Olympiad problems

2007 Moldova Team Selection Test, 1

Tags: geometry , 3d geometry , algebra , factorization , sum of cubes , number theory

Find the least positive integers $m,k$ such that a) There exist $2m+1$ consecutive natural numbers whose sum of cubes is also a cube. b) There exist $2k+1$ consecutive natural numbers whose sum of squares is also a square. The author is Vasile Suceveanu

1979 IMO Longlists, 26

Tags: prime number , factorization , number theory

Let $n$ be a positive integer. If $4^n + 2^n + 1$ is a prime, prove that $n$ is a power of three.

2020 AIME Problems, 12

Tags: factorization , number theory , binomial theorem

Let $n$ be the least positive integer for which $149^n - 2^n$ is divisible by $3^3 \cdot 5^5 \cdot 7^7$. Find the number of positive divisors of $n$.

1992 Baltic Way, 3

Tags: geometry , 3d geometry , arithmetic sequence , algebra , factorization , sum of cubes , number theory

Find an infinite non-constant arithmetic progression of natural numbers such that each term is neither a sum of two squares, nor a sum of two cubes (of natural numbers).

2011 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: factorization , algebra

Factorise $$(a+2b-3c)^3+(b+2c-3a)^3+(c+2a-3b)^3$$

2018 Moscow Mathematical Olympiad, 4

Tags: number theory , algebra , factorization , sum of cubes

Are there natural solution of $$a^3+b^3=11^{2018}$$ ?

2024 Indonesia MO, 2

Tags: number theory , nt construction , factorization , prime

The triplet of positive integers $(a,b,c)$ with $a<b<c$ is called a [i]fatal[/i] triplet if there exist three nonzero integers $p,q,r$ which satisfy the equation $a^p b^q c^r = 1$. As an example, $(2,3,12)$ is a fatal triplet since $2^2 \cdot 3^1 \cdot (12)^{-1} = 1$. The positive integer $N$ is called [i]fatal[/i] if there exists a fatal triplet $(a,b,c)$ satisfying $N=a+b+c$. (a) Prove that 16 is not [i]fatal[/i]. (b) Prove that all integers bigger than 16 which are [b]not[/b] an integer multiple of 6 are fatal.

1982 Bundeswettbewerb Mathematik, 4

Tags: prime number , factorization , number theory

Let $n$ be a positive integer. If $4^n + 2^n + 1$ is a prime, prove that $n$ is a power of three.

2023 Bangladesh Mathematical Olympiad, P3

Tags: number theory , factorization , diophantine equation

Solve the equation for the positive integers: $$(x+2y)^2+2x+5y+9=(y+z)^2$$

2022 Germany Team Selection Test, 1

Tags: number theory , algebra , factorization , combinatorics

Let $a_1, a_2, \ldots, a_n$ be $n$ positive integers, and let $b_1, b_2, \ldots, b_m$ be $m$ positive integers such that $a_1 a_2 \cdots a_n = b_1 b_2 \cdots b_m$. Prove that a rectangular table with $n$ rows and $m$ columns can be filled with positive integer entries in such a way that * the product of the entries in the $i$-th row is $a_i$ (for each $i \in \left\{1,2,\ldots,n\right\}$); * the product of the entries in the $j$-th row is $b_j$ (for each $i \in \left\{1,2,\ldots,m\right\}$).

2007 iTest Tournament of Champions, 4

Tags: algebra , factorization , sum of cubes

For each positive integer $n$, let $S_n = \sum_{k=1}^nk^3$, and let $d(n)$ be the number of positive divisors of $n$. For how many positive integers $m$, where $m\leq 25$, is there a solution $n$ to the equation $d(S_n) = m$?

1996 Hungary-Israel Binational, 2

Tags: geometry , 3d geometry , modular arithmetic , algebra , factorization , difference of cubes , special factorizations

$ n>2$ is an integer such that $ n^2$ can be represented as a difference of cubes of 2 consecutive positive integers. Prove that $ n$ is a sum of 2 squares of positive integers, and that such $ n$ does exist.

1969 IMO, 1

Tags: number theory , sophie germain identity , composite number , factorization

Prove that there are infinitely many positive integers $m$, such that $n^4+m$ is not prime for any positive integer $n$.

2013 AIME Problems, 5

Tags: algebra , polynomial , geometry , 3d geometry , difference of squares , special factorizations , factorization

The real root of the equation $8x^3 - 3x^2 - 3x - 1 = 0$ can be written in the form $\frac{\sqrt[3]a + \sqrt[3]b + 1}{c}$, where $a$, $b$, and $c$ are positive integers. Find $a+b+c$.

2004 National Olympiad First Round, 24

Tags: geometry , 3d geometry , algebra , factorization , sum of cubes

What is the sum of cubes of real roots of the equation $x^3-2x^2-x+1=0$? $ \textbf{(A)}\ -6 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 11 \qquad\textbf{(E)}\ \text{None of above} $

2017 CCA Math Bonanza, I3

Tags: algebra , factorization , sum of cubes

A sequence starts with $2017$ as its first term and each subsequent term is the sum of cubes of the digits in the previous number. What is the $2017$th term of this sequence? [i]2017 CCA Math Bonanza Individual Round #3[/i]

1967 IMO Shortlist, 1

Tags: trigonometric equation , trigonometry , algebra , factorization

Decompose the expression into real factors: \[E = 1 - \sin^5(x) - \cos^5(x).\]

2011 USAMTS Problems, 2

Tags: 3d geometry , geometric series , algebra , factorization , difference of cubes , geometry

Let $x$ be a complex number such that $x^{2011}=1$ and $x\neq 1$. Compute the sum \[\dfrac{x^2}{x-1}+\dfrac{x^4}{x^2-1}+\dfrac{x^6}{x^3-1}+\cdots+\dfrac{x^{4020}}{x^{2010}-1}.\]

2024 Bangladesh Mathematical Olympiad, P1

Tags: diophantine equation , number theory , divisibility , factorization

Find all non-negative integers $x, y$ such that\[x^3y+x+y=xy+2xy^2\]

2015 Middle European Mathematical Olympiad, 4

Tags: number theory , relatively prime , factorization , euclidean algorithm , exponent

Find all pairs of positive integers $(m,n)$ for which there exist relatively prime integers $a$ and $b$ greater than $1$ such that $$\frac{a^m+b^m}{a^n+b^n}$$ is an integer.

1973 IMO Shortlist, 16

Tags: algebra , polynomial , trigonometry , geometry , factorization

Given $a, \theta \in \mathbb R, m \in \mathbb N$, and $P(x) = x^{2m}- 2|a|^mx^m \cos \theta +a^{2m}$, factorize $P(x)$ as a product of $m$ real quadratic polynomials.

2009 Stanford Mathematics Tournament, 2