Olympiad problems

2008 Serbia National Math Olympiad, 5

The sequence $ (a_n)_{n\ge 1}$ is defined by $ a_1 \equal{} 3$, $ a_2 \equal{} 11$ and $ a_n \equal{} 4a_{n\minus{}1}\minus{}a_{n\minus{}2}$, for $ n \ge 3$. Prove that each term of this sequence is of the form $ a^2 \plus{} 2b^2$ for some natural numbers $ a$ and $ b$.

2003 BAMO, 1

Tags: factorial , perfect number , number theory , divisor

An integer is a perfect number if and only if it is equal to the sum of all of its divisors except itself. For example, $28$ is a perfect number since $28 = 1 + 2 + 4 + 7 + 14$. Let $n!$ denote the product $1\cdot 2\cdot 3\cdot ...\cdot n$, where $n$ is a positive integer. An integer is a factorial if and only if it is equal to $n!$ for some positive integer $n$. For example, $24$ is a factorial number since $24 = 4! = 1\cdot 2\cdot 3\cdot 4$. Find all perfect numbers greater than $1$ that are also factorials.

2017 AMC 12/AHSME, 3

Tags: algebra

Suppose that $x$ and $y$ are nonzero real numbers such that \[\frac{3x+y}{x-3y}= -2.\] What is the value of \[\frac{x+3y}{3x-y}?\] $\textbf{(A) } {-3} \qquad \textbf{(B) } {-1} \qquad \textbf{(C) } 1 \qquad \textbf{(D) }2 \qquad \textbf{(E) } 3$

2016 Oral Moscow Geometry Olympiad, 3

Tags: geometry , square , equal area , area

Two squares are arranged as shown in the picture. Prove that the areas of shaded quadrilaterals are equal. [img]https://3.bp.blogspot.com/-W50DOuizFvY/XT6wh3-L6sI/AAAAAAAAKaw/pIW2RKmttrwPAbrKK3bpahJz7hfIZwM8QCK4BGAYYCw/s400/Oral%2BSharygin%2B2016%2B10.11%2Bp3.png[/img]

2012 Polish MO Finals, 1

Tags: number theory , rational number , power of number

Decide, whether exists positive rational number $w$, which isn't integer, such that $w^w$ is a rational number.

1979 Bulgaria National Olympiad, Problem 3

Tags: triangle , combinatorics , geometry , combinatorial geometry

Each side of a triangle $ABC$ has been divided into $n+1$ equal parts. Find the number of triangles with the vertices at the division points having no side parallel to or lying at a side of $\triangle ABC$.

2010 Morocco TST, 3

Tags: function , algebra

Let $G$ be a non-empty set of non-constant functions $f$ such that $f(x)=ax + b$ (where $a$ and $b$ are two reals) and satisfying the following conditions: 1) if $f \in G$ and $g \in G$ then $gof \in G$, 2) if $f \in G$ then $f^ {-1} \in G$, 3) for all $f \in G$ there exists $x_f \in \mathbb{R}$ such that $f(x_f)=x_f$. Prove that there is a real $k$ such that for all $f \in G$ we have $f(k)=k$

2021-IMOC, A8

Tags: functional equation , algebra , inequality , function , inequalities

Find all functions $f : \mathbb{N} \to \mathbb{N}$ with $$f(x) + yf(f(x)) < x(1 + f(y)) + 2021$$ holds for all positive integers $x,y.$

2014 239 Open Mathematical Olympiad, 7

Tags: geometry

A circle $\omega$ is strictly inside triangle $ABC$. The tangents from $A$ to $\omega$ intersect $BC$ in $A_1,A_2$ define $B_1,B_2,C_1,C_2$ similarly. Prove that if five of six points $A_1,A_2,B_1,B_2,C_1,C_2$ lie on a circle the sixth one lie on the circle too.

2023 Novosibirsk Oral Olympiad in Geometry, 6

Tags: geometry , equal segments , isosceles , angle

An isosceles triangle $ABC$ with base $AC$ is given. On the rays $CA$, $AB$ and $BC$, the points $D, E$ and $F$ were marked, respectively, in such a way that $AD = AC$, $BE = BA$ and $CF = CB$. Find the sum of the angles $\angle ADB$, $\angle BEC$ and $\angle CFA$.

1930 Eotvos Mathematical Competition, 2

Tags: combinatorics , combinatorial geometry , chessboard , geometry

A straight line is drawn across an $8\times 8$ chessboard. It is said to [i]pierce [/i]a square if it passes through an interior point of the square. At most how many of the $64$ squares can this line [i]pierce[/i]?

1967 IMO Longlists, 9

Tags: incenter , angle bisector , locus , locus problems , geometry

Circle $k$ and its diameter $AB$ are given. Find the locus of the centers of circles inscribed in the triangles having one vertex on $AB$ and two other vertices on $k.$

2024 HMNT, 2

Tags: guts

Compute the smallest integer $n > 72$ that has the same set of prime divisors as $72.$

1990 IMO Longlists, 83

Tags: geometry

Point $D$ is on the hypotenuse $BC$ of right-angled triangle $ABC$. The inradii of triangles $ADB$ and $ADC$ are equal. Prove that $S_{ABC} = AD^2$, where $S$ is the area function.

