Olympiad problems

1998 Junior Balkan MO, 3

Find all pairs of positive integers $ (x,y)$ such that \[ x^y \equal{} y^{x \minus{} y}. \] [i]Albania[/i]

2008 Thailand Mathematical Olympiad, 4

Tags: algebra , inequalities

Prove that $$\sqrt{a^2 + b^2 -\sqrt2 ab} +\sqrt{b^2 + c^2 -\sqrt2 bc} \ge \sqrt{a^2 + c^2}$$ for all real numbers $a, b, c > 0$

2006 MOP Homework, 2

Tags: inequalities , algebra

Prove that $\frac{a}{(a + 1)(b + 1)} +\frac{ b}{(b + 1)(c + 1)} + \frac{c}{(c + 1)(a + 1)} \ge \frac34$ where $a, b$ and $c$ are positive real numbers satisfying $abc = 1$.

1992 Irish Math Olympiad, 1

Tags: algebra

Let $n > 2$ be an integer and let $m = \sum k^3$, where the sum is taken over all integers $k$ with $1 \leq k < n$ that are relatively prime to $n$. Prove that $n$ divides $m$.

1996 Romania Team Selection Test, 3

Tags: algebra , trigonometry

Let $ x,y\in \mathbb{R} $. Show that if the set $ A_{x,y}=\{ \cos {(n\pi x)}+\cos {(n\pi y)} \mid n\in \mathbb{N}\} $ is finite then $ x,y \in \mathbb{Q} $. [i]Vasile Pop[/i]

1996 South africa National Olympiad, 2

Tags: algebra , trigonometry , inequalities , function

Find all real numbers for which $3^x+4^x=5^x$.

2017 Federal Competition For Advanced Students, P2, 4

Tags: maximum , three variable inequality , rational , algebra , inequalities

(a) Determine the maximum $M$ of $x+y +z$ where $x, y$ and $z$ are positive real numbers with $16xyz = (x + y)^2(x + z)^2$. (b) Prove the existence of infinitely many triples $(x, y, z)$ of positive rational numbers that satisfy $16xyz = (x + y)^2(x + z)^2$ and $x + y + z = M$. Proposed by Karl Czakler

1997 AIME Problems, 1

Tags: difference of squares , 3d geometry , matrix , geometry , algebra , linear algebra , modular arithmetic

How many of the integers between 1 and 1000, inclusive, can be expressed as the difference of the squares of two nonnegative integers?

2023 USA IMO Team Selection Test, 6

Tags: function , algebra

Let $\mathbb{N}$ denote the set of positive integers. Fix a function $f: \mathbb{N} \rightarrow \mathbb{N}$ and for any $m,n \in \mathbb{N}$ define $$\Delta(m,n)=\underbrace{f(f(\ldots f}_{f(n)\text{ times}}(m)\ldots))-\underbrace{f(f(\ldots f}_{f(m)\text{ times}}(n)\ldots)).$$ Suppose $\Delta(m,n) \neq 0$ for any distinct $m,n \in \mathbb{N}$. Show that $\Delta$ is unbounded, meaning that for any constant $C$ there exists $m,n \in \mathbb{N}$ with $\left|\Delta(m,n)\right| > C$.

1999 Harvard-MIT Mathematics Tournament, 8

Tags: algebra

Find all the roots of $(x^2 + 3x + 2)(x^2 - 7x + 12)(x^2- 2x -1) + 24 = 0$.

