Olympiad problems

2013 District Olympiad, 2

Find all real numbers $x$ for which the number $$a =\frac{2x + 1}{x^2 + 2x + 3}$$ is an integer.

2006 JBMO ShortLists, 15

Tags: geometry , prime number , integer , area , minimum

Let $A_1$ and $B_1$ be internal points lying on the sides $BC$ and $AC$ of the triangle $ABC$ respectively and segments $AA_1$ and $BB_1$ meet at $O$. The areas of the triangles $AOB_1,AOB$ and $BOA_1$ are distinct prime numbers and the area of the quadrilateral $A_1OB_1C$ is an integer. Find the least possible value of the area of the triangle $ABC$, and argue the existence of such a triangle.

1980 Swedish Mathematical Competition, 3

Tags: geometry , combinatorics , integer , combinatorial geometry

Let $T(n)$ be the number of dissimilar (non-degenerate) triangles with all side lengths integral and $\leq n$. Find $T(n+1)-T(n)$.

Indonesia Regional MO OSP SMA - geometry, 2005.4

Tags: geometry , integer , perimeter , area

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.

2024 ITAMO, 6

Tags: inequalities , integer , extermal , fraction , maximal

For each integer $n$, determine the smallest real number $M_n$ such that \[\frac{1}{a_1}+\frac{a_1}{a_2}+\frac{a_2}{a_3}+\dots+\frac{a_{n-1}}{a_n} \le M_n\] for any $n$-tuple $(a_1,a_2,\dots,a_n)$ of integers such that $1<a_1<a_2<\dots<a_n$.

2024 Czech-Polish-Slovak Junior Match, 4

Tags: inequalities , integer

Let $a,b,c$ be integers satisfying $a+b+c=1$ and $ab+bc+ca<abc$. Show that $ab+bc+ca<2abc$.

1973 Putnam, B1

Tags: integer

Let $a_1, a_2, \ldots a_{2n+1}$ be a set of integers such that, if any one of them is removed, the remaining ones can be divided into two sets of $n$ integers with equal sums. Prove $a_{1}=a_2 =\cdots=a_{2n+1}.$

2018 Dutch BxMO TST, 2

Tags: integer , sides of a triangle , coprime , geometry

Let $\vartriangle ABC$ be a triangle of which the side lengths are positive integers which are pairwise coprime. The tangent in $A$ to the circumcircle intersects line $BC$ in $D$. Prove that $BD$ is not an integer.

1976 Vietnam National Olympiad, 4

Tags: number theory , integer , integer equation , factorial equation

Find all three digit integers $\overline{abc} = n$, such that $\frac{2n}{3} = a! b! c!$

2015 Costa Rica - Final Round, N4

Tags: coprime , integer , number theory

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.

2018 China Girls Math Olympiad, 6

Tags: set , integer , combinatorics

Given $k \in \mathbb{N}^+$. A sequence of subset of the integer set $\mathbb{Z} \supseteq I_1 \supseteq I_2 \supseteq \cdots \supseteq I_k$ is called a $k-chain$ if for each $1 \le i \le k$ we have (i) $168 \in I_i$; (ii) $\forall x, y \in I_i$, we have $x-y \in I_i$. Determine the number of $k-chain$ in total.

2006 Estonia Team Selection Test, 1

Tags: integer , quadratic , trinomial , integer polynomial , algebra

Let $k$ be any fixed positive integer. Let's look at integer pairs $(a, b)$, for which the quadratic equations $x^2 - 2ax + b = 0$ and $y^2 + 2ay + b = 0$ are real solutions (not necessarily different), which can be denoted by $x_1, x_2$ and $y_1, y_2$, respectively, in such an order that the equation $x_1 y_1 - x_2 y_2 = 4k$. a) Find the largest possible value of the second component $b$ of such a pair of numbers ($a, b)$. b) Find the sum of the other components of all such pairs of numbers.

2018 Rioplatense Mathematical Olympiad, Level 3, 5

Tags: number theory , integer , positive integer , product

Let $n$ be a positive integer. Find all $n$- rows $( a_1 , a_2 ,..., a_n )$ of different positive integers such that $$ \frac{(a_1 + d ) (a_2 + d ) \cdot\cdot\cdot ( a_n + d )}{a_1a_2\cdot \cdot \cdot a_n }$$ is integer for every integer $d\ge 0$

2016 Bangladesh Mathematical Olympiad, 9

Tags: calculus , definite integral , integer , combinatorics

Consider the integral $Z(0)=\int^{\infty}_{-\infty} dx e^{-x^2}= \sqrt{\pi}$. [b](a)[/b] Show that the integral $Z(j)=\int^{\infty}_{-\infty} dx e^{-x^{2}+jx}$, where $j$ is not a function of $x$, is $Z(j)=e^{j^{2}/4a} Z(0)$. [b](b)[/b] Show that $$\dfrac 1 {Z(0)}=\int x^{2n} e^{-x^2}= \dfrac {(2n-1)!!}{2^n},$$ where $(2n-1)!!$ is defined as $(2n-1)(2n-3)\times\cdots\times3\times 1$. [b](c)[/b] What is the number of ways to form $n$ pairs from $2n$ distinct objects? Interpret the previous part of the problem in term of this answer.

1969 Kurschak Competition, 1

Tags: integer , perfect square , number theory

Show that if $2 + 2\sqrt{28n^2 + 1}$ is an integer, then it is a square (for $n$ an integer).

2018 Junior Regional Olympiad - FBH, 2

Tags: integer , frac integer

Find all integers $n$ such that $\frac{n+4}{3n-2}$ is integer

1995 Czech And Slovak Olympiad IIIA, 2

Tags: sum , integer , 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 $66$.

2006 Korea Junior Math Olympiad, 5

Tags: number theory , coprime , integer , positive integer

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.

2010 Hanoi Open Mathematics Competitions, 4

Tags: number theory , integer

How many real numbers $a \in (1,9)$ such that the corresponding number $a- \frac1a$ is an integer? (A): $0$, (B): $1$, (C): $8$, (D): $9$, (E) None of the above.

2000 Estonia National Olympiad, 3

Tags: algebra , polynomial , root , integer

Find all values of $a$ for which the equation $x^3 - x + a = 0$ has three different integer solutions.

2020 Malaysia IMONST 1, 10

Tags: integer

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

2024 Thailand TST, 3

Tags: functional equation , integer

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

2013 Junior Balkan Team Selection Tests - Romania, 5

Tags: set , number theory , integer

a) Prove that for every positive integer n, there exist $a, b \in R - Z$ such that the set $A_n = \{a - b, a^2 - b^2, a^3 - b^3,...,a^n - b^n\}$ contains only positive integers. b) Let $a$ and $b$ be two real numbers such that the set $A = \{a^k - b^k | k \in N*\}$ contains only positive integers. Prove that $a$ and $b$ are integers.

2009 Junior Balkan Team Selection Tests - Romania, 2

Tags: number theory , integer , finite set

Let $a$ and $b$ be positive integers. Consider the set of all non-negative integers $n$ for which the number $\left(a+\frac12\right)^n +\left(b+\frac12\right)^n$ is an integer. Show that the set is finite.

2012 Tournament of Towns, 4

Tags: algebra , integer , number theory

Brackets are to be inserted into the expression $10 \div 9 \div 8 \div 7 \div 6 \div 5 \div 4 \div 3 \div 2$ so that the resulting number is an integer. (a) Determine the maximum value of this integer. (b) Determine the minimum value of this integer.