Olympiad problems

2020 AIME Problems, 12

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 $n$.

1967 IMO Longlists, 42

Tags: trigonometry , algebra , factorization , trigonometric equation

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

1992 Poland - First Round, 8

Tags: factorization , sum of cubes , algebra

Given is a positive integer $n \geq 2$. Determine the maximum value of the sum of natural numbers $k_1,k_2,...,k_n$ satisfying the condition: $k_1^3+k_2^3+ \dots +k_n^3 \leq 7n$.

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 $?

2024-25 IOQM India, 1

Tags: factorization , number theory

The smallest positive integer that does not divide $1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9$ is:

2017 Pan-African Shortlist, N2

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

For which prime numbers $p$ can we find three positive integers $n$, $x$ and $y$ such that $p^n = x^3 + y^3$?

2020 AMC 10, 25

Tags: factorization

Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product$$n = f_1\cdot f_2\cdots f_k,$$where $k\ge1$, the $f_i$ are integers strictly greater than $1$, and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number $6$ can be written as $6$, $2\cdot 3$, and $3\cdot2$, so $D(6) = 3$. What is $D(96)$? $\textbf{(A) } 112 \qquad\textbf{(B) } 128 \qquad\textbf{(C) } 144 \qquad\textbf{(D) } 172 \qquad\textbf{(E) } 184$

1978 IMO Shortlist, 5

Tags: arithmetic sequence , algebra , number theory , 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 $a \in A, b \in B.$

1996 Poland - Second Round, 1

Tags: algebra , polynomial , factorization , sum of cubes

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]

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

2020 AMC 12/AHSME, 24

Tags: factorization

Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product$$n = f_1\cdot f_2\cdots f_k,$$where $k\ge1$, the $f_i$ are integers strictly greater than $1$, and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number $6$ can be written as $6$, $2\cdot 3$, and $3\cdot2$, so $D(6) = 3$. What is $D(96)$? $\textbf{(A) } 112 \qquad\textbf{(B) } 128 \qquad\textbf{(C) } 144 \qquad\textbf{(D) } 172 \qquad\textbf{(E) } 184$

1994 Tuymaada Olympiad, 2

Tags: factor , number theory , 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: polynomial , integer polynomial , greatest common divisor , prime , factorization , algebra

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 $1$.

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} $

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).

2024 Indonesia MO, 2

Tags: factorization , number theory , nt construction , 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.

2018 Moscow Mathematical Olympiad, 4

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

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

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.

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]

2001 Greece JBMO TST, 1

Tags: number theory , algebra , factorization , diophantine equation

a) Factorize $A= x^4+y^4+z^4-2x^2y^2-2y^2z^2-2z^2x^2$ b) Prove that there are no integers $x,y,z$ such that $x^4+y^4+z^4-2x^2y^2-2y^2z^2-2z^2x^2=2000 $

2011 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: factorization , algebra

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

1990 Baltic Way, 15

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

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

2023 Bangladesh Mathematical Olympiad, P3

Tags: number theory , factorization , diophantine equation

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

2003 National Olympiad First Round, 10

Tags: modular arithmetic , geometry , 3d geometry , algebra , 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} $

1978 IMO Longlists, 26

Tags: arithmetic sequence , algebra , number theory , 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 $a \in A, b \in B.$