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 Latvia Baltic Way TST, 3
Tags: algebra , polynomial
Given a polynomial $P$ of degree $2016$ with real coefficients and a quadratic polynomial $Q$ with real coefficients. Is it possible that the roots of the polynomial $P (Q(x))$ are exactly all these numbers: $$-2015, -2014, . . . , -2, -1, 1, 2, . . . , 2016, 2017?$$

2000 China Team Selection Test, 3
Tags: algebra , function
Let $n$ be a positive integer. Denote $M = \{(x, y)|x, y \text{ are integers }, 1 \leq x, y \leq n\}$. Define function $f$ on $M$ with the following properties: [b]a.)[/b] $f(x, y)$ takes non-negative integer value; [b] b.)[/b] $\sum^n_{y=1} f(x, y) = n - 1$ for $1 \eq x \leq n$; [b]c.)[/b] If $f(x_1, y_1)f(x2, y2) > 0$, then $(x_1 - x_2)(y_1 - y_2) \geq 0.$ Find $N(n)$, the number of functions $f$ that satisfy all the conditions. Give the explicit value of $N(4)$.

2008 IMO Shortlist, 1
Tags: functional equation , algebra
Find all functions $ f: (0, \infty) \mapsto (0, \infty)$ (so $ f$ is a function from the positive real numbers) such that \[ \frac {\left( f(w) \right)^2 \plus{} \left( f(x) \right)^2}{f(y^2) \plus{} f(z^2) } \equal{} \frac {w^2 \plus{} x^2}{y^2 \plus{} z^2} \] for all positive real numbers $ w,x,y,z,$ satisfying $ wx \equal{} yz.$ [i]Author: Hojoo Lee, South Korea[/i]

Today's calculation of integrals, 867
Tags: algebra , quadratic , derivative , linear equation , integration , function , calculus
Express $\int_0^2 f(x)dx$ for any quadratic functions $f(x)$ in terms of $f(0),\ f(1)$ and $f(2).$

2009 Dutch IMO TST, 4
Tags: algebra , functional equation , functional equation in z
Find all functions $f : Z \to Z$ satisfying $f(m + n) + f(mn -1) = f(m)f(n) + 2$ for all $m, n \in Z$.

2025 Kyiv City MO Round 2, Problem 2
Tags: algebra
Mykhailo chose three distinct positive real numbers \( a, b, c \) and wrote the following numbers on the board: \[ a + b, \quad b + c, \quad c + a, \quad ab, \quad bc, \quad ca. \] What is the minimum possible number of distinct numbers that can be written on the board? [i]Proposed by Anton Trygub[/i]

2013 Stars Of Mathematics, 1
Tags: algebra , functional inequality , function
Let $\mathcal{F}$ be the family of bijective increasing functions $f\colon [0,1] \to [0,1]$, and let $a \in (0,1)$. Determine the best constants $m_a$ and $M_a$, such that for all $f \in \mathcal{F}$ we have \[m_a \leq f(a) + f^{-1}(a) \leq M_a.\] [i](Dan Schwarz)[/i]

2012 Thailand Mathematical Olympiad, 9
Tags: polynomial , real root , algebra , inequalities
Let $n$ be a positive integer and let $P(x) = x^n + a_{n-1}x^{n-1} +... + a_1x + 1$ be a polynomial with positive real coefficients. Under the assumption that the roots of $P$ are all real, show that $P(x) \ge (x + 1)^n$ for all $x > 0$.

MOAA Team Rounds, 2018.5
Tags: algebra , team
Mr. DoBa likes to listen to music occasionally while he does his math homework. When he listens to classical music, he solves one problem every $3$ minutes. When he listens to rap music, however, he only solves one problem every $5$ minutes. Mr. DoBa listens to a playlist comprised of $60\%$ classical music and $40\%$ rap music. Each song is exactly $4$ minutes long. Suppose that the expected number of problems he solves in an hour does not depend on whether or not Mr. DoBa is listening to music at any given moment, and let $m$ the average number of problems Mr. DoBa solves per minute when he is not listening to music. Determine the value of $1000m$.

2022 Bulgarian Spring Math Competition, Problem 12.3
Tags: algebra , polynomial
Let $P,Q\in\mathbb{R}[x]$, such that $Q$ is a $2021$-degree polynomial and let $a_{1}, a_{2}, \ldots , a_{2022}, b_{1}, b_{2}, \ldots , b_{2022}$ be real numbers such that $a_{1}a_{2}\ldots a_{2022}\neq 0$. If for all real $x$ \[P(a_{1}Q(x) + b_{1}) + \ldots + P(a_{2021}Q(x) + b_{2021}) = P(a_{2022}Q(x) + b_{2022})\] prove that $P(x)$ has a real root.

2017 Purple Comet Problems, 20
Tags: algebra
Let $a$ be a solution to the equation $\sqrt{x^2 + 2} = \sqrt[3]{x^3 + 45}$. Evaluate the ratio of $\frac{2017}{a^2}$ to $a^2 - 15a + 2$.

