Olympiad problems

VII Soros Olympiad 2000 - 01, 10.2

Let $a$ and $ b$ be acute corners. Prove that if $\sin a$, $\sin b$, and $\sin (a + b)$ are rational numbers, then $\cos a$, $\cos b$, and $\cos (a + b)$ are also rational numbers.

2021 China Second Round Olympiad, Problem 11

Tags: algebra , polynomial , function , arithmetic sequence

The function $f(x) = x^2+ax+b$ has two distinct zeros. If $f(x^2+2x-1)=0$ has four distinct zeros $x_1<x_2<x_3<x_4$ that form an arithmetic sequence, compute the range of $a-b$. [i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 11)[/i]

2012 Pre - Vietnam Mathematical Olympiad, 2

Tags: polynomial , algebra

Let $(a_n)$ defined by: $a_0=1, \; a_1=p, \; a_2=p(p-1)$, $a_{n+3}=pa_{n+2}-pa_{n+1}+a_n, \; \forall n \in \mathbb{N}$. Knowing that (i) $a_n>0, \; \forall n \in \mathbb{N}$. (ii) $a_ma_n>a_{m+1}a_{n-1}, \; \forall m \ge n \ge 0$. Prove that $|p-1| \ge 2$.

2015 Princeton University Math Competition, B2

Tags: algebra

Let $f$ be a function which takes in $0, 1, 2$ and returns $0, 1, $ or $2$. The values need not be distinct: for instance we could have $f(0) = 1, f(1) = 1, f(2) = 2$. How many such functions are there which satisfy \[f(2) + f(f(0)) + f(f(f(1))) = 5?\]

2017 South East Mathematical Olympiad, 4

Tags: algebra , maximum value

Let $a_1,a_2,\dots,a_{2017}$ be reals satisfied $a_1=a_{2017}$, $|a_i+a_{i+2}-2a_{i+1}|\le 1$ for all $i=1,2,\dots,2015$. Find the maximum value of $\max_{1\le i<j\le 2017}|a_i-a_j|$.

1996 Tuymaada Olympiad, 1

Tags: algebra , inequalities

Prove the inequality $x_1y_1+x_2y_2+x_2y_1+2x_2y_2\le 1996$ if $x_1^2+2x_1x_2+2x_2^2\le 998$ and $y_1^2+2y_1y_2+2y_2^2\le 3992$.

2021 Peru PAGMO TST, P6

Tags: algebra , function

Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for any real numbers $x$ and $y$ the following is true: $$x^2+y^2+2f(xy)=f(x+y)(f(x)+f(y))$$

2003 Bulgaria Team Selection Test, 2

Tags: functional equation , algebra

Find all $f:R-R$ such that $f(x^2+y+f(y))=2y+f(x)^2$

2002 AMC 10, 25

Tags: algebra , system of equations

When $ 15$ is appended to a list of integers, the mean is increased by $ 2$. When $ 1$ is appended to the enlarged list, the mean of the enlarged list is decreased by $ 1$. How many integers were in the original list? $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8$

1991 Austrian-Polish Competition, 7

Tags: function , max , algebra

For a given positive integer $n$ determine the maximum value of the function $f (x) = \frac{x + x^2 +...+ x^{2n-1}}{(1 + x^n)^2}$ over all $x \ge 0$ and find all positive $x$ for which the maximum is attained.

2008 Indonesia TST, 1

Tags: algebra , polynomial

A polynomial $P(x) = 1 + x^2 + x^5 + x^{n_1} + ...+ x^{n_s} + x^{2008}$ with $n_1, ..., n_s$ are positive integers and $5 < n_1 < ... <n_s < 2008$ are given. Prove that if $P(x)$ has at least a real root, then the root is not greater than $\frac{1-\sqrt5}{2}$

2022 Switzerland Team Selection Test, 7

Tags: algebra , polynomial , real coefficients

Let $n$ be a positive integer. Find all polynomials $P$ with real coefficients such that $$P(x^2+x-n^2)=P(x)^2+P(x)$$ for all real numbers $x$.

