Olympiad problems

2025 Belarusian National Olympiad, 9.6

Tags: algebra , inequality

Numbers $a,b,c$ are lengths of sides of some triangle. Prove the inequality$$\frac{a}{b+c-a}+\frac{b}{c+a-b}+\frac{c}{a+b-c} \geq \frac{a+b}{2c}+\frac{b+c}{2a}+\frac{c+a}{2b}$$ [i]M. Karpuk[/i]

2012 Romania National Olympiad, 1

Tags: trigonometry , algebra

[color=darkred]Let $M=\{x\in\mathbb{C}\, |\, |z|=1,\ \text{Re}\, z\in\mathbb{Q}\}\, .$ Prove that there exist infinitely many equilateral triangles in the complex plane having all affixes of their vertices in the set $M$ .[/color]

2014 ELMO Shortlist, 3

Tags: function , algebra , geometry , reflection

We say a finite set $S$ of points in the plane is [i]very[/i] if for every point $X$ in $S$, there exists an inversion with center $X$ mapping every point in $S$ other than $X$ to another point in $S$ (possibly the same point). (a) Fix an integer $n$. Prove that if $n \ge 2$, then any line segment $\overline{AB}$ contains a unique very set $S$ of size $n$ such that $A, B \in S$. (b) Find the largest possible size of a very set not contained in any line. (Here, an [i]inversion[/i] with center $O$ and radius $r$ sends every point $P$ other than $O$ to the point $P'$ along ray $OP$ such that $OP\cdot OP' = r^2$.) [i]Proposed by Sammy Luo[/i]

2022 MOAA, 13

Tags: floor function , algebra

Determine the number of distinct positive real solutions to $$\lfloor x \rfloor ^{\{x\}} = \frac{1}{2022}x^2$$ . Note: $\lfloor x \rfloor$ is known as the floor function, which returns the greatest integer less than or equal to $x$. Furthermore, $\{x\}$ is defined as $x - \lfloor x \rfloor$.

2005 AIME Problems, 13

Tags: polynomial , quadratic , vieta , function , algebra

Let $P(x)$ be a polynomial with integer coefficients that satisfies $P(17)=10$ and $P(24)=17$. Given that $P(n)=n+3$ has two distinct integer solutions $n_1$ and $n_2$, find the product $n_1\cdot n_2$.

2018 Moscow Mathematical Olympiad, 9

Tags: algebra , number theory , trigonometry

$x$ and $y$ are integer $5$-digits numbers, such that in the decimal notation, all ten digits are used exactly once. Also $\tan{x}-\tan{y}=1+\tan{x}\tan{y}$, where $x,y$ are angles in degrees. Find maximum of $x$

2010 Greece Team Selection Test, 1

Tags: algebra

Solve in positive reals the system: $x+y+z+w=4$ $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{w}=5-\frac{1}{xyzw}$

2011 JBMO Shortlist, 6

Tags: inequalities , algebra

Let $\displaystyle {x_i> 1, \forall i \in \left \{1, 2, 3, \ldots, 2011 \right \}}$. Show that:$$\displaystyle{\frac{x^2_1}{x_2-1}+\frac{x^2_2}{x_3-1}+\frac{x^2_3}{x_4-1}+\ldots+\frac{x^2_{2010}}{x_{2011}-1}+\frac{x^2_{2011}}{x_1-1}\geq 8044}$$ When the equality holds?

2006 Putnam, B5

Tags: function , integration , calculus , quadratic , inequalities , algebra

For each continuous function $f: [0,1]\to\mathbb{R},$ let $I(f)=\int_{0}^{1}x^{2}f(x)\,dx$ and $J(f)=\int_{0}^{1}x\left(f(x)\right)^{2}\,dx.$ Find the maximum value of $I(f)-J(f)$ over all such functions $f.$

2018 Costa Rica - Final Round, F2

Tags: functional equation , functional , algebra , sequence , combinatorics

Consider $f (n, m)$ the number of finite sequences of $ 1$'s and $0$'s such that each sequence that starts at $0$, has exactly n $0$'s and $m$ $ 1$'s, and there are not three consecutive $0$'s or three $ 1$'s. Show that if $m, n> 1$, then $$f (n, m) = f (n-1, m-1) + f (n-1, m-2) + f (n-2, m-1) + f (n-2, m-2)$$

1988 Balkan MO, 2

Tags: polynomial , algebra

Find all polynomials of two variables $P(x,y)$ which satisfy \[P(a,b) P(c,d) = P (ac+bd, ad+bc), \forall a,b,c,d \in \mathbb{R}.\]

2018-2019 Fall SDPC, 6

Tags: quadratic , algebra

Alice and Bob play a game. Alice writes an equation of the form $ax^2 + bx + c =0$, choosing $a$, $b$, $c$ to be real numbers (possibly zero). Bob can choose to add (or subtract) any real number to each of $a$, $b$, $c$, resulting in a new equation. Bob wins if the resulting equation is quadratic and has two distinct real roots; Alice wins otherwise. For which choices of $a$, $b$, $c$ does Alice win, no matter what Bob does?

