Olympiad problems

2011 Cuba MO, 5

Determine all functions $f : R \to R$ such that $$f(x)f(y) = 2f(x + y) + 9xy \ \ \forall x, y \in R.$$

2014 IFYM, Sozopol, 1

Tags: inequality , algebra , inequalities

Prove that for $\forall$ $a,b,c\in [\frac{1}{3},3]$ the following inequality is true: $\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}\geq \frac{7}{5}$.

2007 Hanoi Open Mathematics Competitions, 15

Tags: polynomial , quadratic , algebra

Let $p = \overline{abcd}$ be a $4$-digit prime number. Prove that the equation $ax^3+bx^2+cx+d=0$ has no rational roots.

2022 India National Olympiad, 2

Tags: combinatorics , algebra

Find all natural numbers $n$ for which there is a permutation $\sigma$ of $\{1,2,\ldots, n\}$ that satisfies: \[ \sum_{i=1}^n \sigma(i)(-2)^{i-1}=0 \]

1971 AMC 12/AHSME, 19

Tags: algebra , conic , ellipse , quadratic , quadratic formula

If the line $y=mx+1$ intersects the ellipse $x^2+4y^2=1$ exactly once, then the value of $m^2$ is $\textbf{(A) }\textstyle\frac{1}{2}\qquad\textbf{(B) }\frac{2}{3}\qquad\textbf{(C) }\frac{3}{4}\qquad\textbf{(D) }\frac{4}{5}\qquad \textbf{(E) }\frac{5}{6}$

1983 All Soviet Union Mathematical Olympiad, 370

Tags: decimal representation , digit , algebra , number theory

The infinite decimal notation of the real number $x$ contains all the digits. Let $u_n$ be the number of different $n$-digit segments encountered in $x$ notation. Prove that if for some $n$, $u_n \le (n+8)$, than $x$ is a rational number.

2007 Peru Iberoamerican Team Selection Test, P1

Tags: algebra

Solve in the set of real numbers, the system: $$x(3y^2+1)=y(y^2+3)$$ $$y(3z^2+1)=z(z^2+3)$$ $$z(3x^2+1)=x(x^2+3)$$

2025 Romania EGMO TST, P1

Tags: algebra , polynomial

find all real coefficient polynomial $ P(x)$ such that $ P(x)P(x\plus{}1)\equal{}P(x^2\plus{}x\plus{}1)$ for all $ x$

2009 Canadian Mathematical Olympiad Qualification Repechage, 1

Tags: 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*}

2023/2024 Tournament of Towns, 1

Tags: algebra , polymonial

For every polynomial of degree 45 with coefficients $1,2,3, \ldots, 46$ (in some order) Tom has listed all its distinct real roots. Then he increased each number in the list by 1 . What is now greater: the amount of positive numbers or the amount of negative numbers? Alexey Glebov

1966 German National Olympiad, 5

Tags: trigonometry , equation , trigonometric identities , algebra

Prove that \[\tan 7 30^{\prime }=\sqrt{6}+\sqrt{2}-\sqrt{3}-2.\]

VMEO III 2006 Shortlist, N8

Tags: algebra , number theory

For every positive integer $n$, the symbol $a_n/b_n$ is the simplest form of the fraction $1+1/2+...+1/n$. Prove that for every pair of positive integers $(M, N)$ we can always find a positive integer $m$ where $(a_n, N) = 1$ for all $n = m, m + 1, ...,m + M$.

2006 Petru Moroșan-Trident, 2

Tags: complex number , algebra

Consider $ n\ge 1 $ complex numbers $ z_1,z_2,\ldots ,z_n $ that have the same nonzero modulus, and which verify $$ 0=\Re\left( \sum_{a=1}^n\sum_{b=1}^n\sum_{c=1}^n\sum_{d=1}^n \frac{z_bz_c}{z_az_d} \right) . $$ Prove that $ n\left( -1+\left| z_1 \right|^2 \right) =\sum_{k=1}^n\left| 1-z_k \right| . $ [i]Botea Viorel[/i]

1996 Tournament Of Towns, (494) 1

Tags: algebra

People are asked “Do you think that the new president will be better than the most recent one?” Suppose $a$ people say “better”,$ b$ say “the same” and $c$ “worse”. Sociologists then calculate two measures of “social optimism”: $m =a + \frac{b}{2}$ and $n = a - c$. Suppose exactly $100$ people respond to this survey and it turns out that $m = 40$. Find $n$. (A Kovaldji)

1959 IMO, 3

Tags: algebra , trigonometry , quadratic , trigonometric equation

Let $a,b,c$ be real numbers. Consider the quadratic equation in $\cos{x}$ \[ a \cos^2{x}+b \cos{x}+c=0. \] Using the numbers $a,b,c$ form a quadratic equation in $\cos{2x}$ whose roots are the same as those of the original equation. Compare the equation in $\cos{x}$ and $\cos{2x}$ for $a=4$, $b=2$, $c=-1$.