2011 Indonesia TST, 1

Tags: inequalities , geometric inequalities , algebra

Let $a, b, c$ be the sides of a triangle with $abc = 1$. Prove that $$\frac{\sqrt{b + c -a}}{a}+\frac{\sqrt{c + a - b}}{b}+\frac{\sqrt{a + b - c}}{c} \ge a + b + c$$

2016 South African National Olympiad, 2

Tags: quadratic equation , algebra

Determine all pairs of real numbers $a$ and $b$, $b > 0$, such that the solutions to the two equations $$x^2 + ax + a = b \qquad \text{and} \qquad x^2 + ax + a = -b$$ are four consecutive integers.

1953 AMC 12/AHSME, 4

Tags: algebra , polynomial

The roots of $ x(x^2\plus{}8x\plus{}16)(4\minus{}x)\equal{}0$ are: $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 0,4 \qquad\textbf{(C)}\ 0,4,\minus{}4 \qquad\textbf{(D)}\ 0,4,\minus{}4,\minus{}4 \qquad\textbf{(E)}\ \text{none of these}$

2006 Peru IMO TST, 2

Tags: function , algebra

[color=blue][size=150]PERU TST IMO - 2006[/size] Saturday, may 20.[/color] [b]Question 02[/b] Find all pairs $(a,b)$ real positive numbers $a$ and $b$ such that: $[a[bn]]= n-1,$ for all $n$ positive integer. Note: [x] denotes the integer part of $x$. ---------- [url=http://www.mathlinks.ro/Forum/viewtopic.php?t=88510]Spanish version[/url] $\text{\LaTeX}{}$ed by carlosbr

2021 LMT Spring, B10

Tags: algebra

Let $f (x)$ be a function mapping real numbers to real numbers. Given that $f (f (x)) =\frac{1}{3x}$, and $f (2) =\frac19$, find $ f\left(\frac{1}{6}\right)$. [i]Proposed by Zachary Perry[/i]

2018 Irish Math Olympiad, 6

Tags: algebra , functional equation

Find all real-valued functions $f$ satisfying $f(2x + f(y)) + f(f(y)) = 4x + 8y$ for all real numbers $x$ and $y$.

III Soros Olympiad 1996 - 97 (Russia), 11.8

Tags: algebra , trigonometry , system of equations

Solve the system of equations: $$ 2(3-2\cos y)^2+2(4-2\sin y)^2=2(3-x)^2+32=(x-2\cos y)^2+4\sin^2y$$

2021 Mediterranean Mathematics Olympiad, 1

Tags: polynomial , integer polynomial , algebra

Determine the smallest positive integer $M$ with the following property: For every choice of integers $a,b,c$, there exists a polynomial $P(x)$ with integer coefficients so that $P(1)=aM$ and $P(2)=bM$ and $P(4)=cM$. [i]Proposed by Gerhard Woeginger, Austria[/i]

2012 Ukraine Team Selection Test, 1

Tags: algebra , inequalities

Let $a, b, c$ be positive reals. Prove that $\sqrt{2a^2+bc}+\sqrt{2b^2+ac}+\sqrt{2c^2+ab}\ge 3 \sqrt{ab+bc+ca}$

2003 France Team Selection Test, 3

Tags: algebra , number theory , divisor , prime

Let $p_1,p_2,\ldots,p_n$ be distinct primes greater than $3$. Show that $2^{p_1p_2\cdots p_n}+1$ has at least $4^n$ divisors.

2015 Turkmenistan National Math Olympiad, 3

Tags: algebra

Find the sum : $C^{n}_{1}$ - $\frac{1}{3} \cdot C^{n}_{3}$ + $\frac{1}{9} \cdot C^{n}_{5}$ - $\frac{1}{27} \cdot C^{n}_{9}$ + ...

2014 IMO Shortlist, C4

Tags: combinatorics , algebra , prime number

Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them $S$- and $Z$-tetrominoes, respectively. Assume that a lattice polygon $P$ can be tiled with $S$-tetrominoes. Prove that no matter how we tile $P$ using only $S$- and $Z$-tetrominoes, we always use an even number of $Z$-tetrominoes. [i]Proposed by Tamas Fleiner and Peter Pal Pach, Hungary[/i]

2002 Stanford Mathematics Tournament, 1

Tags: algebra , polynomial

Completely factor the polynomial $x^4-x^3-5x^2+3x+6$