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

2010 Irish Math Olympiad, 2
Tags: polynomial , algebra
For each odd integer $p\ge 3$ find the number of real roots of the polynomial $$f_p(x)=(x-1)(x-2)\cdots (x-p+1)+1.$$

2015 Iberoamerican Math Olympiad, 5
Tags: algebra
Find all pairs of integers $(a,b)$ such that $(b^2+7(a-b))^2=a^{3}b$.

LMT Accuracy Rounds, 2023 S8
Tags: algebra
Ephramis taking his final exams. He has $7$ exams and his school holds finals over $3$ days. For a certain arrangement of finals, let $f$ be the maximum number of finals Ephram takes on any given day. Find the expected value of $f$ .

2011 Paraguay Mathematical Olympiad, 4
Tags: algebra
A positive integer $N$ is divided in $n$ parts inversely proportional to the numbers $2, 6, 12, 20, \ldots$ The smallest part is equal to $\frac{1}{400} N$. Find the value of $n$.

2006 Estonia National Olympiad, 4
Tags: floor function , equation , algebra
Solve the equation $\left[\frac{x}{3}\right]+\left [\frac{2x}{3}\right]=x $

2004 Harvard-MIT Mathematics Tournament, 7
Tags: algebra , geometry
Farmer John is grazing his cows at the origin. There is a river that runs east to west $50$ feet north of the origin. The barn is $100$ feet to the south and $80$ feet to the east of the origin. Farmer John leads his cows to the river to take a swim, then the cows leave the river from the same place they entered and Farmer John leads them to the barn. He does this using the shortest path possible, and the total distance he travels is $d$ feet. Find the value of $d$.

2023 Durer Math Competition Finals, 1
Tags: prime number , number theory , algebra
Nüx has three moira daughters, whose ages are three distinct prime numbers, and the sum of their squares is also a prime number. What is the age of the youngest moira?

2002 Romania Team Selection Test, 2
Tags: algebra , polynomial , number theory
Let $P(x)$ and $Q(x)$ be integer polynomials of degree $p$ and $q$ respectively. Assume that $P(x)$ divides $Q(x)$ and all their coefficients are either $1$ or $2002$. Show that $p+1$ is a divisor of $q+1$. [i]Mihai Cipu[/i]

1998 Switzerland Team Selection Test, 3
Tags: algebra , min , function
Given positive numbers $a,b,c$, find the minimum of the function $f(x) = \sqrt{a^2 +x^2} +\sqrt{(b-x)^2 +c^2}$.

2025 Euler Olympiad, Round 2, 3
Tags: trigonometry , functional equation , algebra
Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that the following two conditions hold: [b]1. [/b] For all real numbers $a$ and $b$ satisfying $a^2 + b^2 = 1$, We have $f(x) + f(y) \geq f(ax + by)$ for all real numbers $x, y$. [b]2.[/b] For all real numbers $x$ and $y$, there exist real numbers $a$ and $b$, such that $a^2 + b^2 = 1$ and $f(x) + f(y) = f(ax + by)$. [i]Proposed by Zaza Melikidze, Georgia[/i]

1987 Tournament Of Towns, (154) 5
Tags: algebra , inequalities
We are given three non-negative numbers $A , B$ and $C$ about which it is known that $$A^4 + B^4 + C^4 \le 2(A^2B^2 + B^2C^2 + C^2A^2)$$ (a) Prove that each of $A, B$ and $C$ is not greater than the sum of the others. (b) Prove that $A^2 + B^2 + C^2 \le 2(AB + BC + CA)$ . (c) Does the original inequality follow from the one in (b)? (V.A. Senderov , Moscow)

2018 Moscow Mathematical Olympiad, 1
Tags: algebra , inequalities
$a_1,a_2,...,a_{81}$ are nonzero, $a_i+a_{i+1}>0$ for $i=1,...,80$ and $a_1+a_2+...+a_{81}<0$. What is sign of $a_1*a_2*...*a_{81}$?

