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: 288

1965 Polish MO Finals, 2
Tags: algebra , number theory , coprime , integer
Prove that if the numbers $ x_1 $ and $ x_2 $ are roots of the equation $ x^2 + px - 1 = 0 $, where $ p $ is an odd number, then for every natural $n$number $ x_1^n + x_2^n $ and $ x_1^{n+1} + x_2^{n+1} $ are integer and coprime.

2011 Greece JBMO TST, 3
Tags: integer , number theory , diophantine equation
Find integer solutions of the equation $8x^3 - 4 = y(6x - y^2)$

2019 Nigerian Senior MO Round 3, 4
Tags: combinatorial geometry , integer , rectangle , grid , combinatorics
A rectangular grid whose side lengths are integers greater than $1$ is given. Smaller rectangles with area equal to an odd integer and length of each side equal to an integer greater than $1$ are cut out one by one. Finally one single unit is left. Find the least possible area of the initial grid before the cuttings. Ps. Collected [url=https://artofproblemsolving.com/community/c949611_2019_nigerian_senior_mo_round_3]here[/url]

2018 Junior Regional Olympiad - FBH, 2
Tags: integer , frac integer
Find all integers $m$ such that $\frac{2m^2+7m-9}{m^2+m+1}$ is integer

2017 India National Olympiad, 6
Tags: binomial coefficient , integer , binomial theorem , geometry , algebra , heron's formula
Let $n\ge 1$ be an integer and consider the sum $$x=\sum_{k\ge 0} \dbinom{n}{2k} 2^{n-2k}3^k=\dbinom{n}{0}2^n+\dbinom{n}{2}2^{n-2}\cdot{}3+\dbinom{n}{4}2^{n-k}\cdot{}3^2 + \cdots{}.$$ Show that $2x-1,2x,2x+1$ form the sides of a triangle whose area and inradius are also integers.

2024 AMC 12/AHSME, 6
Tags: integer
The product of three integers is $60$. What is the least possible positive sum of the three integers? $\textbf{(A) } 2 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 6 \qquad \textbf{(E) } 13$

2024 AMC 12/AHSME, 17
Tags: integer , probability
Integers $a$ and $b$ are randomly chosen without replacement from the set of integers with absolute value not exceeding $10$. What is the probability that the polynomial $x^3 + ax^2 + bx + 6$ has $3$ distinct integer roots? $\textbf{(A)} \frac{1}{240} \qquad \textbf{(B)} \frac{1}{221} \qquad \textbf{(C)} \frac{1}{105} \qquad \textbf{(D)} \frac{1}{84} \qquad \textbf{(E)} \frac{1}{63}$.

2006 Estonia Team Selection Test, 1
Tags: integer , algebra , quadratic , trinomial , integer polynomial
Let $k$ be any fixed positive integer. Let's look at integer pairs $(a, b)$, for which the quadratic equations $x^2 - 2ax + b = 0$ and $y^2 + 2ay + b = 0$ are real solutions (not necessarily different), which can be denoted by $x_1, x_2$ and $y_1, y_2$, respectively, in such an order that the equation $x_1 y_1 - x_2 y_2 = 4k$. a) Find the largest possible value of the second component $b$ of such a pair of numbers ($a, b)$. b) Find the sum of the other components of all such pairs of numbers.

2023 AMC 10, 14
Tags: integer , algebra
How many ordered pairs of integers $(m, n)$ satisfy the equation $m^2+mn+n^2=m^2n^2$? $\textbf{(A) }7\qquad\textbf{(B) }1\qquad\textbf{(C) }3\qquad\textbf{(D) }6\qquad\textbf{(E) }5$

1999 Ukraine Team Selection Test, 2
Tags: integer , algebra
Show that there exist integers $j,k,l,m,n$ greater than $100$ such that $j^2 +k^2 +l^2 +m^2 +n^2 = jklmn-12$.

2000 Greece JBMO TST, 3
Tags: polynomial , integer , number theory
Find $a\in Z$ such that the equation $2x^2+2ax+a-1=0$ has integer solutions, which should be found.

