Olympiad problems

2018 India PRMO, 24

If $N$ is the number of triangles of different shapes (i.e., not similar) whose angles are all integers (in degrees), what is $\frac{N}{100}$?

1954 Putnam, A7

Tags: Putnam , number theory , Integers

Prove that there are no integers $x$ and $y$ for which $$x^2 +3xy-2y^2 =122.$$

1974 Putnam, A1

Tags: Putnam , Integers , relatively prime

Call a set of positive integers "conspiratorial" if no three of them are pairwise relatively prime. What is the largest number of elements in any "conspiratorial" subset of the integers $1$ to $16$?

2022 Indonesia TST, N

Tags: quadratics , greatest common divisor , number theory , Integers , diophantine

For each natural number $n$, let $f(n)$ denote the number of ordered integer pairs $(x,y)$ satisfying the following equation: \[ x^2 - xy + y^2 = n. \] a) Determine $f(2022)$. b) Determine the largest natural number $m$ such that $m$ divides $f(n)$ for every natural number $n$.

1960 Putnam, B1

Tags: Putnam , Integers , equation , number theory

Find all integer solutions $(m,n)$ to $m^{n}=n^{m}.$

2018 Junior Regional Olympiad - FBH, 5

Tags: Integers , exponential , Exponents

Find all integers $x$ and $y$ such that $2^x+1=y^2$

2016 Bangladesh Mathematical Olympiad, 9

Tags: combinatorics , calculus , contest problem , definite integrals , Integers

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.

2002 Rioplatense Mathematical Olympiad, Level 3, 2

Tags: Inequality , Integers , positive integers , number theory , algebra

Let $\lambda$ be a real number such that the inequality $0 <\sqrt {2002} - \frac {a} {b} <\frac {\lambda} {ab}$ holds for an infinite number of pairs $ (a, b)$ of positive integers. Prove that $\lambda \geq 5 $.

2012 Thailand Mathematical Olympiad, 2

Tags: algebra , number theory , Integers

Let $a_1, a_2, ..., a_{2012}$ be pairwise distinct integers. Show that the equation $(x -a_1)(x - a_2)...(x - a_{2012}) = (1006!)^2$ has at most one integral solution.

2025 Greece National Olympiad, 1

Tags: algebra , polynomial , roots , Integers

Let $P(x)=x^4+5x^3+mx^2+5nx+4$ have $2$ distinct real roots, which sum up to $-5$. If $m,n \in \mathbb {Z_+}$, find the values of $m,n$ and their corresponding roots.

2017 South East Mathematical Olympiad, 7

Tags: number theory , Integers

Let $m$ be a given positive integer. Define $a_k=\frac{(2km)!}{3^{(k-1)m}},k=1,2,\cdots.$ Prove that there are infinite many integers and infinite many non-integers in the sequence $\{a_k\}$.

1988 Mexico National Olympiad, 4

Tags: combinatorics , Integers

In how many ways can one select eight integers $a_1,a_2, ... ,a_8$, not necesarily distinct, such that $1 \le a_1 \le ... \le a_8 \le 8$?

Indonesia MO Shortlist - geometry, g8

Tags: geometry , consecutive , equal angles , Integers

Prove that there is only one triangle whose sides are consecutive natural numbers and one of the angles is twice the other angle.

2005 Mexico National Olympiad, 2

Tags: matrix , Integers , combinatorics , number theory

Given several matrices of the same size. Given a positive integer $N$, let's say that a matrix is $N$-balanced if the entries of the matrix are integers and the difference between any two adjacent entries of the matrix is less than or equal to $N$. (i) Show that every $2N$-balanced matrix can be written as a sum of two $N$-balanced matrices. (ii) Show that every $3N$-balanced matrix can be written as a sum of three $N$-balanced matrices.

2008 Greece JBMO TST, 3

Tags: combinatorics , Integers , Difference

Let $x_1,x_2,x_3,...,x_{102}$ be natural numbers such that $x_1<x_2<x_3<...<x_{102}<255$. Prove that among the numbers $d_1=x_2-x_1, d_2=x_3-x_2, ..., d_{101}=x_{102}-x_{101}$ there are at least $26$ equal.

2024 AMC 12/AHSME, 6

Tags: AMC , AMC 10 , AMC 12 , 2024 AMC 10A , 2024 AMC 12A , 2024 AMC , Integers

The product of three integers is $60$. What is the least possible positive sum of the three integers? $\textbf{(A) } 2 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 6 \qquad \textbf{(E) } 13$

2018 Auckland Mathematical Olympiad, 5

Tags: number theory , Integers

Find all possible triples of positive integers, $a, b, c$ so that $\frac{a+1}{b}$, $\frac{b+1}{c}$ and $\frac{c+1}{a}$ are also integers.

1981 Czech and Slovak Olympiad III A, 5

Tags: algebra , inequalities , max , Integers

Let $n$ be a positive integer. Determine the maximum of the sum $x_1+\cdots+x_n$ where $x_1,\ldots,x_n$ are non-negative integers satisfying the condition \[x_1^3+\cdots+x_n^3\le7n.\]

2014 JBMO TST - Macedonia, 1

Tags: Inequality , Integers

Prove that $\frac{1}{1\times2013}+\frac{1}{2\times2012}+\frac{1}{3\times2011}+...+\frac{1}{2012\times2}+\frac{1}{2013\times1}<1$

2024 AMC 12/AHSME, 17

Tags: 2024 AMC , AMC 12 , 2024 AMC 12B , Integers , probability

Integers $a$ and $b$ are randomly chosen without replacement from the set of integers with absolute value not exceeding $10$. What is the probability that the polynomial $x^3 + ax^2 + bx + 6$ has $3$ distinct integer roots? $\textbf{(A)} \frac{1}{240} \qquad \textbf{(B)} \frac{1}{221} \qquad \textbf{(C)} \frac{1}{105} \qquad \textbf{(D)} \frac{1}{84} \qquad \textbf{(E)} \frac{1}{63}$.

Mathley 2014-15, 2

Tags: geometry , Triangle , Integers , Sequence , algebra , area of a triangle

Given the sequence $(t_n)$ defined as $t_0 = 0$, $t_1 = 6$, $t_{n + 2} = 14t_{n + 1} - t_n$. Prove that for every number $n \ge 1$, $t_n$ is the area of a triangle whose lengths are all numbers integers. Dang Hung Thang, University of Natural Sciences, Hanoi National University.