Olympiad problems

1998 Mexico National Olympiad, 4

Find all integers that can be written in the form $\frac{1}{a_1}+\frac{2}{a_2}+...+\frac{9}{a_9}$ where $a_1,a_2, ...,a_9$ are nonzero digits, not necessarily different.

2004 German National Olympiad, 4

Tags: algebra , square roots , Sum , Integers

For a positive integer $n,$ let $a_n$ be the integer closest to $\sqrt{n}.$ Compute $$ \frac{1}{a_1 } + \frac{1}{a_2 }+ \cdots + \frac{1}{a_{2004}}.$$

Indonesia Regional MO OSP SMA - geometry, 2005.4

Tags: geometry , perimeter , area , Integers

The lengths of the three sides $a, b, c$ with $a \le b \le c$, of a right triangle is an integer. Find all the sequences $(a, b, c)$ so that the values of perimeter and area of the triangle are the same.

2015 India PRMO, 3

Tags: Integers , number theory , algebra

$3.$ Positive integers $a$ and $b$ are such that $a+b=\frac{a}{b}+\frac{b}{a}.$ What is the value of $a^2+b^2 ?$

2006 Abels Math Contest (Norwegian MO), 3

Tags: number theory , diophantine , rational , Integers

(a) Let $a$ and $b$ be rational numbers such that line $y = ax + b$ intersects the circle $x^2 + y^2 = 5$ at two different points. Show that if one of the intersections has two rational coordinates, so does the other intersection. (b) Show that there are infinitely many triples ($k, n, m$) that are such that $k^2 + n^2 = 5m^2$, where $k, n$ and $m$ are integers, and not all three have any in common prime factor.

2015 India Regional MathematicaI Olympiad, 2

Tags: polynomial , Integers , quadratic trinomial , algebra

Let $P(x)=x^{2}+ax+b$ be a quadratic polynomial where $a$ is real and $b \neq 2$, is rational. Suppose $P(0)^{2},P(1)^{2},P(2)^{2}$ are integers, prove that $a$ and $b$ are integers.

2013 JBMO Shortlist, 6

Tags: number theory , Diophantine Equations , Integers

Solve in integers the system of equations: $$x^2-y^2=z$$ $$3xy+(x-y)z=z^2$$

1977 Bundeswettbewerb Mathematik, 3

Tags: number theory , infinitely many solutions , Integers , Generalization

Show that there are infinitely many positive integers $a$ that cannot be written as $a = a_{1}^{6}+ a_{2}^{6} + \ldots + a_{7}^{6},$ where the $a_i$ are positive integers. State and prove a generalization.

2022 Turkey EGMO TST, 3

Tags: number theory , Integers

Find all pairs of integers $(a,b)$ satisfying the equation $a^7(a-1)=19b(19b+2)$.

1985 Brazil National Olympiad, 4

Tags: Diophantine equation , number theory , trinomial , Integers

$a, b, c, d$ are integers. Show that $x^2 + ax + b = y^2 + cy + d$ has infinitely many integer solutions iff $a^2 - 4b = c^2 - 4d$.

2020 Malaysia IMONST 1, 10

Tags: Integers

Given positive integers $a, b,$ and $c$ with $a + b + c = 20$. Determine the number of possible integer values for $\frac{a + b}{c}.$

1997 Romania National Olympiad, 1

Tags: linear algebra , matrix , Integers , modulo

Let $m \ge 2$ and $n \ge 1$ be integers and $A=(a_{ij})$ a square matrix of order $n$ with integer entries. Prove that for any permutation $\sigma \in S_n$ there is a function $\varepsilon : \{1,2,\ldots,n\} \to \{0,1\}$ such that replacing the entries $a_{\sigma(1)1},$ $a_{\sigma(2)2}, $ $\ldots,$ $a_{\sigma(n)n}$ of $A$ respectively by $$a_{\sigma(1)1}+\varepsilon(1), ~a_{\sigma(2)2}+\varepsilon(2), ~\ldots, ~a_{\sigma(n)n}+\varepsilon(n),$$ the determinant of the matrix $A_{\varepsilon}$ thus obtained is not divisible by $m.$

2003 Korea Junior Math Olympiad, 5

Tags: number theory , KJMO , Integers

Four odd positive intgers $a, b, c, d (a\leq b \leq c\leq d)$ are given. Choose any three numbers among them and divide their sum by the un-chosen number, and you will always get the remainder as $1$. Find all $(a, b, c, d)$ that satisfies this.

