Found problems: 177
2020 Dutch IMO TST, 4
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.
2012 Romania National Olympiad, 3
We consider the non-zero natural numbers $(m, n)$ such that the numbers $$\frac{m^2 + 2n}{n^2 - 2m} \,\,\,\, and \,\,\, \frac{n^2 + 2m}{m^2-2n}$$ are integers.
a) Show that $|m - n| \le 2$:
b) Find all the pairs $(m, n)$ with the property from hypothesis $a$.
2008 Mathcenter Contest, 4
Let $a,b$ and $c$ be positive integers that $$\frac{a\sqrt{3}+b}{b\sqrt3+c}$$ is a rational number, show that $$\frac{a^2+b^2+c^2}{a+b+ c}$$ is an integer.
[i](Anonymous314)[/i]
2003 Korea Junior Math Olympiad, 2
$a, b$ are odd numbers that satisfy $(a-b)^2 \le 8\sqrt {ab}$. For $n=ab$, show that the equation
$$x^2-2([\sqrt n]+1)x+n=0$$ has two integral solutions. $[r]$ denotes the biggest integer, not strictly bigger than $r$.
2008 Abels Math Contest (Norwegian MO) Final, 1
Let $s(n) = \frac16 n^3 - \frac12 n^2 + \frac13 n$.
(a) Show that $s(n)$ is an integer whenever $n$ is an integer.
(b) How many integers $n$ with $0 < n \le 2008$ are such that $s(n)$ is divisible by $4$?
1934 Eotvos Mathematical Competition, 1
Let $n$ be a given positive integer and
$$A =\frac{1 \cdot 3 \cdot 5 \cdot ... \cdot (2n- 1)}{2 \cdot 4 \cdot 6 \cdot ... \cdot 2n}$$
Prove that at least one term of the sequence $A, 2A,4A,8A,...,2^kA, ... $ is an integer.
2018 Junior Regional Olympiad - FBH, 2
Find all integers $n$ such that $\frac{n+4}{3n-2}$ is integer
1948 Moscow Mathematical Olympiad, 144
Prove that if $\frac{2^n- 2}{n} $ is an integer, then so is $\frac{2^{2^n-1}-2}{2^n - 1}$ .
2001 Nordic, 1
Let ${A}$ be a finite collection of squares in the coordinate plane such that the vertices of all squares that belong to ${A}$ are ${(m, n), (m + 1, n), (m, n + 1)}$, and ${(m + 1, n + 1)}$ for some integers ${m}$ and ${n}$. Show that there exists a subcollection ${B}$ of ${A}$ such that ${B}$ contains at least ${25 \% }$ of the squares in ${A}$, but no two of the squares in ${B}$ have a common vertex.
2003 BAMO, 3
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.)
2013 Singapore Junior Math Olympiad, 4
Let $a,b,$ be positive integers and $a>b>2$. Prove that $\frac{2^a+1}{2^b-1}$ is never an integer
1986 Czech And Slovak Olympiad IIIA, 2
Let $P(x)$ be a polynomial with integer coefficients of degree $n \ge 3$.
If $x_1,...,x_m$ ($n\ge m\ge3$) are different integers such that $P(x_1) = P(x_2) = ... = P(x_m) = 1$, prove that $P$ cannot have integer roots$.
2002 Swedish Mathematical Competition, 4
For which integers $n \ge 8$ is $n^{\frac{1}{n-7}}$ an integer?
1959 Kurschak Competition, 1
$a, b, c$ are three distinct integers and $n$ is a positive integer. Show that $$\frac{a^n}{(a - b)(a - c)}+\frac{ b^n}{(b - a)(b - c)} +\frac{ c^n}{(c - a)(c - b)}$$ is an integer.
1996 Nordic, 2
Determine all real numbers $x$, such that $x^n+x^{-n}$ is an integer for all integers $n$.