2017 India IMO Training Camp, 1

Tags: algebra , polynomial

Suppose $f,g \in \mathbb{R}[x]$ are non constant polynomials. Suppose neither of $f,g$ is the square of a real polynomial but $f(g(x))$ is. Prove that $g(f(x))$ is not the square of a real polynomial.

2012 USA TSTST, 1

Tags: symmetry , pigeonhole principle , arithmetic sequence , additive combinatorics , sequence , algebra

Find all infinite sequences $a_1, a_2, \ldots$ of positive integers satisfying the following properties: (a) $a_1 < a_2 < a_3 < \cdots$, (b) there are no positive integers $i$, $j$, $k$, not necessarily distinct, such that $a_i+a_j=a_k$, (c) there are infinitely many $k$ such that $a_k = 2k-1$.

2024 IFYM, Sozopol, 3

Tags: algebra , functional equation

Find all functions $ f:\mathbb{Z} \to \mathbb{Z} $ such that \[ f(x + f(y) - 2y) + f(f(y)) = f(x) \] for all integers $ x $ and $ y $.

2002 Kazakhstan National Olympiad, 6

Tags: polynomial , algebra

Find all polynomials $ P (x) $ with real coefficients that satisfy the identity $ P (x ^ 2) = P (x) P (x + 1) $.

JOM 2015 Shortlist, A6

Tags: algebra

Let $(a_{n})_{n\ge 0}$ and $(b_{n})_{n\ge 0}$ be two sequences with arbitrary real values $a_0, a_1, b_0, b_1$. For $n\ge 1$, let $a_{n+1}, b_{n+1}$ be defined in this way: $$a_{n+1}=\dfrac{b_{n-1}+b_{n}}{2}, b_{n+1}=\dfrac{a_{n-1}+a_{n}}{2}$$ Prove that for any constant $c>0$ there exists a positive integer $N$ s.t. for all $n>N$, $|a_{n}-b_{n}|<c$.

2024 Regional Competition For Advanced Students, 1

Tags: inequalities , algebra

Let $a$, $b$ and $c$ be real numbers larger than $1$. Prove the inequality $$\frac{ab}{c-1}+\frac{bc}{a - 1}+\frac{ca}{b -1} \ge 12.$$ When does equality hold? [i](Karl Czakler)[/i]

2004 Nicolae Coculescu, 2

Tags: floor function , equation , algebra

Let be a natural number $ n\ge 2. $ Find the real numbers $ a $ that satisfy the equation $$ \lfloor nx \rfloor =\sum_{k=1}^{n} \lfloor x+(k-1)a \rfloor , $$ for any real numbers $ x. $ [i]Marius Perianu[/i]

2011 Kazakhstan National Olympiad, 2

Tags: inequalities , limit , algebra

Given a positive integer $n$. Prove the inequality $\sum\limits_{i=1}^{n}\frac{1}{i(i+1)(i+2)(i+3)(i+4)}<\frac{1}{96}$

2018 Germany Team Selection Test, 1

Tags: algebra , polynomial , ineqstd

Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that $$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$ If $M>1$, prove that the polynomial $$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$ has no positive roots.

2021 Azerbaijan EGMO TST, 4

Tags: algebra

Let $P(x), Q(x)$ be distinct polynomials of degree $2020$ with non-zero coefficients. Suppose that they have $r$ common real roots counting multiplicity and $s$ common coefficients. Determine the maximum possible value of $r + s$. [i]Demetres Christofides, Cyprus[/i]

KoMaL A Problems 2022/2023, A. 835

Tags: algebra , functional equation , polynomial

Let $f^{(n)}(x)$ denote the $n^{\text{th}}$ iterate of function $f$, i.e $f^{(1)}(x)=f(x)$, $f^{(n+1)}(x)=f(f^{(n)}(x))$. Let $p(n)$ be a given polynomial with integer coefficients, which maps the positive integers into the positive integers. Is it possible that the functional equation $f^{(n)}(n)=p(n)$ has exactly one solution $f$ that maps the positive integers into the positive integers? [i]Submitted by Dávid Matolcsi and Kristóf Szabó, Budapest[/i]

2019 Junior Balkan Team Selection Tests - Romania, 4

Tags: algebra , min , inequalities

Let $a$ and $b$ be positive real numbers such that $3(a^2+b^2-1) = 4(a+b$). Find the minimum value of the expression $\frac{16}{a}+\frac{1}{b}$ .

2017 BMT Spring, 8

Tags: algebra

A function $f$ with its domain on the positive integers $N =\{1, 2, ...\}$ satisfies the following conditions: (a) $f(1) = 2017$. (b) $\sum_{i=1}^n f(i) = n^2f(n)$, for every positive integer $n > 1$. What is the value of $f(2017)$?