Olympiad problems

Mathley 2014-15, 4

Tags: algebra , inequalities

Let $S_k$ be the set of all triplets of real numbers $(a, b, c)$ satisfying $a <k (b + c)$, $b <k (c + a)$, and $c <k (a + b)$. For what value of $k$ then $S_k$ is a subset of $\{(a, b, c) | ab + bc + ca> 0\}$ ? Michel Bataille, France

2005 AIME Problems, 8

Tags: quadratic , geometry , trapezoid , number theory , relatively prime , pythagorean theorem , algebra

Circles $C_1$ and $C_2$ are externally tangent, and they are both internally tangent to circle $C_3$. The radii of $C_1$ and $C_2$ are $4$ and $10$, respectively, and the centers of the three circles are all collinear. A chord of $C_3$ is also a common external tangent of $C_1$ and $C_2$. Given that the length of the chord is $\frac{m\sqrt{n}}{p}$ where $m,n,$ and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime, find $m+n+p$.

India EGMO 2021 TST, 2

Tags: algebra , inequality , 4-variable inequality , inequalities

Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

2008 IMO Shortlist, 4

Tags: function , algebra , functional equation , inequality

For an integer $ m$, denote by $ t(m)$ the unique number in $ \{1, 2, 3\}$ such that $ m \plus{} t(m)$ is a multiple of $ 3$. A function $ f: \mathbb{Z}\to\mathbb{Z}$ satisfies $ f( \minus{} 1) \equal{} 0$, $ f(0) \equal{} 1$, $ f(1) \equal{} \minus{} 1$ and $ f\left(2^{n} \plus{} m\right) \equal{} f\left(2^n \minus{} t(m)\right) \minus{} f(m)$ for all integers $ m$, $ n\ge 0$ with $ 2^n > m$. Prove that $ f(3p)\ge 0$ holds for all integers $ p\ge 0$. [i]Proposed by Gerhard Woeginger, Austria[/i]

2010 Postal Coaching, 4

Tags: function , domain , vector , circumcircle , geometry , algebra

Five distinct points $A, B, C, D$ and $E$ lie in this order on a circle of radius $r$ and satisfy $AC = BD = CE = r$. Prove that the orthocentres of the triangles $ACD, BCD$ and $BCE$ are the vertices of a right-angled triangle.

2013 ELMO Shortlist, 2

Tags: algebra , polynomial , 3d geometry , modular arithmetic , number theory , geometry

For what polynomials $P(n)$ with integer coefficients can a positive integer be assigned to every lattice point in $\mathbb{R}^3$ so that for every integer $n \ge 1$, the sum of the $n^3$ integers assigned to any $n \times n \times n$ grid of lattice points is divisible by $P(n)$? [i]Proposed by Andre Arslan[/i]

2017 Federal Competition For Advanced Students, 1

Tags: polynomial , algebra

Determine all polynomials $P(x) \in \mathbb R[x]$ satisfying the following two conditions : (a) $P(2017) = 2016$ and (b) $(P(x) + 1)^2 = P(x^2 + 1)$ for all real numbers $x$. [i]proposed by Walther Janous[/i]

2018 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: inequalities , algebra , logarithm

If numbers $x_1$, $x_2$,...,$x_n$ are from interval $\left( \frac{1}{4},1 \right)$ prove the inequality: $\log _{x_1} {\left(x_2-\frac{1}{4} \right)} + \log _{x_2} {\left(x_3-\frac{1}{4} \right)}+ ... + \log _{x_{n-1}} {\left(x_n-\frac{1}{4} \right)} + \log _{x_n} {\left(x_1-\frac{1}{4} \right)} \geq 2n$

BIMO 2022, 3

Tags: functional equation , algebra

Find all functions $ f : \mathbb{R} \rightarrow \mathbb{R} $ such that for all reals $ x, y $,$$ f(x^2+f(x+y))=y+xf(x+1) $$

2011 Junior Balkan Team Selection Tests - Romania, 4

Tags: rational , sum , algebra

Let $k$ and $n$ be integer numbers with $2 \le k \le n - 1$. Consider a set $A$ of $n$ real numbers such that the sum of any $k$ distinct elements of $A$ is a rational number. Prove that all elements of the set $A$ are rational numbers.

MathLinks Contest 2nd, 6.1

Tags: algebra , floor function , function

Determine the parity of the positive integer $N$, where $$N = \lfloor \frac{2002!}{2001 \cdot2003} \rfloor.$$

1994 Tournament Of Towns, (433) 3

Tags: algebra , system of equations

Let $a, b, c$ and $d$ be real numbers such that $$a^3+b^3+c^3+d^3=a+b+c+d=0$$ Prove that the sum of a pair of these numbers is equal to $0$. (LD Kurliandchik)

2018 India IMO Training Camp, 3

Tags: algebra , functional equation

Find all functions $f: \mathbb{R} \mapsto \mathbb{R}$ such that $$f(x)f\left(yf(x)-1\right)=x^2f(y)-f(x),$$for all $x,y \in \mathbb{R}$.

2012 ELMO Shortlist, 8

Tags: binomial coefficient , algebra , polynomial , ceiling function , inequalities , absolute value , geometric series

Fix two positive integers $a,k\ge2$, and let $f\in\mathbb{Z}[x]$ be a nonconstant polynomial. Suppose that for all sufficiently large positive integers $n$, there exists a rational number $x$ satisfying $f(x)=f(a^n)^k$. Prove that there exists a polynomial $g\in\mathbb{Q}[x]$ such that $f(g(x))=f(x)^k$ for all real $x$. [i]Victor Wang.[/i]

2001 China Team Selection Test, 2

Tags: algebra , combinatorics , number theory

$a$ and $b$ are natural numbers such that $b > a > 1$, and $a$ does not divide $b$. The sequence of natural numbers $\{b_n\}_{n=1}^\infty$ satisfies $b_{n + 1} \geq 2b_n \forall n \in \mathbb{N}$. Does there exist a sequence $\{a_n\}_{n=1}^\infty$ of natural numbers such that for all $n \in \mathbb{N}$, $a_{n + 1} - a_n \in \{a, b\}$, and for all $m, l \in \mathbb{N}$ ($m$ may be equal to $l$), $a_m + a_l \not\in \{b_n\}_{n=1}^\infty$?

2007 Stanford Mathematics Tournament, 11

Tags: algebra , polynomial

The polynomial $R(x)$ is the remainder upon dividing $x^{2007}$ by $x^2-5x+6$. $R(0)$ can be expressed as $ab(a^c-b^c)$. Find $a+c-b$.

1994 Irish Math Olympiad, 1

Tags: induction , algebra

A sequence $ (x_n)$ is given by $ x_1\equal{}2$ and $ nx_n\equal{}2(2n\minus{}1)x_{n\minus{}1}$ for $ n>1$. Prove that $ x_n$ is an integer for every $ n \in \mathbb{N}$.

2025 All-Russian Olympiad, 9.5