Olympiad problems

1990 Hungary-Israel Binational, 3

Tags: induction , algebra

Prove that: \[ \frac{1989}{2}\minus{}\frac{1988}{3}\plus{}\frac{1987}{4}\minus{}\cdots\minus{}\frac{2}{1989}\plus{}\frac{1}{1990}\equal{}\frac{1}{996}\plus{}\frac{3}{997}\plus{}\frac{5}{998}\plus{}\cdots\plus{}\frac{1989}{1990}\]

2015 Indonesia MO Shortlist, A4

Tags: algebra , functional equation , function

Determine all functions $f: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ such that \[ f(x,y) + f(y,z) + f(z,x) = \max \{ x,y,z \} - \min \{ x,y,z \} \] for every $x,y,z \in \mathbb{R}$ and there exists some real $a$ such that $f(x,a) = f(a,x) $ for every $x \in \mathbb{R}$.

2017 HMNT, 5

Tags: algebra

Given that $a,b,c$ are integers with $abc = 60$, and that complex number $\omega \neq 1$ satisfies $\omega^3=1$, find the minimum possible value of $|a + b\omega + c\omega^2|$.

1997 Flanders Math Olympiad, 4

Tags: geometry , geometric transformation , reflection , algebra

Thirteen birds arrive and sit down in a plane. It's known that from each 5-tuple of birds, at least four birds sit on a circle. Determine the greatest $M \in \{1, 2, ..., 13\}$ such that from these 13 birds, at least $M$ birds sit on a circle, but not necessarily $M + 1$ birds sit on a circle. (prove that your $M$ is optimal)

2023 Moldova EGMO TST, 4

Tags: number theory , prime number , algebra , system of equations

Find all triplets of prime numbers $(m, n, p)$, that satisfy the system of equations: $$\left\{\begin{matrix} 2m-n+13p=2072,\\3m+11n+13p=2961.\end{matrix}\right.$$

2016 Mathematical Talent Reward Programme, SAQ: P 1

Tags: algebra , polynomial

Show that there exist a polynomial $P(x)$ whose one cofficient is $\frac{1}{2016}$ and remaining cofficients are rational numbers, such that $P(x)$ is an integer for any integer $x$ .

1995 Korea National Olympiad, Problem 2

Tags: function , algebra

find all functions from the nonegative integers into themselves, such that: $2f(m^2+n^2)=f^2(m)+f^2(n)$ and for $m\geq n$ $f(m^2)\geq f(n^2)$.

2017 All-Russian Olympiad, 1

Tags: algebra , parabola , conic

$f_1(x)=x^2+p_1x+q_1,f_2(x)=x^2+p_2x+q_2$ are two parabolas. $l_1$ and $l_2$ are two not parallel lines. It is knows, that segments, that cuted on the $l_1$ by parabolas are equals, and segments, that cuted on the $l_2$ by parabolas are equals too. Prove, that parabolas are equals.

2016 Czech And Slovak Olympiad III A, 4

Tags: inequalities , maximum value , maximum , algebra

For positive numbers $a, b, c$ holds $(a + c) (b^2 + a c) = 4a$. Determine the maximum value of $b + c$ and find all triplets of numbers $(a, b, c)$ for which expression takes this value

2021 South East Mathematical Olympiad, 3

Tags: number theory , algebra

Let $p$ be an odd prime and $\{u_i\}_{i\ge 0}$be an integer sequence. Let $v_n=\sum_{i=0}^{n} C_{n}^{i} p^iu_i$ where $C_n^i$ denotes the binomial coefficients. If $v_n=0$ holds for infinitely many $n$ , prove that it holds for every positive integer $n$.

2003 India IMO Training Camp, 3

Tags: function , algebra

Find all functions $f: \mathbb R \to \mathbb R$ such that for all reals $x$ and $y$, \[f(x+y)+f(x)f(y)=f(xy)+f(x)+f(y).\]

2009 Dutch IMO TST, 4

Tags: functional equation , functional equation in z , algebra

Find all functions $f : Z \to Z$ satisfying $f(m + n) + f(mn -1) = f(m)f(n) + 2$ for all $m, n \in Z$.

1996 Baltic Way, 13

Tags: function , algebra

Consider the functions $f$ defined on the set of integers such that \[f(x)=f(x^2+x+1)\] for all integer $x$. Find $(a)$ all even functions, $(b)$ all odd functions of this kind.

