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

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Found problems: 15925

2016 Romania Team Selection Tests, 2
Tags: function , number theory , functional equation , algebra
Determine all $f:\mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ such that $f(m)\geq m$ and $f(m+n) \mid f(m)+f(n)$ for all $m,n\in \mathbb{Z}^+$

2020 New Zealand MO, 1
Tags: algebra , polynomial
Let $P(x) = x^3 - 2x + 1$ and let $Q(x) = x^3 - 4x^2 + 4x - 1$. Show that if $P(r) = 0$ then $Q(r^2) = 0$.

2019 Romania National Olympiad, 1
Tags: algebra , inequalities
a) Prove that for $x,y \ge 1$, holds $$x+y - \frac{1}{x}- \frac{1}{y} \ge 2\sqrt{xy} -\frac{2}{\sqrt{xy}}$$ b) Prove that for $a,b,c,d \ge 1$ with $abcd=16$ , holds $$a+b+c+d-\frac{1}{a}-\frac{1}{b}-\frac{1}{c}-\frac{1}{d}\ge 6$$

2007 Hungary-Israel Binational, 2
Tags: algebra , inequalities
Let $ a,b,c,d$ be real numbers, such that $ a^2\le 1, a^2 \plus{} b^2\le 5, a^2 \plus{} b^2 \plus{} c^2\le 14, a^2 \plus{} b^2 \plus{} c^2 \plus{} d^2\le 30$. Prove that $ a \plus{} b \plus{} c \plus{} d\le 10$.

2013 Iran MO (3rd Round), 4
Tags: pell equation , algebra , polynomial , vieta , number theory , diophantine equation
Prime $p=n^2 +1$ is given. Find the sets of solutions to the below equation: \[x^2 - (n^2 +1)y^2 = n^2.\] (25 points)

1977 All Soviet Union Mathematical Olympiad, 236
Tags: combinatorics , combinatorial geometry , algebra
Given several points, not all lying on one straight line. Some number is assigned to every point. It is known, that if a straight line contains two or more points, than the sum of the assigned to those points equals zero. Prove that all the numbers equal to zero.

2017 Saint Petersburg Mathematical Olympiad, 1
Tags: algebra
It’s allowed to replace any of three coefficients of quadratic trinomial by its discriminant. Is it true that from any quadratic trinomial that does not have real roots, we can perform such operation several times to get a quadratic trinomial that have real roots?

2020 LMT Fall, B1
Tags: algebra
Four $L$s are equivalent to three $M$s. Nine $M$s are equivalent to fourteen $T$ s. Seven $T$ s are equivalent to two $W$ s. If Kevin has thirty-six $L$s, how many $W$ s would that be equivalent to?

2016 Tournament Of Towns, 4
Tags: algebra
There are $2016$ red and $2016$ blue cards each having a number written on it. For some $64$ distinct positive real numbers, it is known that the set of numbers on cards of a particular color happens to be the set of their pairwise sums and the other happens to be the set of their pairwise products. Can we necessarily determine which color corresponds to sum and which to product? [i](B. Frenkin)[/i] (Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.[/url])

2011 ELMO Shortlist, 2
Tags: function , induction , functional equation , algebra
Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that whenever $a>b>c>d>0$ and $ad=bc$, \[f(a+d)+f(b-c)=f(a-d)+f(b+c).\] [i]Calvin Deng.[/i]

1967 IMO Longlists, 45
Tags: trigonometry , algebra , trigonometric equation , geometry
[b](i)[/b] Solve the equation: \[ \sin^3(x) + \sin^3\left( \frac{2 \pi}{3} + x\right) + \sin^3\left( \frac{4 \pi}{3} + x\right) + \frac{3}{4} \cos {2x} = 0.\] [b](ii)[/b] Supposing the solutions are in the form of arcs $AB$ with one end at the point $A$, the beginning of the arcs of the trigonometric circle, and $P$ a regular polygon inscribed in the circle with one vertex in $A$, find: 1) The subsets of arcs having the other end in $B$ in one of the vertices of the regular dodecagon. 2) Prove that no solution can have the end $B$ in one of the vertices of polygon $P$ whose number of sides is prime or having factors other than 2 or 3.

2021 LMT Fall, Tie
Tags: algebra
Estimate the value of $e^f$ , where $f = e^e$ .

2021 New Zealand MO, 2
Tags: algebra , inequalities
Prove that $$x^2 +\frac{8}{xy}+ y^2 \ge 8$$ for all positive real numbers $x$ and $y$.

2006 Moldova National Olympiad, 10.4
Tags: parameterization , algebra , logarithm
Find all real values of the real parameter $a$ such that the equation \[ 2x^{2}-6ax+4a^{2}-2a-2+\log_{2}(2x^{2}+2x-6ax+4a^{2})= \] \[ =\log_{2}(x^{2}+2x-3ax+2a^{2}+a+1). \] has a unique solution.

