Olympiad problems

1988 Iran MO (2nd round), 1

[b](a)[/b] Prove that for all positive integers $m,n$ we have \[\sum_{k=1}^n k(k+1)(k+2)\cdots (k+m-1)=\frac{n(n+1)(n+2) \cdots (n+m)}{m+1}\] [b](b)[/b] Let $P(x)$ be a polynomial with rational coefficients and degree $m.$ If $n$ tends to infinity, then prove that \[\frac{\sum_{k=1}^n P(k)}{n^{m+1}}\] Has a limit.

2013 All-Russian Olympiad, 2

Tags: algebra , polynomial , algebra proposed

Peter and Vasil together thought of ten 5-degree polynomials. Then, Vasil began calling consecutive natural numbers starting with some natural number. After each called number, Peter chose one of the ten polynomials at random and plugged in the called number. The results were recorded on the board. They eventually form a sequence. After they finished, their sequence was arithmetic. What is the greatest number of numbers that Vasil could have called out? [i]A. Golovanov[/i]

2006 Romania Team Selection Test, 1

Tags: function , LaTeX , algebra proposed , algebra

Let $r$ and $s$ be two rational numbers. Find all functions $f: \mathbb Q \to \mathbb Q$ such that for all $x,y\in\mathbb Q$ we have \[ f(x+f(y)) = f(x+r)+y+s. \]

2013 Iran Team Selection Test, 16

Tags: function , algebra proposed , algebra , functional equation

The function $f:\mathbb Z \to \mathbb Z$ has the property that for all integers $m$ and $n$ \[f(m)+f(n)+f(f(m^2+n^2))=1.\] We know that integers $a$ and $b$ exist such that $f(a)-f(b)=3$. Prove that integers $c$ and $d$ can be found such that $f(c)-f(d)=1$. [i]Proposed by Amirhossein Gorzi[/i]

2012 Vietnam National Olympiad, 1

Tags: limit , algebra proposed , algebra

Define a sequence $\{x_n\}$ as: $\left\{\begin{aligned}& x_1=3 \\ & x_n=\frac{n+2}{3n}(x_{n-1}+2)\ \ \text{for} \ n\geq 2.\end{aligned}\right.$ Prove that this sequence has a finite limit as $n\to+\infty.$ Also determine the limit.

1986 India National Olympiad, 3

Tags: Pythagorean Theorem , geometry , algebra proposed , algebra

Two circles with radii a and b respectively touch each other externally. Let c be the radius of a circle that touches these two circles as well as a common tangent to the two circles. Prove that \[ \frac{1}{\sqrt{c}}\equal{}\frac{1}{\sqrt{a}}\plus{}\frac{1}{\sqrt{b}}\]

2014 Dutch IMO TST, 3

Tags: algebra proposed , algebra

Let $a$, $b$ and $c$ be rational numbers for which $a+bc$, $b+ac$ and $a+b$ are all non-zero and for which we have \[\frac{1}{a+bc}+\frac{1}{b+ac}=\frac{1}{a+b}.\] Prove that $\sqrt{(c-3)(c+1)}$ is rational.

2010 Contests, 1

Tags: ratio , algebra proposed , algebra

Let $n$ be an integer greater than two, and let $A_1,A_2, \cdots , A_{2n}$ be pairwise distinct subsets of $\{1, 2, ,n\}$. Determine the maximum value of \[\sum_{i=1}^{2n} \dfrac{|A_i \cap A_{i+1}|}{|A_i| \cdot |A_{i+1}|}\] Where $A_{2n+1}=A_1$ and $|X|$ denote the number of elements in $X.$

2024 ITAMO, 1

Tags: algebra , algebra proposed , absolute value , easy , irrational

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 $x_{n+2}=x_n$.

2003 Balkan MO, 3

Tags: function , algebra proposed , algebra

Find all functions $f: \mathbb{Q}\to\mathbb{R}$ which fulfill the following conditions: a) $f(1)+1>0$; b) $f(x+y) -xf(y) -yf(x) = f(x)f(y) -x-y +xy$, for all $x,y\in\mathbb{Q}$; c) $f(x) = 2f(x+1) +x+2$, for every $x\in\mathbb{Q}$.

2010 Moldova Team Selection Test, 2

Tags: inequalities , trigonometry , calculus , derivative , algebra proposed , algebra

Prove that for any real number $ x$ the following inequality is true: $ \max\{|\sin x|, |\sin(x\plus{}2010)|\}>\dfrac1{\sqrt{17}}$

1987 India National Olympiad, 8

Tags: geometry , incenter , algebra proposed , algebra

Three congruent circles have a common point $ O$ and lie inside a given triangle. Each circle touches a pair of sides of the triangle. Prove that the incentre and the circumcentre of the triangle and the common point $ O$ are collinear.

