Olympiad problems

2005 Thailand Mathematical Olympiad, 20

Let $a, b, c, d > 0$ satisfy $36a + 4b + 4c + 3d = 25$. What is the maximum possible value of $ab^{1/2}c^{1/3}d^{1/4}$ ?

2015 South East Mathematical Olympiad, 8

Tags: algebra , number theory

Find all prime number $p$ such that there exists an integer-coefficient polynomial $f(x)=x^{p-1}+a_{p-2}x^{p-2}+…+a_1x+a_0$ that has $p-1$ consecutive positive integer roots and $p^2\mid f(i)f(-i)$, where $i$ is the imaginary unit.

1986 IMO Longlists, 76

Tags: number theory , diophantine equation , geometry , pythagorean theorem , algebra

Let $A, B$, and $C$ be three points on the edge of a circular chord such that $B$ is due west of $C$ and $ABC$ is an equilateral triangle whose side is $86$ meters long. A boy swam from $A$ directly toward $B$. After covering a distance of $x$ meters, he turned and swam westward, reaching the shore after covering a distance of $y$ meters. If $x$ and $y$ are both positive integers, determine $y.$

2012 ELMO Shortlist, 4

Tags: floor function , induction , algebra

Let $a_0,b_0$ be positive integers, and define $a_{i+1}=a_i+\lfloor\sqrt{b_i}\rfloor$ and $b_{i+1}=b_i+\lfloor\sqrt{a_i}\rfloor$ for all $i\ge0$. Show that there exists a positive integer $n$ such that $a_n=b_n$. [i]David Yang.[/i]

1974 IMO Shortlist, 3

Tags: coefficient , algebra , polynomial , number theory , degree

Let $P(x)$ be a polynomial with integer coefficients. We denote $\deg(P)$ its degree which is $\geq 1.$ Let $n(P)$ be the number of all the integers $k$ for which we have $(P(k))^{2}=1.$ Prove that $n(P)- \deg(P) \leq 2.$

2018 Azerbaijan BMO TST, 1

Tags: inequalities , algebra

Problem Shortlist BMO 2017 Let $ a $,$ b$,$ c$, be positive real numbers such that $abc= 1 $. Prove that $$\frac{1}{a^{5}+b^{5}+c^{2}}+\frac{1}{b^{5}+c^{5}+a^{2}}+\frac{1}{c^{5}+b^{5}+b^{2}}\leq 1 . $$

2006 MOP Homework, 4

Tags: function , algebra , increasing function

Assume that $f : [0,1)\to R$ is a function such that $f(x)-x^3$ and $f(x)-3x$ are both increasing functions. Determine if $f(x)-x^2-x$ is also an increasing function.

2018 India IMO Training Camp, 2

Tags: function , algebra

Let $S$ be a finite set, and let $\mathcal{A}$ be the set of all functions from $S$ to $S$. Let $f$ be an element of $\mathcal{A}$, and let $T=f(S)$ be the image of $S$ under $f$. Suppose that $f\circ g\circ f\ne g\circ f\circ g$ for every $g$ in $\mathcal{A}$ with $g\ne f$. Show that $f(T)=T$.

1986 Bulgaria National Olympiad, Problem 6

Tags: inequalities , recurrence relation , algebra , sequence

Let $0<k<1$ be a given real number and let $(a_n)_{n\ge1}$ be an infinite sequence of real numbers which satisfies $a_{n+1}\le\left(1+\frac kn\right)a_n-1$. Prove that there is an index $t$ such that $a_t<0$.

2012 Albania Team Selection Test, 5

Tags: function , algebra , functional equation

Let $f:\mathbb R^+ \to \mathbb R^+$ be a function such that: \[ x,y > 0 \qquad f(x+f(y)) = yf(xy+1). \] a) Show that $(y-1)*(f(y)-1) \le 0$ for $y>0$. b) Find all such functions that require the given condition.

2005 India IMO Training Camp, 2

Tags: function , number theory , algebra , divisibility , functional equation

Find all functions $ f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying \[ \left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}\] for any two positive integers $ m$ and $ n$. [i]Remark.[/i] The abbreviation $ \mathbb{N^{*}}$ stands for the set of all positive integers: $ \mathbb{N^{*}}=\left\{1,2,3,...\right\}$. By $ f^{2}\left(m\right)$, we mean $ \left(f\left(m\right)\right)^{2}$ (and not $ f\left(f\left(m\right)\right)$). [i]Proposed by Mohsen Jamali, Iran[/i]

1997 German National Olympiad, 4

Tags: system of equations , algebra

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

2011 Saudi Arabia Pre-TST, 4.3

Tags: algebra , inequalities

Let $x_1,x_2,...,x_n$ be positive real numbers for which $$\frac{1}{1+x_1}+\frac{1}{1+x_2}+...+\frac{1}{1+x_n}=1$$ Prove that $x_1x_2...x_n \ge (n -1)^n$.