2024 AMC 12/AHSME, 9
Tags: algebra , special factorizations
Let $M$ be the greatest integer such that both $M + 1213$ and $M + 3773$ are perfect squares. What is the units digit of $M$? $ \textbf{(A) }1 \qquad \textbf{(B) }2 \qquad \textbf{(C) }3 \qquad \textbf{(D) }6 \qquad \textbf{(E) }8 \qquad $

2015 Harvard-MIT Mathematics Tournament, 4
Tags: algebra , divisibility , sequence
Compute the number of sequences of integers $(a_1,\ldots,a_{200})$ such that the following conditions hold. [list] [*] $0\leq a_1<a_2<\cdots<a_{200}\leq 202.$ [*] There exists a positive integer $N$ with the following property: for every index $i\in\{1,\ldots,200\}$ there exists an index $j\in\{1,\ldots,200\}$ such that $a_i+a_j-N$ is divisible by $203$. [/list]

2019 Teodor Topan, 3
Tags: function , algebra , functional equation , find all functions
Let be a positive real number $ r, $ a natural number $ n, $ and a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ satisfying $ f(rxy)=(f(x)f(y))^n, $ for any real numbers $ x,y. $ [b]a)[/b] Give three distinct examples of what $ f $ could be if $ n=1. $ [b]b)[/b] For a fixed $ n\ge 2, $ find all possibilities of what $ f $ could be. [i]Bogdan Blaga[/i]

1997 AIME Problems, 1
Tags: modular arithmetic , linear algebra , matrix , geometry , 3d geometry , algebra , difference of squares
How many of the integers between 1 and 1000, inclusive, can be expressed as the difference of the squares of two nonnegative integers?

1991 Putnam, B2
Tags: function , functional equation , algebra , superior algebra
Define functions $f$ and $g$ as nonconstant, differentiable, real-valued functions on $R$. If $f(x+y)=f(x)f(y)-g(x)g(y)$, $g(x+y)=f(x)g(y)+g(x)f(y)$, and $f'(0)=0$, prove that $\left(f(x)\right)^2+\left(g(x)\right)^2=1$ for all $x$.

1999 Korea - Final Round, 2
Tags: function , algebra , functional inequality
Suppose $f(x)$ is a function satisfying $\left | f(m+n)-f(m) \right | \leq \frac{n}{m}$ for all positive integers $m$,$n$. Show that for all positive integers $k$: \[\sum_{i=1}^{k}\left |f(2^k)-f(2^i) \right |\leq \frac{k(k-1)}{2}\].

2009 IMC, 4
Tags: modular arithmetic , polynomial , algebra
Let $p$ be a prime number and $\mathbf{W}\subseteq \mathbb{F}_p[x]$ be the smallest set satisfying the following : [list] (a) $x+1\in \mathbf{W}$ and $x^{p-2}+x^{p-3}+\cdots +x^2+2x+1\in \mathbf{W}$ (b) For $\gamma_1,\gamma_2$ in $\mathbf{W}$, we also have $\gamma(x)\in \mathbf{W}$, where $\gamma(x)$ is the remainder $(\gamma_1\circ \gamma_2)(x)\pmod {x^p-x}$.[/list] How many polynomials are in $\mathbf{W}?$

2011 Postal Coaching, 1
Tags: algebra , polynomial , number theory
Prove that, for any positive integer $n$, there exists a polynomial $p(x)$ of degree at most $n$ whose coefficients are all integers such that, $p(k)$ is divisible by $2^n$ for every even integer $k$, and $p(k) -1$ is divisible by $2^n$ for every odd integer $k$.

2002 Austrian-Polish Competition, 8
Tags: trigonometry , limit , inequalities , algebra
Determine the number of real solutions of the system \[\left\{ \begin{aligned}\cos x_{1}&= x_{2}\\ &\cdots \\ \cos x_{n-1}&= x_{n}\\ \cos x_{n}&= x_{1}\\ \end{aligned}\right.\]

1989 Nordic, 1
Tags: integer polynomial , integer root , algebra
Find a polynomial $P$ of lowest possible degree such that (a) $P$ has integer coefficients, (b) all roots of $P$ are integers, (c) $P(0) = -1$, (d) $P(3) = 128$.

2022 German National Olympiad, 1
Tags: system of equations , quadratic equation , parameterization , algebra
Determine all real numbers $a$ for which the system of equations \begin{align*} 3x^2+2y^2+2z^2&=a\\ 4x^2+4y^2+5z^2&=1-a \end{align*} has at least one solution $(x,y,z)$ in the real numbers.