2005 Georgia Team Selection Test, 4

Tags: algebra , polynomial , algebra proposed

Find all polynomials with real coefficients, for which the equality \[ P(2P(x)) \equal{} 2P(P(x)) \plus{} 2(P(x))^{2}\] holds for any real number $ x$.

2014 BMO TST, 1

Tags: inequalities , induction , algebra proposed , algebra

Prove that for $n\ge 2$ the following inequality holds: $$\frac{1}{n+1}\left(1+\frac{1}{3}+\ldots +\frac{1}{2n-1}\right) >\frac{1}{n}\left(\frac{1}{2}+\ldots+\frac{1}{2n}\right).$$

2007 Kazakhstan National Olympiad, 1

Tags: algebra , polynomial , calculus , derivative , algebra proposed

Zeros of a fourth-degree polynomial $f (x)$ form an arithmetic progression. Prove that the zeros of $f '(x)$ also form an arithmetic progression.

2007 CentroAmerican, 3

Tags: quadratics , algebra , polynomial , algebra proposed

Let $S$ be a finite set of integers. Suppose that for every two different elements of $S$, $p$ and $q$, there exist not necessarily distinct integers $a \neq 0$, $b$, $c$ belonging to $S$, such that $p$ and $q$ are the roots of the polynomial $ax^{2}+bx+c$. Determine the maximum number of elements that $S$ can have.

1996 Turkey Team Selection Test, 1

Tags: algebra proposed , algebra

Let $ \prod_{n=1}^{1996}{(1+nx^{3^n})}= 1+ a_{1}x^{k_{1}}+ a_{2}x^{k_{2}}+...+ a_{m}x^{k_{m}}$ where $a_{1}, a_{1}, . . . , a_{m}$ are nonzero and $k_{1} < k_{2} <...< k_{m}$. Find $a_{1996}$.

2005 Morocco TST, 1

Tags: function , calculus , derivative , induction , limit , algebra proposed , algebra

Find all the functions $f: \mathbb R \rightarrow \mathbb R$ satisfying : $(x+y)(f(x)-f(y))=(x-y)f(x+y)$ for all $x,y \in \mathbb R$

1999 Irish Math Olympiad, 1

Tags: algebra , system of equations , algebra proposed

Solve the system of equations: $ y^2\equal{}(x\plus{}8)(x^2\plus{}2),$ $ y^2\minus{}(8\plus{}4x)y\plus{}(16\plus{}16x\minus{}5x^2)\equal{}0.$

1983 IMO Longlists, 59

Tags: trigonometry , algebra proposed , algebra

Solve the equation \[\tan^2(2x) + 2 \tan(2x) \cdot \tan(3x) -1 = 0.\]

2008 IMAR Test, 4

Tags: function , algebra proposed , algebra

Show that for any function $ f: (0,\plus{}\infty)\to (0,\plus{}\infty)$ there exist real numbers $ x>0$ and $ y>0$ such that: $ f(x\plus{}y)<yf(f(x)).$ [b]Dan Schwarz[/b]

2003 Federal Competition For Advanced Students, Part 1, 3

Tags: quadratics , algebra , polynomial , algebra proposed

Given a positive real number $t$, find the number of real solutions $a, b, c, d$ of the system \[a(1 - b^2) = b(1 -c^2) = c(1 -d^2) = d(1 - a^2) = t.\]

2010 Iran MO (3rd Round), 1

Tags: algebra , polynomial , inequalities , function , algebra proposed

[b]two variable ploynomial[/b] $P(x,y)$ is a two variable polynomial with real coefficients. degree of a monomial means sum of the powers of $x$ and $y$ in it. we denote by $Q(x,y)$ sum of monomials with the most degree in $P(x,y)$. (for example if $P(x,y)=3x^4y-2x^2y^3+5xy^2+x-5$ then $Q(x,y)=3x^4y-2x^2y^3$.) suppose that there are real numbers $x_1$,$y_1$,$x_2$ and $y_2$ such that $Q(x_1,y_1)>0$ , $Q(x_2,y_2)<0$ prove that the set $\{(x,y)|P(x,y)=0\}$ is not bounded. (we call a set $S$ of plane bounded if there exist positive number $M$ such that the distance of elements of $S$ from the origin is less than $M$.) time allowed for this question was 1 hour.

2006 Moldova MO 11-12, 1

Tags: limit , trigonometry , algebra proposed , algebra

Let $n\in\mathbb{N}^*$. Prove that \[ \lim_{x\to 0}\frac{ \displaystyle (1+x^2)^{n+1}-\prod_{k=1}^n\cos kx}{ \displaystyle x\sum_{k=1}^n\sin kx}=\frac{2n^2+n+12}{6n}. \]

2010 Contests, 1

Tags: function , induction , algebra proposed , algebra

Determine all strictly increasing functions $f: \mathbb{N}\to\mathbb{N}$ satisfying $nf(f(n))=f(n)^2$ for all positive integers $n$. [i]Carl Lian and Brian Hamrick.[/i]