2009 Sharygin Geometry Olympiad, 7
Given points $O, A_1, A_2, ..., A_n$ on the plane. For any two of these points the square of distance between them is natural number. Prove that there exist two vectors $\vec{x}$ and $\vec{y}$, such that for any point $A_i$, $\vec{OA_i }= k\vec{x}+l \vec{y}$, where $k$ and $l$ are some integer numbers.
(A.Glazyrin)
1986 All Soviet Union Mathematical Olympiad, 422
Prove that it is impossible to draw a convex quadrangle, with one diagonal equal to doubled another, the angle between them $45$ degrees, on the coordinate plane, so, that all the vertices' coordinates would be integers.
1960 Kurschak Competition, 2
Let $a_1 = 1, a_2, a_3,...$: be a sequence of positive integers such that $$a_k < 1 + a_1 + a_2 +... + a_{k-1}$$ for all $k > 1$. Prove that every positive integer can be expressed as a sum of $a_i$s.
2014 Hanoi Open Mathematics Competitions, 5
The first two terms of a sequence are $2$ and $3$. Each next term thereafter is the sum of the nearestly previous two terms if their sum is not greather than $10, 0$ otherwise. The $2014$th term is:
(A): $0$, (B): $8$, (C): $6$, (D): $4$, (E) None of the above.
2010 District Olympiad, 2
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.
2010 Dutch Mathematical Olympiad, 3
Consider a triangle $XYZ$ and a point $O$ in its interior. Three lines through $O$ are drawn, parallel to the respective sides of the triangle. The intersections with the sides of the triangle determine six line segments from $O$ to the sides of the triangle. The lengths of these segments are integer numbers $a, b, c, d, e$ and $f$ (see figure).
Prove that the product $a \cdot b \cdot c\cdot d \cdot e \cdot f$ is a perfect square.
[asy]
unitsize(1 cm);
pair A, B, C, D, E, F, O, X, Y, Z;
X = (1,4);
Y = (0,0);
Z = (5,1.5);
O = (1.8,2.2);
A = extension(O, O + Z - X, X, Y);
B = extension(O, O + Y - Z, X, Y);
C = extension(O, O + X - Y, Y, Z);
D = extension(O, O + Z - X, Y, Z);
E = extension(O, O + Y - Z, Z, X);
F = extension(O, O + X - Y, Z, X);
draw(X--Y--Z--cycle);
draw(A--D);
draw(B--E);
draw(C--F);
dot("$A$", A, NW);
dot("$B$", B, NW);
dot("$C$", C, SE);
dot("$D$", D, SE);
dot("$E$", E, NE);
dot("$F$", F, NE);
dot("$O$", O, S);
dot("$X$", X, N);
dot("$Y$", Y, SW);
dot("$Z$", Z, dir(0));
label("$a$", (A + O)/2, SW);
label("$b$", (B + O)/2, SE);
label("$c$", (C + O)/2, SE);
label("$d$", (D + O)/2, SW);
label("$e$", (E + O)/2, SE);
label("$f$", (F + O)/2, NW);
[/asy]
2013 India PRMO, 3
It is given that the equation $x^2 + ax + 20 = 0$ has integer roots. What is the sum of all possible values of $a$?
2000 Greece JBMO TST, 3
Find $a\in Z$ such that the equation $2x^2+2ax+a-1=0$ has integer solutions, which should be found.
2015 Balkan MO Shortlist, N3
Let $a$ be a positive integer. For all positive integer n, we define $ a_n=1+a+a^2+\ldots+a^{n-1}. $
Let $s,t$ be two different positive integers with the following property:
If $p$ is prime divisor of $s-t$, then $p$ divides $a-1$.
Prove that number $\frac{a_{s}-a_{t}}{s-t}$ is an integer.
(FYROM)
2021 239 Open Mathematical Olympiad, 1
You are given $n$ different primes $p_1, p_2,..., p_n$. Consider the polynomial $$x^n + a_1x^{n -1} + a_2x^{n - 2} + ...+ a_{n - 1}x + a_n$$, where $a_i$ is the product of the first $i$ given prime numbers. For what $n$ can it have an integer root?