Olympiad problems

2009 Cuba MO, 5

Prove that there are infinitely many positive integers $n$ such that $\frac{5^n-1}{n+2}$ is an integer.

2017 Peru IMO TST, 10

Tags: squarefree , polynomial , Integer , primes , number theory

Let $P (n)$ and $Q (n)$ be two polynomials (not constant) whose coefficients are integers not negative. For each positive integer $n$, define $x_n = 2016^{P (n)} + Q (n)$. Prove that there exist infinite primes $p$ for which there is a positive integer $m$, squarefree, such that $p | x_m$. Clarification: A positive integer is squarefree if it is not divisible by the square of any prime number.

2020 HK IMO Preliminary Selection Contest, 2

Tags: Integer , algebra

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

1997 Dutch Mathematical Olympiad, 3

Tags: game , Integer , integer solutions , polynomial , winning strategy , algebra

a. View the second-degree quadratic equation $x^2+? x +? = 0$ Two players successively put an integer each at the location of a question mark. Show that the second player can always ensure that the quadratic gets two integer solutions. Note: we say that the quadratic also has two integer solutions, even when they are equal (for example if they are both equal to $3$). b.View the third-degree equation $x^3 +? x^2 +? x +? = 0$ Three players successively put an integer each at the location of a question mark. The equation appears to have three integer (possibly again the same) solutions. It is given that two players each put a $3$ in the place of a question mark. What number did the third player put? Determine that number and the place where it is placed and prove that only one number is possible.

2017 QEDMO 15th, 12

Tags: algebra , Integer

Let $a$ be a real number such that $\left(a + \frac{1}{a}\right)^2=11$. For which $n\in N$ is $a^n + \frac{1}{a^n}$ an integer? Does this depend on the exact value of $a$?

1985 All Soviet Union Mathematical Olympiad, 400

Tags: trinomial , Integer , maximum , polynomial , algebra

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 $|P(x)|<50$.

2012 Tournament of Towns, 4

Tags: number theory , algebra , Integer

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.

2013 Junior Balkan Team Selection Tests - Romania, 5

Tags: Sets , Integer , number theory

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.

1979 Chisinau City MO, 169

Tags: algebra , number theory , Integer

Prove that the number $x^8+\frac{1}{x^8}$ is an integer if $x+\frac{1}{x }$ is an integer.

2014 India PRMO, 6

Tags: trinomial , Integer , roots , minimum

What is the smallest possible natural number $n$ for which the equation $x^2 -nx + 2014 = 0$ has integer roots?

2000 Estonia National Olympiad, 3

Tags: polynomial , roots , Integer , algebra

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

2014 India PRMO, 17

Tags: algebra , Integer , roots , minimum

For a natural number $b$, let $N(b)$ denote the number of natural numbers $a$ for which the equation $x^2 + ax + b = 0$ has integer roots. What is the smallest value of $b$ for which $N(b) = 20$?

2011 Saudi Arabia Pre-TST, 2.1

Tags: number theory , Integer

Let $n$ be a positive integer. Prove that the interval $I_n= \left( \frac{1+\sqrt{8n+1}}{2}, \frac{1+\sqrt{8n+9}}{2}\right)$ does not contain any integer.

2018 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: Integer , Perfect Squares , number theory

Find all positive integers $n$ such that number $n^4-4n^3+22n^2-36n+18$ is perfect square of positive integer

2014 India PRMO, 13

Tags: number theory , Integer , divides

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

1976 Chisinau City MO, 121

Tags: algebra , odd , polynomial , Integer

Prove that the polynomial $P (x)$ with integer coefficients, taking odd values for $x = 0$ and $x= 1$, has no integer roots.

2008 Austria Beginners' Competition, 1

Tags: number theory , Integer

Determine all positive integers $n$ such that $\frac{2^n}{n^2}$ is an integer.

2017 Hanoi Open Mathematics Competitions, 10

Tags: trinomial , Integer , roots , algebra

Find all non-negative integers $a, b, c$ such that the roots of equations: $\begin{cases}x^2 - 2ax + b = 0 \\ x^2- 2bx + c = 0 \\ x^2 - 2cx + a = 0 \end{cases}$ are non-negative integers.

1992 Swedish Mathematical Competition, 1

Tags: number theory , Integer

Is $\frac{19^{92} - 91^{29}}{90}$ an integer?

2018 Dutch BxMO TST, 2

Tags: geometry , Integer , Sides of a triangle , coprime

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.

1994 Nordic, 4

Tags: number theory , Perfect Square , Integer , Pell equations

Determine all positive integers $n < 200$, such that $n^2 + (n+ 1)^2$ is the square of an integer.

2015 Bangladesh Mathematical Olympiad, 2

Tags: number theory , Bdmo , contest problem , Contest Preparation , Integer , solutions

[b][u]BdMO National Higher Secondary Problem 3[/u][/b] Let $N$ be the number if pairs of integers $(m,n)$ that satisfies the equation $m^2+n^2=m^3$ Is $N$ finite or infinite?If $N$ is finite,what is its value?

2002 Moldova Team Selection Test, 4

Tags: Integer , Sum , distance , arc , projection , geometry , number theory

Let $C$ be the circle with center $O(0,0)$ and radius $1$, and $A(1,0), B(0,1)$ be points on the circle. Distinct points $A_1,A_2, ....,A_{n-1}$ on $C$ divide the smaller arc $AB$ into $n$ equal parts ($n \ge 2$). If $P_i$ is the orthogonal projection of $A_i$ on $OA$ ($i =1, ... ,n-1$), find all values of $n$ such that $P_1A^{2p}_1 +P_2A^{2p}_2 +...+P_{n-1}A^{2p}_{n-1}$ is an integer for every positive integer $p$.

2000 All-Russian Olympiad Regional Round, 11.3

Tags: algebra , number theory , Integer

Sequence of real numbers $a_1, a_2, . . . , a_{2000}$ is such that for any natural number $n$, $1\le n \le 2000$, the equality $$a^3_1+ a^3_2+... + a^3_n = (a_1 + a_2 +...+ a_n)^2.$$ Prove that all terms of this sequence are integers.

2020 Turkey Junior National Olympiad, 2

Tags: algebra , Integer

If the ratio $$\frac{17m+43n}{m-n}$$ is an integer where $m$ and $n$ positive integers, let's call $(m,n)$ is a special pair. How many numbers can be selected from $1,2,..., 2021$, any two of which do not form a special pair?