2002 Poland - Second Round, 1

Tags: function , algebra

Prove that all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying, for all real $x$, \[ f(x)=f(2x)=f(1-x)\] are periodic.

2012 Kyoto University Entry Examination, 3

Tags: function , geometry , geometric transformation , rotation , algebra

When real numbers $x,\ y$ moves in the constraint with $x^2+xy+y^2=6.$ Find the range of $x^2y+xy^2-x^2-2xy-y^2+x+y.$ 30 points

2022 Mexican Girls' Contest, 5

Tags: algebra

A biologist found a pond with frogs. When classifying them by their mass, he noticed the following: [i]The $50$ lightest frogs represented $30\%$ of the total mass of all the frogs in the pond, while the $44$ heaviest frogs represented $27\%$ of the total mass. [/i]As fate would have it, the frogs escaped and the biologist only has the above information. How many frogs were in the pond?

2017 Pan-African Shortlist, N2

Tags: number theory , prime number , sum of cubes , algebra , factorization

For which prime numbers $p$ can we find three positive integers $n$, $x$ and $y$ such that $p^n = x^3 + y^3$?

2006 District Olympiad, 3

Tags: algebra , polynomial

Prove that there exists an infinity of irrational numbers $x,y$ such that the number $x+y=xy$ is a nonnegative integer.

2023 Korea Summer Program Practice Test, P2

Tags: functional equation , function , algebra

Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that $$f(f(x)^2 + |y|) = x^2 + f(y)$$

2015 CHMMC (Fall), 4

Tags: number theory , algebra

Let $P(x) = x^{16}-x^{15}+·...-x+ 1$, and let p be a prime such that $p-1$ is divisible by $34$ ($p = 103$ is an example). How many integers a between $1$ and $ p-1$ inclusive satisfy the property that $P(a)$ is divisible by $p$?

2018 Taiwan TST Round 2, 2

Tags: algebra

Find all functions $ f: \mathbb{Z} \to \mathbb{Z} $ such that $$ f\left(x+f\left(y\right)\right)f\left(y+f\left(x\right)\right)=\left(2x+f\left(y-x\right)\right)\left(2y+f\left(x-y\right)\right) $$ holds for all integers $ x,y $

2004 All-Russian Olympiad Regional Round, 10.5

Tags: algebra , polynomial , coprime , integer polynomial

Equation $$x^n + a_1x^{n-1} + a_2x^{n-2} +...+ a_{n-1}x + a_n = 0$$ with integer non-zero coefficients $a_1$, $a_2$, $...$ , $a_n$ has $n$ different integer roots. Prove that if any two roots are relatively prime, then the numbers $a_{n-1}$ and $a_n$ are coprime.

2011 Romanian Masters In Mathematics, 2

Tags: algebra , polynomial , modular arithmetic , function , number theory , greatest common divisor , system of equations

Determine all positive integers $n$ for which there exists a polynomial $f(x)$ with real coefficients, with the following properties: (1) for each integer $k$, the number $f(k)$ is an integer if and only if $k$ is not divisible by $n$; (2) the degree of $f$ is less than $n$. [i](Hungary) Géza Kós[/i]

ICMC 5, 2

Tags: algebra , analysis , calculus

Evaluate \[\frac{1/2}{1+\sqrt2}+\frac{1/4}{1+\sqrt[4]2}+\frac{1/8}{1+\sqrt[8]2}+\frac{1/16}{1+\sqrt[16]2}+\cdots\] [i]Proposed by Ethan Tan[/i]

2017 AMC 12/AHSME, 21

Tags: algebra , polynomial

A set $S$ is constructed as follows. To begin, $S=\{0,10\}$. Repeatedly, as long as possible, if $x$ is an integer root of some polynomial $a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ for some $n\geq 1$, all of whose coefficients $a_i$ are elements of $S$, then $x$ is put into $S$. When no more elements can be added to $S$, how many elements does $S$ have? $\textbf{(A) } 4 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 7 \qquad \textbf{(D) } 9 \qquad \textbf{(E) } 11$