This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 15925

2020 Jozsef Wildt International Math Competition, W28
Tags: algebra , summation , binomial coefficient
For positive integers $j\le n$, prove that $$\sum_{k=j}^n\binom{2n}{2k}\binom kj=\frac{n\cdot4^{n-j}}j\binom{2n-j-1}{j-1}.$$ [i]Proposed by Ángel Plaza[/i]

2009 Bulgaria National Olympiad, 4
Tags: polynomial , algebra
Let $ n\ge 3$ be a natural number. Find all nonconstant polynomials with real coeficcietns $ f_{1}\left(x\right),f_{2}\left(x\right),\ldots,f_{n}\left(x\right)$, for which \[ f_{k}\left(x\right)f_{k+ 1}\left(x\right) = f_{k +1}\left(f_{k + 2}\left(x\right)\right), \quad 1\le k\le n,\] for every real $ x$ (with $ f_{n +1}\left(x\right)\equiv f_{1}\left(x\right)$ and $ f_{n + 2}\left(x\right)\equiv f_{2}\left(x\right)$).

1969 IMO Longlists, 61
Tags: function , algebra , number theory , binomial theorem , power of 2
$(SWE 4)$ Let $a_0, a_1, a_2, \cdots$ be determined with $a_0 = 0, a_{n+1} = 2a_n + 2^n$. Prove that if $n$ is power of $2$, then so is $a_n$

1983 IMO Longlists, 39
Tags: algebra
If $\alpha $ is the real root of the equation \[E(x) = x^3 - 5x -50 = 0\] such that $x_{n+1} = (5x_n + 50)^{1/3}$ and $x_1 = 5$, where $n$ is a positive integer, prove that: [b](a)[/b] $x_{n+1}^3 - \alpha^3 = 5(x_n - \alpha)$ [b](b)[/b] $\alpha < x_{n+1} < x_n.$

2020 AIME Problems, 14
Tags: polynomial , algebra
Let $P(x)$ be a quadratic polynomial with complex coefficients whose $x^2$ coefficient is $1$. Suppose the equation $P(P(x))=0$ has four distinct solutions, $x=3,4,a,b$. Find the sum of all possible values of $(a+b)^2$.

2017 China Team Selection Test, 4
Tags: polynomial , algebra
Find out all the integer pairs $(m,n)$ such that there exist two monic polynomials $P(x)$ and $Q(x)$ ,with $\deg{P}=m$ and $\deg{Q}=n$,satisfy that $$P(Q(t))\not=Q(P(t))$$ holds for any real number $t$.

2020 BMT Fall, 9
Tags: algebra , diophantine equation
There is a unique triple $(a,b,c)$ of two-digit positive integers $a,\,b,$ and $c$ that satisfy the equation $$a^3+3b^3+9c^3=9abc+1.$$ Compute $a+b+c$.

2019 Kosovo National Mathematical Olympiad, 3
Tags: inequalities , algebra
Show that for any non-negative real numbers $a,b,c,d$ such that $a^2+b^2+c^2+d^2=1$ the following inequality hold: $$a+b+c+d-1\geq 16abcd$$ When does equality hold?

2011 Bosnia And Herzegovina - Regional Olympiad, 1
Tags: algebra , factorization
Factorise $$(a+2b-3c)^3+(b+2c-3a)^3+(c+2a-3b)^3$$

2020 Dutch IMO TST, 2
Tags: polynomial , algebra
Determine all polynomials $P (x)$ with real coefficients that apply $P (x^2) + 2P (x) = P (x)^2 + 2$.

2007 Nicolae Coculescu, 1
Tags: algebra , function
Let $w\in \mathbb{C}\setminus \mathbb{R}$, $|w|\neq 1$. Prove that $f\colon \mathbb{C} \to \mathbb{C}$, given by $f(z)= z+w\overline{z}$, is a bijection, and find its inverse.

2025 Israel National Olympiad (Gillis), P7
Tags: algebra , number theory
For a positive integer $n$, let $A_n$ be the set of quadruplets $(a,b,c,d)$ of integers, satisfying the following properties simultaneously: [list] [*] $0\le a\le c\le n,$ [*] $0\le b\le d\le n,$ [*] $c+d>n,$ and [*] $bc=ad+1.$ [/list] Moreover, define $$\alpha_n=\sum_{(a,b,c,d)\in A_n}\frac{1}{ab+cd}.$$ Find all real numbers $t$ such that $\alpha_n>t$ for every positive integer $n$.

1977 USAMO, 1
Tags: modular arithmetic , algebra , polynomial
Determine all pairs of positive integers $ (m,n)$ such that $ (1\plus{}x^n\plus{}x^{2n}\plus{}\cdots\plus{}x^{mn})$ is divisible by $ (1\plus{}x\plus{}x^2\plus{}\cdots\plus{}x^{m})$.

