Olympiad problems

2024 ITAMO, 1

Let $x_0=2024^{2024}$ and $x_{n+1}=|x_n-\pi|$ for $n \ge 0$. Show that there exists a value of $n$ such that $x_{n+2}=x_n$.

2007 Nicolae Coculescu, 1

Tags: algebra , floor , inequalities , three variable inequality , geometry

Calculate $ \left\lfloor \frac{(a^2+b^2+c^2)(a+b+c)}{a^3+b^3+c^3} \right\rfloor , $ where $ a,b,c $ are the lengths of the side of a triangle. [i]Costel Anghel[/i]

2020 Jozsef Wildt International Math Competition, W42

Tags: inequalities

If $a,b,c$ are non-negative real numbers such that $a+b+c=3m,(m\ge1)$ then prove that $$(a^a+b^a+c^a)(a^b+b^b+c^b)(a^c+b^c+c^c)\ge27m^{3m}$$ [i]Proposed by Dorin Mărghidanu[/i]

2024 Bulgarian Winter Tournament, 11.3

Tags: number theory

Let $q>3$ be a rational number, such that $q^2-4$ is a perfect square of a rational number. The sequence $a_0, a_1, \ldots$ is defined by $a_0=2, a_1=q, a_{i+1}=qa_i-a_{i-1}$ for all $i \geq 1$. Is it true that there exist a positive integer $n$ and nonzero integers $b_0, b_1, \ldots, b_n$ with sum zero, such that if $\sum_{i=0}^{n} a_ib_i=\frac{A} {B}$ for $(A, B)=1$, then $A$ is squarefree?

2003 IMO Shortlist, 6

Tags: modular arithmetic , combinatorics , decimal representation , counting , function

Let $f(k)$ be the number of integers $n$ satisfying the following conditions: (i) $0\leq n < 10^k$ so $n$ has exactly $k$ digits (in decimal notation), with leading zeroes allowed; (ii) the digits of $n$ can be permuted in such a way that they yield an integer divisible by $11$. Prove that $f(2m) = 10f(2m-1)$ for every positive integer $m$. [i]Proposed by Dirk Laurie, South Africa[/i]

2006 AIME Problems, 11

Tags: geometry

A sequence is defined as follows $a_1=a_2=a_3=1$, and, for all positive integers $n$, $a_{n+3}=a_{n+2}+a_{n+1}+a_n$. Given that $a_{28}=6090307$, $a_{29}=11201821$, and $a_{30}=20603361$, find the remainder when $\displaystyle \sum^{28}_{k=1} a_k$ is divided by 1000.

1993 Cono Sur Olympiad, 3

Tags: combinatorics

Prove that, given a positive integrer $n$, there exists a positive integrer $k_n$ with the following property: Given any $k_n$ points in the space, $4$ by $4$ non-coplanar, and associated integrer numbers between $1$ and $n$ to each sharp edge that meets $2$ of this points, there's necessairly a triangle determined by $3$ of them, whose sharp edges have associated the same number.

2009 Sharygin Geometry Olympiad, 3

Tags: trapezoid , parallelogram , rectangle , incenter , geometry

The bisectors of trapezoid's angles form a quadrilateral with perpendicular diagonals. Prove that this trapezoid is isosceles.

1996 Czech and Slovak Match, 5

Tags: set , interval , combinatorics

Two sets of intervals $A ,B$ on the line are given. The set $A$ contains $2m-1$ intervals, every two of which have an interior point in common. Moreover, every interval from $A$ contains at least two disjoint intervals from $B$. Show that there exists an interval in $B$ which belongs to at least $m$ intervals from $A$ .

2012 Miklós Schweitzer, 3

Tags: combinatorics

There is a simple graph which chromatic number is equal to $k$. We painted all of the edges of graph using two colors. Prove that there exist a monochromatic tree with $k$ vertices

1978 USAMO, 1

Tags: inequalities , vector , marvio

Given that $a,b,c,d,e$ are real numbers such that $a+b+c+d+e=8$, $a^2+b^2+c^2+d^2+e^2=16$. Determine the maximum value of $e$.

2019 Moldova EGMO TST, 7

Tags: combinatorics

Let $A{}$ be a subset formed of $16$ elements of the set $B=\{1, 2, 3, \ldots, 105, 106\}$ such that the difference between every two elements from $A$ is different from $6, 9, 12, 15, 18, 21$. Prove that there are two elements in $A{}$ whose difference is $3$.

