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.

AND:
OR:
NO:

Found problems: 288

2010 District Olympiad, 2
Tags: algebra , number theory , integer
Let $x, y$ be distinct positive integers. Show that the number $$\frac{(x + y)^2}{x^3 + xy^2- x^2y -y^3}$$ is not an integer.

2015 India PRMO, 11
Tags: number theory , algebra , integer , diophantine equation
$11.$ Let $a,$ $b,$ and $c$ be real numbers such that $a-7b+8c=4.$ and $8a+4b-c=7.$ What is the value of $a^2-b^2+c^2 ?$

2001 Tuymaada Olympiad, 7
Tags: integer , fractional part , algebra
Several rational numbers were written on the blackboard. Dima wrote off their fractional parts on paper. Then all the numbers on the board squared, and Dima wrote off another paper with fractional parts of the resulting numbers. It turned out that on Dima's papers were written the same sets of numbers (maybe in different order). Prove that the original numbers on the board were integers. (The fractional part of a number $x$ is such a number $\{x\}, 0 \le \{x\} <1$, that $x-\{x\}$ is an integer.)

2020 New Zealand MO, 1
Tags: number theory , integer
What is the maximum integer $n$ such that $\frac{50!}{2^n}$ is an integer?

2014 India PRMO, 6
Tags: minimum , trinomial , integer , root
What is the smallest possible natural number $n$ for which the equation $x^2 -nx + 2014 = 0$ has integer roots?

2006 Tournament of Towns, 4
Tags: number theory , geometric progression , arithmetic progression , sequence , integer
Every term of an infinite geometric progression is also a term of a given infinite arithmetic progression. Prove that the common ratio of the geometric progression is an integer. (4)

2010 Ukraine Team Selection Test, 6
Tags: number theory , integer , divisibility
Find all pairs of odd integers $a$ and $b$ for which there exists a natural number$ c$ such that the number $\frac{c^n+1}{2^na+b}$ is integer for all natural $n$.

2018 India PRMO, 9
Tags: root , integer , trinomial , maximum
Suppose $a, b$ are integers and $a+b$ is a root of $x^2 +ax+b = 0$. What is the maximum possible value of $b^2$?

2021 Turkey Junior National Olympiad, 1
Tags: number theory , integer
Find all $(m, n)$ positive integer pairs such that both $\frac{3n^2}{m}$ and $\sqrt{n^2+m}$ are integers.

Ukrainian TYM Qualifying - geometry, 2019.8
Tags: integer , isosceles , geometry
Hannusya, Petrus and Mykolka drew independently one isosceles triangle $ABC$, all angles of which are measured as a integer number of degrees. It turned out that the bases $AC$ of these triangles are equals and for each of them on the ray $BC$ there is a point $E$ such that $BE=AC$, and the angle $AEC$ is also measured by an integer number of degrees. Is it in necessary that: a) all three drawn triangles are equal to each other? b) among them there are at least two equal triangles?

2014 Hanoi Open Mathematics Competitions, 7
Tags: floor function , algebra , integer
Determine the integral part of $A$, where $A =\frac{1}{672}+\frac{1}{673}+... +\frac{1}{2014}$

2003 Korea Junior Math Olympiad, 4
Tags: multiple , integer , combinatorics
When any $11$ integers are given, prove that you can always choose $6$ integers among them so that the sum of the chosen numbers is a multiple of $6$. The $11$ integers aren't necessarily different.

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? $\textbf{(A)} \frac{1}{240} \qquad \textbf{(B)} \frac{1}{221} \qquad \textbf{(C)} \frac{1}{105} \qquad \textbf{(D)} \frac{1}{84} \qquad \textbf{(E)} \frac{1}{63}$.

2003 Estonia National Olympiad, 2
Tags: integer , number theory
Find all possible integer values of $\frac{m^2+n^2}{mn}$ where m and n are integers.

1988 Mexico National Olympiad, 4
Tags: integer , combinatorics
In how many ways can one select eight integers $a_1,a_2, ... ,a_8$, not necesarily distinct, such that $1 \le a_1 \le ... \le a_8 \le 8$?

1972 Putnam, B4
Tags: polynomial , integer
Show that for $n > 1$ we can find a polynomial $P(a, b, c)$ with integer coefficients such that $$P(x^{n},x^{n+1},x+x^{n+2})=x.$$

2019 Nigerian Senior MO Round 3, 4
Tags: combinatorial geometry , integer , grid , rectangle , 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]

2008 Cuba MO, 3
Tags: number theory , integer
Prove that there are infinitely many ordered pairs of positive integers $(m, n)$ such that $\frac{m+1}{n}+\frac{n+1}{m}$ is a positive integer.

2011 Saudi Arabia Pre-TST, 2.1
Tags: number theory , integer
Let $n$ be a positive integer. Prove that the interval $I_n= \left( \frac{1+\sqrt{8n+1}}{2}, \frac{1+\sqrt{8n+9}}{2}\right)$ does not contain any integer.

2003 BAMO, 3
Tags: combinatorial geometry , lattice points , integer , area , geometry
A lattice point is a point $(x, y)$ with both $x$ and $y$ integers. Find, with proof, the smallest $n$ such that every set of $n$ lattice points contains three points that are the vertices of a triangle with integer area. (The triangle may be degenerate, in other words, the three points may lie on a straight line and hence form a triangle with area zero.)

2012 Thailand Mathematical Olympiad, 2
Tags: algebra , integer , number theory
Let $a_1, a_2, ..., a_{2012}$ be pairwise distinct integers. Show that the equation $(x -a_1)(x - a_2)...(x - a_{2012}) = (1006!)^2$ has at most one integral solution.

2011 Saudi Arabia Pre-TST, 3
Tags: number theory , radical , integer
Find all integers $n \ge 2$ for which $\sqrt[n]{3^n+ 4^n+5^n+8^n+10^n}$ is an integer.

2015 Bundeswettbewerb Mathematik Germany, 2
Tags: number theory , sum , integer , positive
A sum of $335$ pairwise distinct positive integers equals $100000$. a) What is the least number of uneven integers in that sum? b) What is the greatest number of uneven integers in that sum?

2007 Postal Coaching, 2
Tags: number theory , algebra , integer
Let $a, b, c$ be nonzero integers such that $M = \frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ and $N =\frac{a}{c}+\frac{b}{a}+\frac{c}{b}$ are both integers. Find $M$ and $N$.

2013 JBMO Shortlist, 6
Tags: number theory , integer , diophantine equation
Solve in integers the system of equations: $$x^2-y^2=z$$ $$3xy+(x-y)z=z^2$$