Olympiad problems

1999 Baltic Way, 1

Determine all real numbers $a,b,c,d$ that satisfy the following equations \[\begin{cases} abc + ab + bc + ca + a + b + c = 1\\ bcd + bc + cd + db + b + c + d = 9\\ cda + cd + da + ac + c + d + a = 9\\ dab + da + ab + bd + d + a + b = 9\end{cases}\]

2012 ELMO Shortlist, 8

Tags: function , modular arithmetic , induction , algebra , functional equation , algebra proposed

Find all functions $f : \mathbb{Q} \to \mathbb{R}$ such that $f(x)f(y)f(x+y) = f(xy)(f(x) + f(y))$ for all $x,y\in\mathbb{Q}$. [i]Sammy Luo and Alex Zhu.[/i]

2010 ELMO Shortlist, 4

Tags: trigonometry , floor function , algebra proposed , algebra

Let $-2 < x_1 < 2$ be a real number and define $x_2, x_3, \ldots$ by $x_{n+1} = x_n^2-2$ for $n \geq 1$. Assume that no $x_n$ is $0$ and define a number $A$, $0 \leq A \leq 1$ in the following way: The $n^{\text{th}}$ digit after the decimal point in the binary representation of $A$ is a $0$ if $x_1x_2\cdots x_n$ is positive and $1$ otherwise. Prove that $A = \frac{1}{\pi}\cos^{-1}\left(\frac{x_1}{2}\right)$. [i]Evan O' Dorney.[/i]

2010 Romanian Masters In Mathematics, 2

Tags: function , inequalities , Cauchy Inequality , absolute value , algebra proposed , algebra

For each positive integer $n$, find the largest real number $C_n$ with the following property. Given any $n$ real-valued functions $f_1(x), f_2(x), \cdots, f_n(x)$ defined on the closed interval $0 \le x \le 1$, one can find numbers $x_1, x_2, \cdots x_n$, such that $0 \le x_i \le 1$ satisfying \[|f_1(x_1)+f_2(x_2)+\cdots f_n(x_n)-x_1x_2\cdots x_n| \ge C_n\] [i]Marko Radovanović, Serbia[/i]

2011 Paraguay Mathematical Olympiad, 4

Tags: algebra proposed , algebra

A positive integer $N$ is divided in $n$ parts inversely proportional to the numbers $2, 6, 12, 20, \ldots$ The smallest part is equal to $\frac{1}{400} N$. Find the value of $n$.

2001 Iran MO (3rd Round), 2

Tags: search , algebra proposed , algebra

Does there exist a sequence $ \{b_{i}\}_{i=1}^\infty$ of positive real numbers such that for each natural $ m$: \[ b_{m}+b_{2m}+b_{3m}+\dots=\frac1m\]

2024 Polish MO Finals, 4

Tags: algebra , system of equations , parameterization , algebra proposed

Do there exist real numbers $a,b,c$ such that the system of equations \begin{align*} x+y+z&=a\\ x^2+y^2+z^2&=b\\ x^4+y^4+z^4&=c \end{align*} has infinitely many real solutions $(x,y,z)$?

2008 Romania National Olympiad, 1

Tags: function , induction , algebra proposed , algebra

Find functions $ f: \mathbb{N} \rightarrow \mathbb{N}$, such that $ f(x^2 \plus{} f(y)) \equal{} xf(x) \plus{} y$, for $ x,y \in \mathbb{N}$.

1997 Pre-Preparation Course Examination, 1

Tags: function , algebra proposed , algebra

Let $f: \mathbb R \to\mathbb R$ be a function such that $|f(x)| \leq 1$ for all $x \in \mathbb R$ and \[f \left( x + \frac{13}{42} \right) + f(x) = f \left( x + \frac 17 \right) + f \left( x + \frac 16 \right), \quad \forall x \in \mathbb R.\] Show that $f$ is a periodic function.

2002 ITAMO, 4

Tags: algebra proposed , algebra

Find all values of $n$ for which all solutions of the equation $x^3-3x+n=0$ are integers.

2022 German National Olympiad, 6

Tags: function , algebra , algebra proposed , functional equation , Functional inequality , functional inequalities

Consider functions $f$ satisfying the following four conditions: (1) $f$ is real-valued and defined for all real numbers. (2) For any two real numbers $x$ and $y$ we have $f(xy)=f(x)f(y)$. (3) For any two real numbers $x$ and $y$ we have $f(x+y) \le 2(f(x)+f(y))$. (4) We have $f(2)=4$. Prove that: a) There is a function $f$ with $f(3)=9$ satisfying the four conditions. b) For any function $f$ satisfying the four conditions, we have $f(3) \le 9$.

2010 Contests, 1

Tags: algebra , polynomial , function , algebra proposed

The real numbers $a$, $b$, $c$, $d$ satisfy simultaneously the equations \[abc -d = 1, \ \ \ bcd - a = 2, \ \ \ cda- b = 3, \ \ \ dab - c = -6.\] Prove that $a + b + c + d \not = 0$.