2008 National Olympiad First Round, 26

Tags: factorial

Let $A=\frac{2^2+3\cdot 2 + 1}{3! \cdot 4!} + \frac{3^2+3\cdot 3 + 1}{4! \cdot 5!} + \frac{4^2+3\cdot 4 + 1}{5! \cdot 6!} + \dots + \frac{10^2+3\cdot 10 + 1}{11! \cdot 12!}$. What is the remainder when $11!\cdot 12! \cdot A$ is divided by $11$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ 10 $

2013 ELMO Problems, 4

Tags: geometry , incenter , angle bisector , perpendicular bisector

Triangle $ABC$ is inscribed in circle $\omega$. A circle with chord $BC$ intersects segments $AB$ and $AC$ again at $S$ and $R$, respectively. Segments $BR$ and $CS$ meet at $L$, and rays $LR$ and $LS$ intersect $\omega$ at $D$ and $E$, respectively. The internal angle bisector of $\angle BDE$ meets line $ER$ at $K$. Prove that if $BE = BR$, then $\angle ELK = \tfrac{1}{2} \angle BCD$. [i]Proposed by Evan Chen[/i]

1988 Swedish Mathematical Competition, 3

Tags: inequalities , sum , algebra

Show that if $x_1+x_2+x_3 = 0$ for real numbers $x_1,x_2,x_3$, then $x_1x_2+x_2x_3+x_3x_1\le 0$. Find all $n \ge 4$ for which $x_1+x_2+...+x_n = 0$ implies $x_1x_2+x_2x_3+...+x_{n-1}x_n+x_nx_1 \le 0$.

2018 Caucasus Mathematical Olympiad, 2

Tags: chessboard , combinatorics

On a chessboard $8\times 8$, $n>6$ Knights are placed so that for any 6 Knights there are two Knights that attack each other. Find the greatest possible value of $n$.

1996 Italy TST, 3

Tags: geometry

3.Let ABCD be a parallelogram with side AB longer than AD and acute angle $\angle DAB$. The bisector of ∠DAB meets side CD at L and line BC at K. If O is the circumcenter of triangle LCK, prove that the points B,C,O,D lie on a circle.

2015 VJIMC, 2

Tags:

[b]Problem 2[/b] Consider the infinite chessboard whose rows and columns are indexed by positive integers. Is it possible to put a single positive rational number into each cell of the chessboard so that each positive rational number appears exactly once and the sum of every row and of every column is finite?

1995 Belarus Team Selection Test, 2

Tags: common tangent , geometry , circles , tangent circle

Circles $S,S_1,S_2$ are given in a plane. $S_1$ and $S_2$ touch each other externally, and both touch $S$ internally at $A_1$ and $A_2$ respectively. The common internal tangent to $S_1$ and $S_2$ meets $S$ at $P$ and $Q.$ Let $B_1$ and $B_2$ be the intersections of $PA_1$ and $PA_2$ with $S_1$ and $S_2$, respectively. Prove that $B_1B_2$ is a common tangent to $S_1,S_2$

1999 Taiwan National Olympiad, 2

Tags: number theory

Let $a_{1},a_{2},...,a_{1999}$ be a sequence of nonnegative integers such that for any $i,j$ with $i+j\leq 1999$ , $a_{i}+a_{j}\leq a_{i+j}\leq a_{i}+a_{j}+1$. Prove that there exists a real number $x$ such that $a_{n}=[nx]\forall n$.

2011 Greece JBMO TST, 3

Tags: integer , number theory , diophantine equation

Find integer solutions of the equation $8x^3 - 4 = y(6x - y^2)$

2010 Iran MO (3rd Round), 3

Tags: group theory , abstract algebra , algebra

suppose that $G<S_n$ is a subgroup of permutations of $\{1,...,n\}$ with this property that for every $e\neq g\in G$ there exist exactly one $k\in \{1,...,n\}$ such that $g.k=k$. prove that there exist one $k\in \{1,...,n\}$ such that for every $g\in G$ we have $g.k=k$.(20 points)

2023 Korea National Olympiad, 8

Tags: combinatorics

For a positive integer $n$, if $n$ is a product of two different primes and $n \equiv 2 \pmod 3$, then $n$ is called "special number." For example, $14, 26, 35, 38$ is only special numbers among positive integers $1$ to $50$. Prove that for any finite set $S$ with special numbers, there exist two sets $A, B$ such that [list] [*] $A \cap B = \emptyset, A \cup B = S$ [*] $||A| - |B|| \leq 1$ [*] For all primes $p$, the difference between number of elements in $A$ which is multiple of $p$ and number of elements in $B$ which is multiple of $p$ is less than or equal to $1$. [/list]