Olympiad problems

1996 Poland - Second Round, 1

Can every polynomial with integer coefficients be expressed as a sum of cubes of polynomials with integer coefficients? [hide]I found the following statement that can be linked to this problem: "It is easy to see that every polynomial in F[x] is sum of cubes if char (F)$\ne$3 and card (F)=2,4"[/hide]

2024 Bangladesh Mathematical Olympiad, P1

Tags: number theory , diophantine equation , divisibility , factorization

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

2022 Germany Team Selection Test, 1

Tags: combinatorics , factorization , number theory , algebra

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 $i \in \left\{1,2,\ldots,m\right\}$).

2023 Thailand Mathematical Olympiad, 9

Tags: number theory , factorization

Prove that there exists an infinite sequence of positive integers $a_1,a_2,a_3,\dots$ such that for any positive integer $k$, $a_k^2+a_k+2023$ has at least $k$ distinct positive divisors.

1973 IMO Shortlist, 16

Tags: trigonometry , algebra , polynomial , factorization , geometry

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.

2024 Mozambican National MO Selection Test, P3

Tags: number theory , nt , diophantine equation , factorization , prime number , positive integer

Find all triples of positive integers $(a,b,c)$ such that: $a^2bc-2ab^2c-2abc^2+b^3c+bc^3+2b^2c^2=11$

1969 IMO, 1

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

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

2022 HMNT, 5

Tags: prime number , factorization

Alice is once again very bored in class. On a whim, she chooses three primes $p$, $q$, $r$ independently and uniformly at random from the set of primes at most 30. She then calculates the roots of $px^2+qx+r$. What is the probability that at least one of her roots is an integer?

1994 Tuymaada Olympiad, 2

Tags: number theory , factor , product , factorization

The set of numbers $M=\{4k-3 | k\in N\}$ is considered. A number of of this set is called “simple” if it is impossible to put in the form of a product of numbers from $M$ other than $1$. Show that in this set, the decomposition of numbers in the product of "simple" factors is ambiguous.

1995 Korea National Olympiad, Day 2

Tags: greatest common divisor , algebra , integer polynomial , prime , factorization , polynomial

Let $a,b$ be integers and $p$ be a prime number such that: (i) $p$ is the greatest common divisor of $a$ and $b$; (ii) $p^2$ divides $a$. Prove that the polynomial $x^{n+2}+ax^{n+1}+bx^{n}+a+b$ cannot be decomposed into the product of two polynomials with integer coefficients and degree greater than $1$.

2015 Junior Balkan Team Selection Tests - Romania, 2

Tags: factorization , algebra , system of equations

Find all the triplets of real numbers $(x , y , z)$ such that : $y=\frac{x^3+12x}{3x^2+4}$ , $z=\frac{y^3+12y}{3y^2+4}$ , $x=\frac{z^3+12z}{3z^2+4}$

2003 National Olympiad First Round, 10

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

Which of the followings is congruent (in $\bmod{25}$) to the sum in of integers $0\leq x < 25$ such that $x^3+3x^2-2x+4 \equiv 0 \pmod{25}$? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 17 \qquad\textbf{(D)}\ 22 \qquad\textbf{(E)}\ \text{None of the preceding} $

2018 Moscow Mathematical Olympiad, 4

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

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

2017 Turkey Junior National Olympiad, 4

Tags: inequalities , factorization

If real numbers $a>b>1$ satisfy the inequality$$(ab+1)^2+(a+b)^2\leq 2(a+b)(a^2-ab+b^2+1)$$what is the minimum possible value of $\dfrac{\sqrt{a-b}}{b-1}$

1978 IMO Shortlist, 5

Tags: number theory , algebra , arithmetic sequence , factorization , partition

For every integer $d \geq 1$, let $M_d$ be the set of all positive integers that cannot be written as a sum of an arithmetic progression with difference $d$, having at least two terms and consisting of positive integers. Let $A = M_1$, $B = M_2 \setminus \{2 \}, C = M_3$. Prove that every $c \in C$ may be written in a unique way as $c = ab$ with $a \in A, b \in B.$

2019 BMT Spring, 7

Tags: algebra , factorization , sum of cubes

Let $ r_1 $, $ r_2 $, $ r_3 $ be the (possibly complex) roots of the polynomial $ x^3 + ax^2 + bx + \dfrac{4}{3} $. How many pairs of integers $ a $, $ b $ exist such that $ r_1^3 + r_2^3 + r_3^3 = 0 $?

2011 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: algebra , factorization

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

1967 IMO Longlists, 42

Tags: trigonometry , factorization , trigonometric equation , algebra

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

2013 AIME Problems, 5

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

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 $a+b+c$.

1990 Baltic Way, 15

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

Prove that none of the numbers $2^{2^n}+ 1$, $n = 0, 1, 2, \dots$ is a perfect cube.

1996 Hungary-Israel Binational, 2

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

$ 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.

2008 International Zhautykov Olympiad, 2

Tags: polynomial , algebra , factorization , sum of cubes

A polynomial $ P(x)$ with integer coefficients is called good,if it can be represented as a sum of cubes of several polynomials (in variable $ x$) with integer coefficients.For example,the polynomials $ x^3 \minus{} 1$ and $ 9x^3 \minus{} 3x^2 \plus{} 3x \plus{} 7 \equal{} (x \minus{} 1)^3 \plus{} (2x)^3 \plus{} 2^3$ are good. a)Is the polynomial $ P(x) \equal{} 3x \plus{} 3x^7$ good? b)Is the polynomial $ P(x) \equal{} 3x \plus{} 3x^7 \plus{} 3x^{2008}$ good? Justify your answers.

2007 Moldova Team Selection Test, 1

Tags: algebra , geometry , 3d geometry , 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

2024 Middle European Mathematical Olympiad, 4

Tags: number theory , polynomial , sum of divisors , divisibility , factorization

Determine all polynomials $P(x)$ with integer coefficients such that $P(n)$ is divisible by $\sigma(n)$ for all positive integers $n$. (As usual, $\sigma(n)$ denotes the sum of all positive divisors of $n$.)