1999 Mongolian Mathematical Olympiad, Problem 2

Tags: geometry

The rays $l_1,l_2,\ldots,l_{n-1}$ divide a given angle $ABC$ into $n$ equal parts. A line $l$ intersects $AB$ at $A_1$, $BC$ at $A_{n+1}$, and $l_i$ at $A_{i+1}$ for $i=1,\ldots,n-1$. Show that the quantity $$\left(\frac1{BA_1}+\frac1{BA_{n+1}}\right)\left(\frac1{BA_1}+\frac1{BA_2}+\ldots+\frac1{BA_{n+1}}\right)^{-1}$$is independent of the line $l$, and compute its value if $\angle ABC=\phi$.

2013 Dutch IMO TST, 3

Tags: prime number , number theory

Fix a sequence $a_1,a_2,a_3\ldots$ of integers satisfying the following condition:for all prime numbers $p$ and all positive integers $k$,we have $a_{pk+1}=pa_k-3a_p+13$.Determine all possible values of $a_{2013}$.

1984 IMO Shortlist, 6

Tags: linear recurrence , algebra , sequence , recurrence relation

Let $c$ be a positive integer. The sequence $\{f_n\}$ is defined as follows: \[f_1 = 1, f_2 = c, f_{n+1} = 2f_n - f_{n-1} + 2 \quad (n \geq 2).\] Show that for each $k \in \mathbb N$ there exists $r \in \mathbb N$ such that $f_kf_{k+1}= f_r.$

1987 Czech and Slovak Olympiad III A, 4

Tags: algebra , number theory , inequalities , inequality , prime number , factorial

Given an integer $n\ge3$ consider positive integers $x_1,\ldots,x_n$ such that $x_1<x_2<\cdots<x_n<2x_1$. If $p$ is a prime and $r$ is a positive integer such that $p^r$ divides the product $x_1\cdots x_n$, prove that $$\frac{x_1\cdots x_n}{p^r}>n!.$$