1995 China Team Selection Test, 2
Tags: algebra , polynomial
$ A$ and $ B$ play the following game with a polynomial of degree at least 4: \[ x^{2n} \plus{} \_x^{2n \minus{} 1} \plus{} \_x^{2n \minus{} 2} \plus{} \ldots \plus{} \_x \plus{} 1 \equal{} 0 \] $ A$ and $ B$ take turns to fill in one of the blanks with a real number until all the blanks are filled up. If the resulting polynomial has no real roots, $ A$ wins. Otherwise, $ B$ wins. If $ A$ begins, which player has a winning strategy?

1998 North Macedonia National Olympiad, 5
Tags: algebra , inequalities , sequence , recurrence relation
The sequence $(a_n)$ is defined by $a_1 =\sqrt2$ and $a_{n+1} =\sqrt{2-\sqrt{4-a_n^2}}$. Let $b_n =2^{n+1}a_n$. Prove that $b_n \le 7$ and $b_n < b_{n+1}$ for all $n$.

2004 Romania Team Selection Test, 3
Tags: function , induction , algebra
Find all one-to-one mappings $f:\mathbb{N}\to\mathbb{N}$ such that for all positive integers $n$ the following relation holds: \[ f(f(n)) \leq \frac {n+f(n)} 2 . \]

1978 Austrian-Polish Competition, 4
Tags: combinatorics , ratio , algebra
Let $c\neq 1$ be a positive rational number. Show that it is possible to partition $\mathbb{N}$, the set of positive integers, into two disjoint nonempty subsets $A,B$ so that $x/y\neq c$ holds whenever $x$ and $y$ lie both in $A$ or both in $B$.

1971 IMO, 3
Tags: linear algebra , algebra , matrix , combinatorial inequality , combinatorics
Let $ A \equal{} (a_{ij})$, where $ i,j \equal{} 1,2,\ldots,n$, be a square matrix with all $ a_{ij}$ non-negative integers. For each $ i,j$ such that $ a_{ij} \equal{} 0$, the sum of the elements in the $ i$th row and the $ j$th column is at least $ n$. Prove that the sum of all the elements in the matrix is at least $ \frac {n^2}{2}$.

2021 Baltic Way, 3
Tags: algebra , sequence
Determine all infinite sequences $(a_1,a_2,\dots)$ of positive integers satisfying \[a_{n+1}^2=1+(n+2021)a_n\] for all $n \ge 1$.

2021 Azerbaijan Senior NMO, 5
Tags: polynomial , algebra , integer coefficients , polynomial with integer coeffi
Define $P(x)=((x-a_1)(x-a_2)...(x-a_n))^2 +1$, where $a_1,a_2...,a_n\in\mathbb{Z}$ and $n\in\mathbb{N^+}$. Prove that $P(x)$ couldn't be expressed as product of two non-constant polynomials with integer coefficients.

1969 IMO Shortlist, 24
Tags: number theory , polynomial , divisibility , algebra
$(GBR 1)$ The polynomial $P(x) = a_0x^k + a_1x^{k-1} + \cdots + a_k$, where $a_0,\cdots, a_k$ are integers, is said to be divisible by an integer $m$ if $P(x)$ is a multiple of $m$ for every integral value of $x$. Show that if $P(x)$ is divisible by $m$, then $a_0 \cdot k!$ is a multiple of $m$. Also prove that if $a, k,m$ are positive integers such that $ak!$ is a multiple of $m$, then a polynomial $P(x)$ with leading term $ax^k$can be found that is divisible by $m.$

2014 Saudi Arabia BMO TST, 5
Tags: inequalities , induction , algebra
Find all positive integers $n$ such that \[3^n+4^n+\cdots+(n+2)^n=(n+3)^n.\]

1993 Chile National Olympiad, 5
Tags: inequalities , algebra
Let $a,b,c$ three positive numbers less than $ 1$. Prove that cannot occur simultaneously these three inequalities: $$a (1- b)>\frac14$$ $$b (1-c)>\frac14 $$ $$c (1-a)>\frac14$$

Russian TST 2015, P3
Tags: functional equation , algebra
Find all functions $f : \mathbb{Z} \to\mathbb{ Z}$ such that \[ n^2+4f(n)=f(f(n))^2 \] for all $n\in \mathbb{Z}$. [i]Proposed by Sahl Khan, UK[/i]

VI Soros Olympiad 1999 - 2000 (Russia), 10.1
Tags: algebra , number theory , trigonometry
For a real number $a$, denote by $(a]$ the smallest integer that is not less than $a$. Find all real values of $x$ for which holds the equality $$(\sin x]^2 + (\cos x]^2 =|tg x| +|ctg x|.$$