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

2015 QEDMO 14th, 6
Tags: algebra , complex , complex number
Let $n\ge 2$ be an integer. Let $z_1, z_2,..., z_n$ be complex numbers in such a way that for all integers $k$ with $1\le k\le n$: $$\Pi_{i = 1,i\ne k}^{n} (z_k- z_i) = \Pi_{i = 1,i\ne k}^{n} (z_k+ z_i).$$ Show that two of them are the same.

1996 Poland - Second Round, 1
Tags: factorization , sum of cubes , algebra , polynomial
Can every polynomial with integer coefficients be expressed as a sum of cubes of polynomials with integer coefficients? [hide]I found the following statement that can be linked to this problem: "It is easy to see that every polynomial in F[x] is sum of cubes if char (F)$\ne$3 and card (F)=2,4"[/hide]

2009 Harvard-MIT Mathematics Tournament, 4
Tags: algebra , least common multiple , greatest common divisor , quadratic , number theory , polynomial
Suppose $a$, $b$ and $c$ are integers such that the greatest common divisor of $x^2+ax+b$ and $x^2+bx+c$ is $x+1$ (in the set of polynomials in $x$ with integer coefficients), and the least common multiple of $x^2+ax+b$ and $x^2+bx+c$ $x^3-4x^2+x+6$. Find $a+b+c$.

2008 Princeton University Math Competition, 2
Tags: algebra
Find $\log_2 3 * \log_3 4 * \log_4 5 * ... * \log_{62} 63 * \log_{63} 64$ .

2018 Tajikistan Team Selection Test, 3
Tags: algebra
Problem 3. Suppose that the equation x^3-ax^2+bx-a=0 has three positive real roots (b>0). Find the minimum value of the expression: (b-a)(b^3+3a^3)

1993 Romania Team Selection Test, 2
Tags: algebra , polynomial , divisor
For coprime integers $m > n > 1$ consider the polynomials $f(x) = x^{m+n} -x^{m+1} -x+1$ and $g(x) = x^{m+n} +x^{n+1} -x+1$. If $f$ and $g$ have a common divisor of degree greater than $1$, find this divisor.

2021 Baltic Way, 1
Tags: algebra , functional equation
Let $n$ be a positive integer. Find all functions $f\colon \mathbb{R}\rightarrow \mathbb{R}$ that satisfy the equation $$ (f(x))^n f(x+y) = (f(x))^{n+1} + x^n f(y) $$ for all $x ,y \in \mathbb{R}$.

2024 Indonesia TST, 2
Tags: algebra , functional inequality
Let $\mathbb R_{>0}$ be the set of positive real numbers. Determine all functions $f \colon \mathbb R_{>0} \to \mathbb R_{>0}$ such that \[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\] for every $x, y \in \mathbb R_{>0}$.

MathLinks Contest 4th, 2.3
Tags: algebra
Let $m \ge 2n$ be two positive integers. Find a closed form for the following expression: $$E(m, n) = \sum_{k=0}^{n} (-1)^k {{m- k} \choose n} { n \choose k}$$

1993 IMO, 5
Tags: algebra , functional equation , function
Let $\mathbb{N} = \{1,2,3, \ldots\}$. Determine if there exists a strictly increasing function $f: \mathbb{N} \mapsto \mathbb{N}$ with the following properties: (i) $f(1) = 2$; (ii) $f(f(n)) = f(n) + n, (n \in \mathbb{N})$.

2019 Rioplatense Mathematical Olympiad, Level 3, 2
Tags: algebra , function
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f(f(x)^2+f(y^2))=(x-y)f(x-f(y))$

2012 IFYM, Sozopol, 4
Tags: maximum value , algebra
The numbers $x_i,i=1,2…6\in \mathbb{R}^+$ are such that $x_1+x_2+...+x_6=1$ and $x_1 x_3 x_5+x_2 x_4 x_6\geq \frac{1}{540}$. Let $S=x_1 x_2 x_3+x_2 x_3 x_4+...+x_6 x_1 x_2$. If $max\, S=\frac{p}{q}$ , where $gcd(p,q)=1$, find $p+q$.