1997 Tournament Of Towns, (547) 1

Tags: algebra

On an escalator which is not moving, a person descends faster than he ascends. Is it faster for this person to descend and ascend once on an upward-moving escalator, or to descend and ascend once on a downward-moving escalator? (It is assumed that all the speeds mentioned here are constant, that the speed of the escalator is the same no matter if it is moving up or down and that the speed of the person relative to the escalator is always greater than the speed of the escalator.) (Folklore)

1990 China National Olympiad, 4

Tags: algebra , system of equations , number theory

Given a positive integer number $a$ and two real numbers $A$ and $B$, find a necessary and sufficient condition on $A$ and $B$ for the following system of equations to have integer solution: \[ \left\{\begin{array}{cc} x^2+y^2+z^2=(Ba)^2\\ x^2(Ax^2+By^2)+y^2(Ay^2+Bz^2)+z^2(Az^2+Bx^2)=\dfrac{1}{4}(2A+B)(Ba)^4\end{array}\right. \]

1989 Greece National Olympiad, 1

Tags: system of equations , algebra

Find all real solutions of $$ \begin{matrix} \sqrt{9+x_1}+ \sqrt{9+x_2}+...+ \sqrt{9+x_{100}}=100\sqrt{10}\\ \sqrt{16-x_1}+ \sqrt{16-x_2}+...+ \sqrt{16-x_{100}}=100\sqrt{15} \end{matrix}$$

1983 IMO Longlists, 32

Tags: logarithm , inequalities , function , algebra

Let $a, b, c$ be positive real numbers and let $[x]$ denote the greatest integer that does not exceed the real number $x$. Suppose that $f$ is a function defined on the set of non-negative integers $n$ and taking real values such that $f(0) = 0$ and \[f(n) \leq an + f([bn]) + f([cn]), \qquad \text{ for all } n \geq 1.\] Prove that if $b + c < 1$, there is a real number $k$ such that \[f(n) \leq kn \qquad \text{ for all } n \qquad (1)\] while if $b + c = 1$, there is a real number $K$ such that $f(n) \leq K n \log_2 n$ for all $n \geq 2$. Show that if $b + c = 1$, there may not be a real number $k$ that satisfies $(1).$

2006 Thailand Mathematical Olympiad, 7

Tags: inequalities , min , algebra

Let $x, y, z$ be reals summing to $1$ which minimizes $2x^2 + 3y^2 + 4z^2$. Find $x$.

2023 Princeton University Math Competition, A3

Tags: algebra , polynomial

Let $f(X)$ be a monic irreducible polynomial over $\mathbb{Z}$; therefore, by Gauss's Lemma, $f$ is also irreducible over $\mathbb{Q}$ (you may assume this). Moreover, assume $f(X) \mid f\left(X^2+n\right)$ where $n$ is an integer such that $n \notin\{-1,0,1\}$. Show that $n^2 \nmid f(0)$.

2006 Iran Team Selection Test, 2

Tags: logarithm , ceiling function , algebra

Let $n$ be a fixed natural number. [b]a)[/b] Find all solutions to the following equation : \[ \sum_{k=1}^n [\frac x{2^k}]=x-1 \] [b]b)[/b] Find the number of solutions to the following equation ($m$ is a fixed natural) : \[ \sum_{k=1}^n [\frac x{2^k}]=x-m \]

2011 Mongolia Team Selection Test, 2

Tags: combinatorics , linear equation , algebra

Let $r$ be a given positive integer. Is is true that for every $r$-colouring of the natural numbers there exists a monochromatic solution of the equation $x+y=3z$? (proposed by B. Batbaysgalan, folklore)

2021 Peru EGMO TST, 6

Tags: functional inequality , functional equation , function , algebra

Find all functions $f : R \to R$ such that $$f(x + y) \ge xf(x) + yf(y)$$, for all $x, y \in R$ .

Kvant 2022, M2682

Tags: algebra , logic

Six real numbers $x_1<x_2<x_3<x_4<x_5<x_6$ are given. For each triplet of distinct numbers of those six Vitya calculated their sum. It turned out that the $20$ sums are pairwise distinct; denote those sums by $$s_1<s_2<s_3<\cdots<s_{19}<s_{20}.$$ It is known that $x_2+x_3+x_4=s_{11}$, $x_2+x_3+x_6=s_{15}$ and $x_1+x_2+x_6=s_{m}$. Find all possible values of $m$.