Olympiad problems

2019 Durer Math Competition Finals, 2

Tags: number theory , right triangle , geometry , sidelengths , prime , integer

Prove that if a triangle has integral side lengths and its circumradius is a prime number then the triangle is right-angled.

2012 Romania National Olympiad, 3

Tags: number theory , integer

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 $a$.

2014 India PRMO, 13

Tags: number theory , integer , divide

For how many natural numbers $n$ between $1$ and $2014$ (both inclusive) is $\frac{8n}{9999-n}$ an integer?

2004 Peru MO (ONEM), 4

Tags: minimum , area of a triangle , integer , sides of a triangle , geometry

Find the smallest real number $x$ for which exist two non-congruent triangles, whose sides have integer lengths and the numerical value of the area of each triangle is $x$.

1985 All Soviet Union Mathematical Olympiad, 400

Tags: algebra , polynomial , trinomial , integer , maximum

The senior coefficient $a$ in the square polynomial $$P(x) = ax^2 + bx + c$$ is more than $100$. What is the maximal number of integer values of $x$, such that $|P(x)|<50$.

2023 AMC 10, 14

Tags: algebra , integer

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

2013 Grand Duchy of Lithuania, 1

Tags: functional , increasing , integer

Let $f : R \to R$ and $g : R \to R$ be strictly increasing linear functions such that $f(x)$ is an integer if and only if $g(x)$ is an integer. Prove that $f(x) - g(x)$ is an integer for any $x \in R$.

1998 Bosnia and Herzegovina Team Selection Test, 5

Tags: number theory , integer

Let $a$, $b$ and $c$ be integers such that $$bc+ad=1$$ $$ac+2bd=1$$ Prove that $a^2+c^2=2b^2+2d^2$

2002 Singapore Team Selection Test, 2

Tags: number theory , floor function , integer , divide , divisible

For each real number $x$, $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$. For example $\lfloor 2.8 \rfloor = 2$. Let $r \ge 0$ be a real number such that for all integers $m, n, m|n$ implies $\lfloor mr \rfloor| \lfloor nr \rfloor$. Prove that $r$ is an integer.

2016 Romania National Olympiad, 1

Tags: number theory , integer

Find all non-negative integers $n$ so that $\sqrt{n + 3}+ \sqrt{n +\sqrt{n + 3}} $ is an integer.

2020 HK IMO Preliminary Selection Contest, 2

Tags: algebra , integer

Let $x$, $y$, $z$ be positive integers satisfying $x<y<z$ and $x+xy+xyz=37$. Find the greatest possible value of $x+y+z$.

1953 Putnam, B2

Tags: integer , polynomial

Let $a_0 ,a_1 , \ldots, a_n$ be real numbers and let $f(x) =a_n x^n +\ldots +a_1 x +a_0.$ Suppose that $f(i)$ is an integer for all $i.$ Prove that $n! \cdot a_k$ is an integer for each $k.$

2010 Hanoi Open Mathematics Competitions, 6

Tags: number theory , quadratic , integer

Let $a,b$ be the roots of the equation $x^2-px+q = 0$ and let $c, d$ be the roots of the equation $x^2 - rx + s = 0$, where $p, q, r,s$ are some positive real numbers. Suppose that $M =\frac{2(abc + bcd + cda + dab)}{p^2 + q^2 + r^2 + s^2}$ is an integer. Determine $a, b, c, d$.