Olympiad problems

2003 Greece JBMO TST, 2

Tags: algebra , sum

Calculate if $n\in N$ with $n>2$ the value of $$B=\sqrt{1+\frac{1}{2^2}+\frac{1}{3^2}}+\sqrt{1+\frac{1}{3^2}+\frac{1}{4^2}}+...+\sqrt{1+\frac{1}{(n-1)^2}+\frac{1}{n^2}} $$

2012 China Team Selection Test, 2

Tags: floor function , function , pigeonhole principle , difference of squares , inequalities , algebra , ceiling function

Prove that there exists a positive real number $C$ with the following property: for any integer $n\ge 2$ and any subset $X$ of the set $\{1,2,\ldots,n\}$ such that $|X|\ge 2$, there exist $x,y,z,w \in X$(not necessarily distinct) such that \[0<|xy-zw|<C\alpha ^{-4}\] where $\alpha =\frac{|X|}{n}$.

1986 Dutch Mathematical Olympiad, 1

Tags: functional , algebra

$f(x) = \frac{12x+9}{19x+86}, \,\, x \ne -\frac{86}{19}$ Prove that $\exists ! \,\,\, {x_o \in R} \,\,\, \forall h_1,h_2 \in R [f(x_0+h_1)f(x_0-h_1)=f(x_0+h_2)f(x_0-h_2)]$, and calculate $x_0$.

2013 Turkey Team Selection Test, 1

Tags: greatest common divisor , modular arithmetic , number theory , polynomial , algebra

Find all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$.

2001 China Team Selection Test, 3

Tags: calculus , algebra , polynomial , function , minimization

Let $F = \max_{1 \leq x \leq 3} |x^3 - ax^2 - bx - c|$. When $a$, $b$, $c$ run over all the real numbers, find the smallest possible value of $F$.

1955 Moscow Mathematical Olympiad, 316

Tags: polynomial , algebra , divisor , integer polynomial , rational

Prove that if $\frac{p}{q}$ is an irreducible rational number that serves as a root of the polynomial $f(x) = a_0x^n + a_1x^{n-1} + ... + a_n$ with integer coefficients, then $p - kq$ is a divisor of $f(k)$ for any integer $k$.

2013 Bosnia and Herzegovina Junior BMO TST, 2

Tags: inequalities , inequality , positive real , algebra

Let $a$, $b$ and $c$ be positive real numbers such that $a^2+b^2+c^2=3$. Prove the following inequality: $\frac{a}{3c(a^2-ab+b^2)} + \frac{b}{3a(b^2-bc+c^2)} + \frac{c}{3b(c^2-ca+a^2)} \leq \frac{1}{abc}$

2012-2013 SDML (High School), 8

Tags: algebra , polynomial

A polynomial $P$ with degree exactly $3$ satisfies $P\left(0\right)=1$, $P\left(1\right)=3$, and $P\left(3\right)=10$. Which of these cannot be the value of $P\left(2\right)$? $\text{(A) }2\qquad\text{(B) }3\qquad\text{(C) }4\qquad\text{(D) }5\qquad\text{(E) }6$

1995 China National Olympiad, 2

Tags: functional equation , algebra , function

Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function satisfying the following conditions: (1) $f(1)=1$; (2) $\forall n\in \mathbb{N}$, $3f(n) f(2n+1) =f(2n) ( 1+3f(n) )$; (3) $\forall n\in \mathbb{N}$, $f(2n) < 6 f(n)$. Find all solutions of equation $f(k) +f(l)=293$, where $k<l$. ($\mathbb{N}$ denotes the set of all natural numbers).

PEN C Problems, 6

Tags: algebra , quadratic residue , absolute value , polynomial

Let $a, b, c$ be integers and let $p$ be an odd prime with \[p \not\vert a \;\; \text{and}\;\; p \not\vert b^{2}-4ac.\] Show that \[\sum_{k=1}^{p}\left( \frac{ak^{2}+bk+c}{p}\right) =-\left( \frac{a}{p}\right).\]

2002 AMC 12/AHSME, 12

Tags: algebra , polynomial , quadratic

For how many positive integers $n$ is $n^3-8n^2+20n-13$ a prime number? $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }\text{more than 4}$

II Soros Olympiad 1995 - 96 (Russia), 10.3

Tags: algebra

Solve the equation $$(12x-1)(6x-1)(4x-1)(3x -1) = 5.$$

2005 Morocco TST, 1

Tags: limit , calculus , function , derivative , induction , 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 $x,y \in \mathbb R$

1988 Austrian-Polish Competition, 4

Tags: algebra , functional equation , functional , increasing

Determine all strictly increasing functions $f: R \to R$ satisfying $f (f(x) + y) = f(x + y) + f (0)$ for all $x,y \in R$.

2017 Romania EGMO TST, P3

Tags: functional equation , function , algebra

Determine all functions $f:\mathbb R\to\mathbb R$ such that \[f(xy-1)+f(x)f(y)=2xy-1,\]for any real numbers $x{}$ and $y{}.$

2008 Nordic, 1

Tags: algebra , function

Find all reals $A,B,C$ such that there exists a real function $f$ satisfying $f(x+f(y))= Ax+By+C$ for all reals $x,y$.

1985 Brazil National Olympiad, 5

Tags: equation , algebra

$A, B$ are reals. Find a necessary and sufficient condition for $Ax + B[x] = Ay + B[y]$ to have no solutions except $x = y$.

2023 LMT Spring, 1

Tags: algebra , number theory

Given the following system of equations: $$\begin{cases} R I +G +SP = 50 \\ R I +T + M = 63 \\ G +T +SP = 25 \\ SP + M = 13 \\ M +R I = 48 \\ N = 1 \end{cases}$$ Find the value of L that makes $LMT +SPR I NG = 2023$ true.

2012 Online Math Open Problems, 25

Tags: algebra , function , vieta , polynomial

Let $a,b,c$ be the roots of the cubic $x^3 + 3x^2 + 5x + 7$. Given that $P$ is a cubic polynomial such that $P(a)=b+c$, $P(b) = c+a$, $P(c) = a+b$, and $P(a+b+c) = -16$, find $P(0)$. [i]Author: Alex Zhu[/i]

MathLinks Contest 4th, 6.3

Tags: algebra , inequalities

If $n>2$ is an integer and $x_1, \ldots ,x_n$ are positive reals such that \[ \frac 1{x_1} + \frac 1{x_2} + \cdots + \frac 1{x_n} = n \] then the following inequality takes place \[ \frac{x_2^2+\cdots+x_n^2}{n-1}\cdot \frac {x_1^2+x_3^2+\cdots +x_n^2} {n-1} \cdots \frac{x_1^2+\cdots+x_{n-1}^2}{n-1}\geq \left(\frac{x_1^2+...+x_n^2}{n}\right)^{n-1}. \]

2013 South East Mathematical Olympiad, 1

Tags: algebra , cubic function , root , inequalities

Let $a,b$ be real numbers such that the equation $x^3-ax^2+bx-a=0$ has three positive real roots . Find the minimum of $\frac{2a^3-3ab+3a}{b+1}$.

2024 Kyiv City MO Round 1, Problem 1

Tags: mean , algebra

Four positive integers $a, b, c, d$ satisfy the condition: $a < b < c < d$. For what smallest possible value of $d$ could the following condition be true: the arithmetic mean of numbers $a, b, c$ is twice smaller than the arithmetic mean of numbers $a, b, c, d$?