Found problems: 112
2021 Indonesia TST, N
For every positive integer $n$, let $p(n)$ denote the number of sets $\{x_1, x_2, \dots, x_k\}$ of integers with $x_1 > x_2 > \dots > x_k > 0$ and $n = x_1 + x_3 + x_5 + \dots$ (the right hand side here means the sum of all odd-indexed elements). As an example, $p(6) = 11$ because all satisfying sets are as follows: $$\{6\}, \{6, 5\}, \{6, 4\}, \{6, 3\}, \{6, 2\}, \{6, 1\}, \{5, 4, 1\}, \{5, 3, 1\}, \{5, 2, 1\}, \{4, 3, 2\}, \{4, 3, 2, 1\}.$$ Show that $p(n)$ equals to the number of partitions of $n$ for every positive integer $n$.
2019 Czech-Polish-Slovak Junior Match, 1
Rational numbers $a, b$ are such that $a+b$ and $a^2+b^2$ are integers. Prove that $a, b$ are integers.
2024 AMC 10, 7
The product of three integers is $60$. What is the least possible positive sum of the three integers?
$\textbf{(A) } 2 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 6 \qquad \textbf{(E) } 13$
1980 Bundeswettbewerb Mathematik, 4
A sequence of integers $a_1,a_2,\ldots $ is defined by $a_1=1,a_2=2$ and for $n\geq 1$,
$$a_{n+2}=\left\{\begin{array}{cl}5a_{n+1}-3a_{n}, &\text{if}\ a_n\cdot a_{n+1}\ \text{is even},\\ a_{n+1}-a_{n}, &\text{if}\ a_n\cdot a_{n+1}\ \text{is odd},\end{array}\right. $$
(a) Prove that the sequence contains infinitely many positive terms and infinitely many negative terms.
(b) Prove that no term of the sequence is zero.
(c) Show that if $n = 2^k - 1$ for $k\geq 2$, then $a_n$ is divisible by $7$.
1968 Czech and Slovak Olympiad III A, 2
Show that for any integer $n$ the number \[a_n=\frac{\bigl(2+\sqrt3\bigr)^n-\bigl(2-\sqrt3\bigr)^n}{2\sqrt3}\] is also integer. Determine all integers $n$ such that $a_n$ is divisible by 3.
1994 Denmark MO - Mohr Contest, 4
In a right-angled triangle in which all side lengths are integers, one has a cathetus length $1994$. Determine the length of the hypotenuse.
1998 Austrian-Polish Competition, 5
Determine all pairs $(a, b)$ of positive integers for which the equation $x^3 - 17x^2 + ax - b^2 = 0$ has three integer roots (not necessarily different).
2015 Costa Rica - Final Round, N4
Show that there are no triples $(a, b, c)$ of positive integers such that
a) $a + c, b + c, a + b$ do not have common multiples in pairs.
b)$\frac{c^2}{a + b},\frac{b^2}{a + c},\frac{a^2}{c + b}$ are integer numbers.
2006 All-Russian Olympiad Regional Round, 10.7
For what positive integers $n$ are there positive rational, but not integer, numbers $a$ and $b$ such that both numbers $a + b$ and $a^n + b^n$ are integers?
1960 Putnam, A1
Let $n$ be a given positive integer. How many solutions are there in ordered positive integer pairs $(x,y)$ to the equation
$$\frac{xy}{x+y}=n?$$
2017 India National Olympiad, 6
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.
1977 Bundeswettbewerb Mathematik, 3
The number $50$ is written as a sum of several positive integers (not necessarily distinct) whose product is divisible by $100.$ What is the largest possible value of this product?