2014 Harvard-MIT Mathematics Tournament, 4

Tags: hmmt , algebra , polynomial , vieta , sum of roots

Let $b$ and $c$ be real numbers and define the polynomial $P(x)=x^2+bx+c$. Suppose that $P(P(1))=P(P(2))=0$, and that $P(1) \neq P(2)$. Find $P(0)$.

2014 China Girls Math Olympiad, 5

Tags: algebra , polynomial , number theory

Let $a$ be a positive integer, but not a perfect square; $r$ is a real root of the equation $x^3-2ax+1=0$. Prove that $ r+\sqrt{a}$ is an irrational number.

2013 AMC 12/AHSME, 7

Tags:

The sequence $S_1, S_2, S_3, \cdots, S_{10}$ has the property that every term beginning with the third is the sum of the previous two. That is, \[ S_n = S_{n-2} + S_{n-1} \text{ for } n \ge 3. \] Suppose that $S_9 = 110$ and $S_7 = 42$. What is $S_4$? $ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 16\qquad $

2025 Iran MO (2nd Round), 1

Tags: number theory

Find all positive integers n>2 such that sum of n and any of its prime divisors is a perfect square.

2014 PUMaC Number Theory B, 4

Tags:

Find the number of fractions in the following list that is in its lowest form. (ie. for $\tfrac pq$, $\gcd(p,q) = 1$.) \[\frac{1}{2014}, \frac{2}{2013}, \dots, \frac{1007}{1008}\]

1997 All-Russian Olympiad Regional Round, 10.1

Tags: algebra , combinatorics

The microcalculator ''MK-97'' can work out the numbers entered in memory, perform only three operations: a) check whether the selected two numbers are equal; b) add the selected numbers; c) using the selected numbers $a$ and $b$, find the equation $x^2 +ax+b = 0$, and if there are no roots, display a message about this. The results of all actions are stored in memory. Initially, one number $x$ is stored in memory. How to use ''MK-97'' to find out whether is this number one?

2022 Romania Team Selection Test, 4

Tags: perfect square , number theory

Any positive integer $N$ which can be expressed as the sum of three squares can obviously be written as \[N=\frac{a^2+b^2+c^2+d^2}{1+abcd}\]where $a,b,c,d$ are nonnegative integers. Is the mutual assertion true?

2021 Regional Competition For Advanced Students, 2

Tags: geometry , tangent , isosceles

Let $ABC$ be an isosceles triangle with $AC = BC$ and circumcircle $k$. The point $D$ lies on the shorter arc of $k$ over the chord $BC$ and is different from $B$ and $C$. Let $E$ denote the intersection of $CD$ and $AB$. Prove that the line through $B$ and $C$ is a tangent of the circumcircle of the triangle $BDE$. (Karl Czakler)

2011 Middle European Mathematical Olympiad, 3

Tags: combinatorics

For an integer $n \geq 3$, let $\mathcal M$ be the set $\{(x, y) | x, y \in \mathbb Z, 1 \leq x \leq n, 1 \leq y \leq n\}$ of points in the plane. What is the maximum possible number of points in a subset $S \subseteq \mathcal M$ which does not contain three distinct points being the vertices of a right triangle?

2011 IMAC Arhimede, 3

Tags: combinatorics

Place $n$ points on a circle and draw all possible chord joining these points. If no three chord are concurent, find the number of disjoint regions created. [color=#008000]Moderator says: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=151&t=260926&hilit=circle+points+segments+regions[/color]

2018 Latvia Baltic Way TST, P4

Tags: function , inequalities , algebra

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function that satisfies $$\sqrt{2f(x)}-\sqrt{2f(x)-f(2x)}\ge 2$$ for all real $x$. Prove for all real $x$: [i](a)[/i] $f(x)\ge 4$; [i](b)[/i] $f(x)\ge 7.$

2021 China Team Selection Test, 4

Tags: algebra , polynomial , number theory , modular arithmetic

Let $f(x),g(x)$ be two polynomials with integer coefficients. It is known that for infinitely many prime $p$, there exist integer $m_p$ such that $$f(a) \equiv g(a+m_p) \pmod p$$ holds for all $a \in \mathbb{Z}.$ Prove that there exists a rational number $r$ such that $$f(x)=g(x+r).$$

1959 Putnam, A3

Tags: function , functional equation , complex

Find all complex-valued functions $f$ of a complex variable such that $$f(z)+zf(1-z)=1+z$$ for all $z\in \mathbb{C}$.