2016 Saudi Arabia GMO TST, 1

Tags: sum , algebra , inequalities

Let $S = x + y +z$ where $x, y, z$ are three nonzero real numbers satisfying the following system of inequalities: $$xyz > 1$$ $$x + y + z >\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$$ Prove that $S$ can take on any real values when $x, y, z$ vary

2014 APMO, 4

Tags: combinatorics , number theory , pigeonhole principle , floor function , algebra

Let $n$ and $b$ be positive integers. We say $n$ is $b$-discerning if there exists a set consisting of $n$ different positive integers less than $b$ that has no two different subsets $U$ and $V$ such that the sum of all elements in $U$ equals the sum of all elements in $V$. (a) Prove that $8$ is $100$-discerning. (b) Prove that $9$ is not $100$-discerning. [i]Senior Problems Committee of the Australian Mathematical Olympiad Committee[/i]

2009 Putnam, B5

Tags: domain , limit , integration , geometry , function , algebra

Let $ f: (1,\infty)\to\mathbb{R}$ be a differentiable function such that \[ f'(x)\equal{}\frac{x^2\minus{}\left(f(x)\right)^2}{x^2\left(\left(f(x)\right)^2\plus{}1\right)}\quad\text{for all }x>1.\] Prove that $ \displaystyle\lim_{x\to\infty}f(x)\equal{}\infty.$

2003 India IMO Training Camp, 3

Tags: algebra , function

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

2013 Stanford Mathematics Tournament, 6

Tags: algebra , polynomial

Compute the largest root of $x^4-x^3-5x^2+2x+6$.

2016 Abels Math Contest (Norwegian MO) Final, 4

Tags: algebra , functional equation

Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that \[ f(x) f(y) = |x - y| \cdot f \left( \frac{xy + 1}{x - y} \right) \] Holds for all $x \not= y \in \mathbb{R}$

1997 German National Olympiad, 4

Tags: system of equations , algebra

Find all real solutions $(x,y,z)$ of the system of equations $$\begin{cases} x^3 = 2y-1 \\y^3 = 2z-1\\ z^3 = 2x-1\end{cases}$$

2022 BMT, 16

Tags: algebra

A street on Stanford can be modeled by a number line. Four Stanford students, located at positions $1$, $9$, $25$ and $49$ along the line, want to take an UberXL to Berkeley, but are not sure where to meet the driver. Find the smallest possible total distance walked by the students to a single position on the street. (For example, if they were to meet at position $46$, then the total distance walked by the students would be $45 + 37 + 21 + 3 = 106$, where the distances walked by the students at positions $1$, $9$, $25$ and $49$ are summed in that order.)

2005 Thailand Mathematical Olympiad, 3

Tags: algebra , functional equation , functional

Does there exist a function $f : Z^+ \to Z^+$ such that $f(f(n)) = 2n$ for all positive integers $n$? Justify your answer, and if the answer is yes, give an explicit construction.

1987 IMO Longlists, 26

Tags: inequalities , algebra

Prove that if $x, y, z$ are real numbers such that $x^2+y^2+z^2 = 2$, then \[x + y + z \leq xyz + 2.\]

1985 Traian Lălescu, 2.1

Tags: floor , equation , algebra

Solve $ \quad 5\lfloor x^2\rfloor -2\lfloor x\rfloor +2=0. $

2014 Albania Round 2, 2

Tags: arithmetic progression , sequence , algebra

Sides of a triangle form an arithmetic sequence with common difference $2$, and its area is $6 \text{ cm }^2$. Find its sides.

1999 Romania Team Selection Test, 9

Tags: reflection , geometry , trigonometry , inequalities , parallelogram , algebra , geometric transformation

Let $O,A,B,C$ be variable points in the plane such that $OA=4$, $OB=2\sqrt3$ and $OC=\sqrt {22}$. Find the maximum value of the area $ABC$. [i]Mihai Baluna[/i]

1997 Romania Team Selection Test, 1

Tags: polynomial , algebra , inequalities

Let $P(X),Q(X)$ be monic irreducible polynomials with rational coefficients. suppose that $P(X)$ and $Q(X)$ have roots $\alpha$ and $\beta$ respectively, such that $\alpha + \beta $ is rational. Prove that $P(X)^2-Q(X)^2$ has a rational root. [i]Bogdan Enescu[/i]

2023 Indonesia TST, A

Tags: algebra , inequalities

Let $a,b,c$ positive real numbers and $a+b+c = 1$. Prove that \[a^2 + b^2 + c^2 + \frac{3}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}} \ge 2(ab + bc + ac)\]