1987 IMO Longlists, 73

Tags: function , calculus , derivative , algebra proposed , algebra

Let $f(x)$ be a periodic function of period $T > 0$ defined over $\mathbb R$. Its first derivative is continuous on $\mathbb R$. Prove that there exist $x, y \in [0, T )$ such that $x \neq y$ and \[f(x)f'(y)=f'(x)f(y).\]

2002 National Olympiad First Round, 28

Tags: algebra , polynomial , algebra proposed

How many positive roots does polynomial $x^{2002} + a_{2001}x^{2001} + a_{2000}x^{2000} + \cdots + a_1x + a_0$ have such that $a_{2001} = 2002$ and $a_k = -k - 1$ for $0\leq k \leq 2000$? $ \textbf{a)}\ 0 \qquad\textbf{b)}\ 1 \qquad\textbf{c)}\ 2 \qquad\textbf{d)}\ 1001 \qquad\textbf{e)}\ 2002 $

2010 Iran MO (3rd Round), 4

Tags: group theory , abstract algebra , vector , algebra proposed , algebra

a) prove that every discrete subgroup of $(\mathbb R^2,+)$ is in one of these forms: i-$\{0\}$. ii-$\{mv|m\in \mathbb Z\}$ for a vector $v$ in $\mathbb R^2$. iii-$\{mv+nw|m,n\in \mathbb Z\}$ for tho linearly independent vectors $v$ and $w$ in $\mathbb R^2$.(lattice $L$) b) prove that every finite group of symmetries that fixes the origin and the lattice $L$ is in one of these forms: $\mathcal C_i$ or $\mathcal D_i$ that $i=1,2,3,4,6$ ($\mathcal C_i$ is the cyclic group of order $i$ and $\mathcal D_i$ is the dyhedral group of order $i$).(20 points)

2009 Canadian Mathematical Olympiad Qualification Repechage, 1

Tags: algebra proposed , algebra

Determine all solutions to the system of equations \begin{align*}x+y+z=2 \\ x^2-y^2-z^2=2 \\ x-3y^2+z=0\end{align*}

2019 IMEO, 4

Tags: polynomial , prime numbers , algebra , algebra proposed

Call a two-element subset of $\mathbb{N}$ [i]cute[/i] if it contains exactly one prime number and one composite number. Determine all polynomials $f \in \mathbb{Z}[x]$ such that for every [i]cute[/i] subset $ \{ p,q \}$, the subset $ \{ f(p) + q, f(q) + p \} $ is [i]cute[/i] as well. [i]Proposed by Valentio Iverson (Indonesia)[/i]

1993 Baltic Way, 7

Tags: algebra proposed , algebra

Solve the system of equations in integers: \[\begin{cases}z^x=y^{2x}\\ 2^z=4^x\\ x+y+z=20.\end{cases}\]

2010 Contests, 1

Tags: algebra , polynomial , limit , algebra proposed

suppose that polynomial $p(x)=x^{2010}\pm x^{2009}\pm...\pm x\pm 1$ does not have a real root. what is the maximum number of coefficients to be $-1$?(14 points)

2007 All-Russian Olympiad Regional Round, 9.1

Tags: quadratics , algebra , polynomial , geometry , geometric transformation , algebra proposed

Pete chooses $ 1004$ monic quadratic polynomial $ f_{1},\cdots,f_{1004}$, such that each integer from $ 0$ to $ 2007$ is a root of at least one of them. Vasya considers all equations of the form $ f_{i}\equal{}f_{j}(i\not \equal{}j)$ and computes their roots; for each such root , Pete has to pay to Vasya $ 1$ ruble . Find the least possible value of Vasya's income.

1998 Baltic Way, 10

Tags: algebra proposed , algebra

Let $n\ge 4$ be an even integer. A regular $n$-gon and a regular $(n-1)$-gon are inscribed into the unit circle. For each vertex of the $n$-gon consider the distance from this vertex to the nearest vertex of the $(n-1)$-gon, measured along the circumference. Let $S$ be the sum of these $n$ distances. Prove that $S$ depends only on $n$, and not on the relative position of the two polygons.

2014 Contests, 1

Tags: algebra proposed , algebra

Let $A$ be a subset of the irrational numbers such that the sum of any two distinct elements of it be a rational number. Prove that $A$ has two elements at most.

2007 Indonesia TST, 2

Tags: function , algebra proposed , algebra

Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying \[ f(f(x \plus{} y)) \equal{} f(x \plus{} y) \plus{} f(x)f(y) \minus{} xy\] for all real numbers $x$ and $y$.

2013 Serbia National Math Olympiad, 1

Tags: algebra proposed , algebra

Let $k$ be a natural number. Bijection $f:\mathbb{Z} \rightarrow \mathbb{Z}$ has the following property: for any integers $i$ and $j$, $|i-j|\leq k$ implies $|f(i) - f(j)|\leq k$. Prove that for every $i,j\in \mathbb{Z}$ it stands: \[|f(i)-f(j)|= |i-j|.\]

2008 ISI B.Math Entrance Exam, 6

Tags: algebra proposed , algebra

Let $\dbinom{n}{k}$ denote the binomial coefficient $\frac{n!}{k!(n-k)!}$ , and $F_m$ be the $m^{th}$ Fibonacci number given by $F_1=F_2=1$ and $F_{m+2}=F_m+F_{m+1}$ for all $m\geq 1$. Show that $\sum \dbinom{n}{k}=F_{m+1}$ for all $m\geq 1$ . Here the above sum is over all pairs of integers $n\geq k\geq 0$ with $n+k=m$ .