2018 IMO Shortlist, A1
Tags: functional equation , algebra
Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$

1996 Cono Sur Olympiad, 3
Tags: algebra , combinatorics
A shop sells bottles with this capacity: $1L, 2L, 3L,..., 1996L$, the prices of bottles satifies this $2$ conditions: $1$. Two bottles have the same price, if and only if, your capacities satifies $m - n = 1000$ $2$. The price of bottle $m$($1001>m>0$) is $1996 - m$ dollars. Find all pair(s) $m$ and $n$ such that: a) $m + n = 1000$ b) the cost is smallest possible!!! c) with the pair, the shop can measure $k$ liters, with $0<k<1996$(for all $k$ integer) Note: The operations to measure are: i) To fill or empty any one of two bottles ii)Pass water of a bottle for other bottle We can measure $k$ liters when the capacity of one bottle plus the capacity of other bottle is equal to $k$

2018 Canada National Olympiad, 4
Tags: number theory , polynomial , algebra
Find all polynomials $p(x)$ with real coefficients that have the following property: there exists a polynomial $q(x)$ with real coefficients such that $$p(1) + p(2) + p(3) +\dots + p(n) = p(n)q(n)$$ for all positive integers $n$.

2011 IFYM, Sozopol, 8
Tags: fraction , algebra
The fraction $\frac{1}{p}$, where $p$ is a prime number coprime with 10, is presented as an infinite periodic fraction. Prove that, if the number of digits in the period is even, then the arithmetic mean of the digits in the period is equal to $\frac{9}{2}$.

2012 Mathcenter Contest + Longlist, 6 sl14
Tags: algebra , max , inequalities
For a real number $a,b,c&gt;0$ where $bc-ca-ab=1$ find the maximum value of $$P=\frac{4024}{1+a^2}-\frac{4024}{1+b^2}-\frac{2555}{1+c^2}$$ and find out when that holds . [i](PP-nine)[/i]

2022 Saudi Arabia IMO TST, 1
Tags: floor function , algebra , summation , number theory
Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?

2018-2019 Winter SDPC, 1
Tags: vieta , algebra , polynomial
Let $r_1$, $r_2$, $r_3$ be the distinct real roots of $x^3-2019x^2-2020x+2021=0$. Prove that $r_1^3+r_2^3+r_3^3$ is an integer multiple of $3$.

2019 Taiwan TST Round 3, 2
Tags: algebra , function
Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.

2010 China Western Mathematical Olympiad, 4
Tags: algebra
Let $a_1,a_2,..,a_n,b_1,b_2,...,b_n$ be non-negative numbers satisfying the following conditions simultaneously: (1) $\displaystyle\sum_{i=1}^{n} (a_i + b_i) = 1$; (2) $\displaystyle\sum_{i=1}^{n} i(a_i - b_i) = 0$; (3) $\displaystyle\sum_{i=1}^{n} i^2(a_i + b_i) = 10$. Prove that $\text{max}\{a_k,b_k\} \le \dfrac{10}{10+k^2}$ for all $1 \le k \le n$.

2019 Miklós Schweitzer, 7
Tags: algebra , complex analysis
Given a polynomial $P$, assume that $L = \{z \in \mathbb{C}: |P(z)| = 1\}$ is a Jordan curve. Show that the zeros of $P'$ are in the interior of $L$.

1973 IMO Shortlist, 11
Tags: algebra , root , minimum value , coefficient , polynomial
Determine the minimum value of $a^{2} + b^{2}$ when $(a,b)$ traverses all the pairs of real numbers for which the equation \[ x^{4} + ax^{3} + bx^{2} + ax + 1 = 0 \] has at least one real root.

1983 IMO Longlists, 54
Tags: algebra , system of equations
Find all solutions of the following system of $n$ equations in $n$ variables: \[\begin{array}{c}\ x_1|x_1| - (x_1 - a)|x_1 - a| = x_2|x_2|,x_2|x_2| - (x_2 - a)|x_2 - a| = x_3|x_3|,\ \vdots \ x_n|x_n| - (x_n - a)|x_n - a| = x_1|x_1|\end{array}\] where $a$ is a given number.

2004 Regional Olympiad - Republic of Srpska, 1
Tags: algebra , polynomial
Define the sequence $(a_n)_{n\geq 1}$ by $a_1=1$, $a_2=p$ and \[a_{n+1}=pa_n-a_{n-1} \textrm { for all } n>1.\] Prove that for $n>1$ the polynomial $x^n-a_nx+a_{n-1}$ is divisible by $x^2-px+1$. Using this result, solve the equation \[x^4-56x+15=0.\]

1978 IMO Longlists, 33
Tags: algebra , logarithm , function , inequalities
A sequence $(a_n)^{\infty}_0$ of real numbers is called [i]convex[/i] if $2a_n\le a_{n-1}+a_{n+1}$ for all positive integers $n$. Let $(b_n)^{\infty}_0$ be a sequence of positive numbers and assume that the sequence $(\alpha^nb_n)^{\infty}_0$ is convex for any choice of $\alpha > 0$. Prove that the sequence $(\log b_n)^{\infty}_0$ is convex.