2008 Tournament Of Towns, 5

Tags: combinatorics , prime , Integers

The positive integers are arranged in a row in some order, each occuring exactly once. Does there always exist an adjacent block of at least two numbers somewhere in this row such that the sum of the numbers in the block is a prime number?

2014 Ukraine Team Selection Test, 5

Tags: Integers , number theory , modulo

Find all positive integers $n \ge 2$ such that equality $i+j \equiv C_{n}^{i} + C_{n}^{j}$ (mod $2$) is true for arbitrary $0 \le i \le j \le n$.

2015 Indonesia MO Shortlist, N5

Tags: number theory , Diophantine equation , Integers

Given a prime number $n \ge 5$. Prove that for any natural number $a \le \frac{n}{2} $, we can search for natural number $b \le \frac{n}{2}$ so the number of non-negative integer solutions $(x, y)$ of the equation $ax+by=n$ to be odd*. Clarification: * For example when $n = 7, a = 3$, we can choose$ b = 1$ so that there number of solutions og $3x + y = 7$ to be $3$ (odd), namely: $(0, 7), (1, 4), (2, 1)$

2024 Thailand TST, 3

Tags: functional equation , IMO Shortlist , Integers

Determine all functions $f\colon\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ such that, for all positive integers $a$ and $b$, \[ f^{bf(a)}(a+1)=(a+1)f(b). \]

1999 Ukraine Team Selection Test, 2

Tags: Integers , algebra

Show that there exist integers $j,k,l,m,n$ greater than $100$ such that $j^2 +k^2 +l^2 +m^2 +n^2 = jklmn-12$.

1998 Romania National Olympiad, 1

Tags: algebra , Integers , combinatorics

Let $n \ge 2$ be an integer and $M= \{1,2,\ldots,n\}.$ For each $k \in \{1,2,\ldots,n-1\}$ we define $$x_k= \frac{1}{n+1} \sum_{\substack{A \subset M \\ |A|=k}} (\min A + \max A).$$ Prove that the numbers $x_k$ are integers and not all of them are divisible by $4.$ [hide=Notations]$|A|$ is the cardinal of $A$ $\min A$ is the smallest element in $A$ $\max A$ is the largest element in $A$[/hide]

1995 Czech And Slovak Olympiad IIIA, 2

Tags: Sum , Integers , algebra

Find the positive real numbers $x,y$ for which $\frac{x+y}{2},\sqrt{xy},\frac{2xy}{x+y},\sqrt{\frac{x^2 +y^2}{2}}$ are integers whose sum is $66$.

1977 Bundeswettbewerb Mathematik, 1

Tags: Integers , algebra , number theory , Sum , divisible

Among $2000$ distinct positive integers, there are equally many even and odd ones. The sum of the numbers is less than $3000000.$ Show that at least one of the numbers is divisible by $3.$

2006 Korea Junior Math Olympiad, 5

Tags: number theory , coprime , positive integers , Integers

Find all positive integers that can be written in the following way $\frac{m^2 + 20mn + n^2}{m^3 + n^3}$ Also, $m,n$ are relatively prime positive integers.

2019 Hanoi Open Mathematics Competitions, 12

Tags: polynomial , algebra , combinatorics , Integers

Given an expression $x^2 + ax + b$ where $a,b$ are integer coefficients. At any step, one can change the expression by adding either $1$ or $-1$ to only one of the two coefficients $a, b$. a) Suppose that the initial expression has $a =-7$ and $b = 19$. Show your modification steps to obtain a new expression that has zero value at some integer value of $x$. b) Starting from the initial expression as above, one gets the expression $x^2 - 17x + 9$ after $m$ modification steps. Prove that at a certain step $k$ with $k < m$, the obtained expression has zero value at some integer value of $x$.

2008 Hanoi Open Mathematics Competitions, 1

Tags: Integers , number theory , algebra

How many integers are there in $(b,2008b]$, where $b$ ($b > 0$) is given.

2005 Abels Math Contest (Norwegian MO), 1b

Tags: geometry , 3D geometry , pyramid , Even , Integers , Volume

In a pyramid, the base is a right-angled triangle with integer sides. The height of the pyramid is also integer. Show that the volume of the pyramid is even.