2022 AMC 10, 24
Tags: algebra , functional equation
Consider functions $f$ that satisfy $|f(x)-f(y)|\leq \frac{1}{2}|x-y|$ for all real numbers $x$ and $y$. Of all such functions that also satisfy the equation $f(300) = f(900)$, what is the greatest possible value of $$f(f(800))-f(f(400))?$$ $ \textbf{(A)}\ 25 \qquad \textbf{(B)}\ 50 \qquad \textbf{(C)}\ 100 \qquad \textbf{(D)}\ 150 \qquad \textbf{(E)}\ 200$

KoMaL A Problems 2018/2019, A. 735
Tags: algebra , function
For any function $f:[0,1]\to [0,1]$, let $P_n (f)$ denote the number of fixed points of the function $\underbrace{f(f(\dotsc f}_{n} (x)\dotsc )$, i.e., the number of points $x\in [0,1]$ satisfying $\underbrace{f(f(\dotsc f}_{n} (x)\dotsc )=x$. Construct a piecewise linear, continuous, surjective function $f:[0,1] \to [0,1]$ such that for a suitable $2<A<3$, the sequence $\frac{P_n(f)}{A^n}$ converges. [i]Based on the 8th problem of the Miklós Schweitzer competition, 2018[/i]

2018 Lusophon Mathematical Olympiad, 1
Tags: algebra , sum
Fill in the corners of the square, so that the sum of the numbers in each one of the $5$ lines of the square is the same and the sum of the four corners is $123$.

1978 Miklós Schweitzer, 5
Tags: algebra , polynomial , function , rational function , complex analysis
Suppose that $ R(z)= \sum_{n=-\infty}^{\infty} a_nz^n$ converges in a neighborhood of the unit circle $ \{ z : \;|z|=1\ \}$ in the complex plane, and $ R(z)=P(z) / Q(z)$ is a rational function in this neighborhood, where $ P$ and $ Q$ are polynomials of degree at most $ k$. Prove that there is a constant $ c$ independent of $ k$ such that \[ \sum_{n=-\infty} ^{\infty} |a_n| \leq ck^2 \max_{|z|=1} |R(z)|.\] [i]H. S. Shapiro, G. Somorjai[/i]

2019 China National Olympiad, 6
Tags: algebra , geometry , trigonometry , inequalities
The point $P_1, P_2,\cdots ,P_{2018} $ is placed inside or on the boundary of a given regular pentagon. Find all placement methods are made so that $$S=\sum_{1\leq i<j\leq 2018}|P_iP_j| ^2$$takes the maximum value.

2003 Bulgaria National Olympiad, 3
Tags: algebra , limit , induction , polynomial
Determine all polynomials $P(x)$ with integer coefficients such that, for any positive integer $n$, the equation $P(x)=2^n$ has an integer root.

2007 Moldova Team Selection Test, 2
Tags: algebra , derivative , calculus , polynomial , induction
If $b_{1}, b_{2}, \ldots, b_{n}$ are non-negative reals not all zero, then prove that the polynomial \[x^{n}-b_{1}x^{n-1}-b_{2}x^{n-2}-\ldots-b_{n}=0\] has only one positive root $p$, which is simple. Moreover prove that any root of the polynomial does not exceed $p$ in absolute value.

2015 Romania National Olympiad, 1
Tags: complex number , absolute value , algebra
Find all triplets $ (a,b,c) $ of nonzero complex numbers having the same absolute value and which verify the equality: $$ \frac{a}{b} +\frac{b}{c}+\frac{c}{a} =-1 $$

2007 Indonesia TST, 2
Tags: parameterization , algebra
Solve the equation \[ x\plus{}a^3\equal{}\sqrt[3]{a\minus{}x}\] where $ a$ is a real parameter.

1991 IMO Shortlist, 30
Tags: game , game strategy , combinatorics , algebra
Two students $ A$ and $ B$ are playing the following game: Each of them writes down on a sheet of paper a positive integer and gives the sheet to the referee. The referee writes down on a blackboard two integers, one of which is the sum of the integers written by the players. After that, the referee asks student $ A:$ “Can you tell the integer written by the other student?” If A answers “no,” the referee puts the same question to student $ B.$ If $ B$ answers “no,” the referee puts the question back to $ A,$ and so on. Assume that both students are intelligent and truthful. Prove that after a finite number of questions, one of the students will answer “yes.”

2020 Korean MO winter camp, #2
Tags: algebra
$X$ is a set of $2020$ distinct real numbers. Prove that there exist $a,b\in \mathbb{R}$ and $A\subset X$ such that $$\sum_{x\in A}(x-a)^2 +\sum_{x\in X\backslash A}(x-b)^2\le \frac{1009}{1010}\sum_{x\in X}x^2$$

2009 Abels Math Contest (Norwegian MO) Final, 4a
Tags: algebra , inequalities
Show that $\left(\frac{2010}{2009}\right)^{2009}> 2$.

2011 IMO Shortlist, 5
Tags: triangle inequality , trigonometry , triangle , algebra
Prove that for every positive integer $n,$ the set $\{2,3,4,\ldots,3n+1\}$ can be partitioned into $n$ triples in such a way that the numbers from each triple are the lengths of the sides of some obtuse triangle. [i]Proposed by Canada[/i]