2018 IOM, 1
Tags: algebra
Solve the system of equations in real numbers: \[ \begin{cases*} (x - 1)(y - 1)(z - 1) = xyz - 1,\\ (x - 2)(y - 2)(z - 2) = xyz - 2.\\ \end{cases*} \] [i]Vladimir Bragin[/i]

2004 Thailand Mathematical Olympiad, 4
Tags: equation , radical , algebra
Find all real solutions $x$ to the equation $$x =\sqrt{x -\frac{1}{x}} +\sqrt{1 -\frac{1}{x}}$$

2007 Putnam, 1
Tags: algebra , polynomial , function
Let $ f$ be a polynomial with positive integer coefficients. Prove that if $ n$ is a positive integer, then $ f(n)$ divides $ f(f(n)\plus{}1)$ if and only if $ n\equal{}1.$

2021/2022 Tournament of Towns, P1
Tags: combinatorics , algebra
Alice wrote a sequence of $n > 2$ nonzero nonequal numbers such that each is greater than the previous one by the same amount. Bob wrote the inverses of those n numbers in some order. It so happened that each number in his row also is greater than the previous one by the same amount, possibly not the same as in Alice’s sequence. What are the possible values of $n{}$? [i]Alexey Zaslavsky[/i]

2004 China Team Selection Test, 3
Tags: integration , calculus , number theory , polynomial , algebra
Given arbitrary positive integer $ a$ larger than $ 1$, show that for any positive integer $ n$, there always exists a n-degree integral coefficient polynomial $ p(x)$, such that $ p(0)$, $ p(1)$, $ \cdots$, $ p(n)$ are pairwise distinct positive integers, and all have the form of $ 2a^k\plus{}3$, where $ k$ is also an integer.

2016 BMT Spring, 15
Tags: algebra
Let $s_1, s_2, s_3$ be the three roots of $x^3 + x^2 +\frac92x + 9$. $$\prod_{i=1}^{3}(4s^4_i + 81)$$ can be written as $2^a3^b5^c$. Find $a + b + c$.

2008 Purple Comet Problems, 7
Tags: polynomial , parabola , analytic geometry , graphing lines , slope , conic , algebra
A line through the origin passes through the curve whose equation is $5y=2x^2-9x+10$ at two points whose $x-$coordinates add up to $77.$ Find the slope of the line.

2015 Caucasus Mathematical Olympiad, 2
Tags: trinomial , inequalities , algebra
The equation $(x+a) (x+b) = 9$ has a root $a+b$. Prove that $ab\le 1$.

1967 IMO Shortlist, 3
Tags: equation , algebra , summation , diophantine equation , polynomial
Suppose that $p$ and $q$ are two different positive integers and $x$ is a real number. Form the product $(x+p)(x+q).$ Find the sum $S(x,n) = \sum (x+p)(x+q),$ where $p$ and $q$ take values from 1 to $n.$ Does there exist integer values of $x$ for which $S(x,n) = 0.$

2010 Saudi Arabia IMO TST, 3
Tags: functional , function , functional equation , algebra
Let $f : N \to N$ be a strictly increasing function such that $f(f(n))= 3n$, for all $n \in N$. Find $f(2010)$. Note: $N = \{0,1,2,...\}$

2021 LMT Fall, 1
Tags: algebra
Kevin writes the multiples of three from $1$ to $100$ on the whiteboard. How many digits does he write?

2023 Hong Kong Team Selection Test, Problem 1
Tags: inequality , easy , algebra
Suppose $a$, $b$ and $c$ are nonzero real numberss satisfying $abc=2$. Prove that among the three numbers $2a-\frac{1}{b}$, $2b-\frac{1}{c}$ and $2c-\frac{1}{a}$, at most two of them are greater than $2$.

2008 JBMO Shortlist, 6
Tags: inequalities , algebra
If the real numbers $a, b, c, d$ are such that $0 < a,b,c,d < 1$, show that $1 + ab + bc + cd + da + ac + bd > a + b + c + d$.