2000 All-Russian Olympiad Regional Round, 11.3
Tags: number theory , integer , algebra
Sequence of real numbers $a_1, a_2, . . . , a_{2000}$ is such that for any natural number $n$, $1\le n \le 2000$, the equality $$a^3_1+ a^3_2+... + a^3_n = (a_1 + a_2 +...+ a_n)^2.$$ Prove that all terms of this sequence are integers.

2012 Tournament of Towns, 4
Tags: number theory , algebra , integer
Brackets are to be inserted into the expression $10 \div 9 \div 8 \div 7 \div 6 \div 5 \div 4 \div 3 \div 2$ so that the resulting number is an integer. (a) Determine the maximum value of this integer. (b) Determine the minimum value of this integer.

2017 Singapore Senior Math Olympiad, 5
Tags: integer , arithmetic progression , algebra , combinatorics , sequence , number theory
Given $7$ distinct positive integers, prove that there is an infinite arithmetic progression of positive integers $a, a + d, a + 2d,..$ with $a < d$, that contains exactly $3$ or $4$ of the $7$ given integers.

2025 Greece National Olympiad, 1
Tags: algebra , polynomial , root , integer
Let $P(x)=x^4+5x^3+mx^2+5nx+4$ have $2$ distinct real roots, which sum up to $-5$. If $m,n \in \mathbb {Z_+}$, find the values of $m,n$ and their corresponding roots.

1989 Nordic, 4
Tags: integer , combinatorics
For which positive integers $n$ is the following statement true: if $a_1, a_2, ... , a_n$ are positive integers, $a_k \le n$ for all $k$ and $\sum\limits_{k=1}^{{n}}{a_k}=2n$ then it is always possible to choose $a_{i1} , a_{i2} , ..., a_{ij}$ in such a way that the indices $i_1, i_2,... , i_j$ are different numbers, and $\sum\limits_{k=1}^{{{j}}}{a_{ik}}=n$?

2020 Dutch IMO TST, 4
Tags: number theory , integer
Let $a, b \ge 2$ be positive integers with $gcd (a, b) = 1$. Let $r$ be the smallest positive value that $\frac{a}{b}- \frac{c}{d}$ can take, where $c$ and $d$ are positive integers satisfying $c \le a$ and $d \le b$. Prove that $\frac{1}{r}$ is an integer.

VMEO III 2006 Shortlist, N12
Tags: number theory , integer
Given a positive integer $n > 1$. Find the smallest integer of the form $\frac{n^a-n^b}{n^c-n^d}$ for all positive integers $a,b,c,d$.

1998 Singapore Senior Math Olympiad, 1
Tags: integer , divisible , divide , number theory
Prove that $1998! \left( 1+ \frac12 + \frac13 +...+\frac{1}{1998}\right)$ is an integer divisible by $1999$.

1993 Bundeswettbewerb Mathematik, 1
Tags: integer , combinatorics , sum , summand
Every positive integer $n>2$ can be written as a sum of distinct positive integers. Let $A(n)$ be the maximal number of summands in such a representation. Find a formula for $A(n).$

2005 Bosnia and Herzegovina Junior BMO TST, 2
Tags: perfect square , integer , number theory
Let n be a positive integer. Prove the following statement: ”If $2 + 2\sqrt{1 + 28n^2}$ is an integer, then it is the square of an integer.”

2008 Greece JBMO TST, 3
Tags: integer , combinatorics , difference
Let $x_1,x_2,x_3,...,x_{102}$ be natural numbers such that $x_1<x_2<x_3<...<x_{102}<255$. Prove that among the numbers $d_1=x_2-x_1, d_2=x_3-x_2, ..., d_{101}=x_{102}-x_{101}$ there are at least $26$ equal.

2011 Saudi Arabia Pre-TST, 2.3
Tags: number theory , integer
Let $x, y$ be distinct positive integers. Prove that the number $$\frac{(x+y)^2}{ x^3 + xy^2 - x^2y - y^3}$$ is not an integer

1948 Moscow Mathematical Olympiad, 144
Tags: integer , number theory
Prove that if $\frac{2^n- 2}{n} $ is an integer, then so is $\frac{2^{2^n-1}-2}{2^n - 1}$ .

2005 Bosnia and Herzegovina Team Selection Test, 6
Tags: integer , number theory , perfect cube
Let $a$, $b$ and $c$ are integers such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=3$. Prove that $abc$ is a perfect cube of an integer.