2021 Estonia Team Selection Test, 2

Tags: algebra , inequalities

Positive real numbers $a, b, c$ satisfy $abc = 1$. Prove that $$\frac{a}{1+b}+\frac{b}{1+c}+\frac{c}{1+a} \ge \frac32$$

2015 NIMO Summer Contest, 15

Tags: funky , system of equations , algebra

Suppose $x$ and $y$ are real numbers such that \[x^2+xy+y^2=2\qquad\text{and}\qquad x^2-y^2=\sqrt5.\] The sum of all possible distinct values of $|x|$ can be written in the form $\textstyle\sum_{i=1}^n\sqrt{a_i}$, where each of the $a_i$ is a rational number. If $\textstyle\sum_{i=1}^na_i=\frac mn$ where $m$ and $n$ are positive realtively prime integers, what is $100m+n$? [i] Proposed by David Altizio [/i]

2005 Baltic Way, 4

Tags: polynomial , algebra

Find three different polynomials $P(x)$ with real coefficients such that $P\left(x^2 + 1\right) = P(x)^2 + 1$ for all real $x$.

1992 IMO Shortlist, 6

Tags: function , algebra , functional equation

Let $\,{\mathbb{R}}\,$ denote the set of all real numbers. Find all functions $\,f: {\mathbb{R}}\rightarrow {\mathbb{R}}\,$ such that \[ f\left( x^{2}+f(y)\right) =y+\left( f(x)\right) ^{2}\hspace{0.2in}\text{for all}\,x,y\in \mathbb{R}. \]

1997 Austrian-Polish Competition, 7

Tags: inequalities , max , algebra

(a) Prove that $p^2 + q^2 + 1 > p(q + 1)$ for any real numbers $p, q$, . (b) Determine the largest real constant $b$ such that the inequality $p^2 + q^2 + 1 \ge bp(q + 1)$ holds for all real numbers $p, q$ (c) Determine the largest real constant c such that the inequality $p^2 + q^2 + 1 \ge cp(q + 1)$ holds for all integers $p, q$.

2010 China Team Selection Test, 2

Tags: algebra , polynomial , floor function , triangle inequality , inequalities

Given positive integer $n$, find the largest real number $\lambda=\lambda(n)$, such that for any degree $n$ polynomial with complex coefficients $f(x)=a_n x^n+a_{n-1} x^{n-1}+\cdots+a_0$, and any permutation $x_0,x_1,\cdots,x_n$ of $0,1,\cdots,n$, the following inequality holds $\sum_{k=0}^n|f(x_k)-f(x_{k+1})|\geq \lambda |a_n|$, where $x_{n+1}=x_0$.

1899 Eotvos Mathematical Competition, 1

Tags: algebra , geometry

The points $A_0, A_1, A_2, A_3, A_4$ divide a unit circle (circle of radius $1$) into five equal parts. Prove that the chords $A_0, A_1, A_0, A_2$ satisfy $$(A_0A_1 \cdot A_0A_2)^2= 5$$

2018-2019 Fall SDPC, 1

Tags: algebra

An isosceles triangle $T$ has the following property: it is possible to draw a line through one of the three vertices of $T$ that splits it into two smaller isosceles triangles $R$ and $S$, neither of which are similar to $T$. Find all possible values of the vertex (apex) angle of $T$.

2012 Princeton University Math Competition, A8

Tags: floor function , algebra

If $n$ is an integer such that $n \ge 2^k$ and $n < 2^{k+1}$, where $k = 1000$, compute the following: $$n - \left( \lfloor \frac{n -2^0}{2^1} \rfloor + \lfloor \frac{n -2^1}{2^2} \rfloor + ...+ \lfloor \frac{n -2^{k-1}}{2^k} \rfloor \right)$$

2008 Cuba MO, 4

Tags: algebra , functional

Determine all functions $f : R \to R$ such that $f(xy + f(x)) =xf(y) + f(x)$ for all real numbers $x, y$.

1967 IMO Shortlist, 4

Tags: parameterization , calculus , trigonometry , algebra , trigonometric identities

Find values of the parameter $u$ for which the expression \[y = \frac{ \tan(x-u) + \tan(x) + \tan(x+u)}{ \tan(x-u)\tan(x)\tan(x+u)}\] does not depend on $x.$

2019 IOM, 6

Tags: algebra , polynomial

Let $p$ be a prime and let $f(x)$ be a polynomial of degree $d$ with integer coefficients. Assume that the numbers $f(1),f(2),\dots,f(p)$ leave exactly $k$ distinct remainders when divided by $p$, and $1<k<p$. Prove that \[ \frac{p-1}{d}\leq k-1\leq (p-1)\left(1-\frac1d \right) .\] [i] Dániel Domán, Gauls Károlyi, and Emil Kiss [/i]