Olympiad problems

1941 Moscow Mathematical Olympiad, 084

a) Find an integer $a$ for which $(x - a)(x - 10) + 1$ factors in the product $(x + b)(x + c)$ with integers $b$ and $c$. b) Find nonzero and nonequal integers $a, b, c$ so that $x(x - a)(x - b)(x - c) + 1$ factors into the product of two polynomials with integer coefficients.

1939 Moscow Mathematical Olympiad, 048

Tags: algebra , factoring , factor

Factor $a^{10} + a^5 + 1$ into nonconstant polynomials with integer coefficients

1949-56 Chisinau City MO, 12

Tags: factoring , algebra

Factor the polynomial $bc (b+c) +ca (c-a)-ab(a + b)$.

1999 Israel Grosman Mathematical Olympiad, 4

Tags: polynomial , integer polynomial , factoring , algebra

Consider a polynomial $f(x) = x^4 +ax^3 +bx^2 +cx+d$ with integer coefficients. Prove that if $f(x)$ has exactly one real root, then it can be factored into nonconstant polynomials with rational coefficients

1949-56 Chisinau City MO, 13

Tags: factoring , algebra

Factor the polynomial $(a+b+c)^3- a^3 -b^3 -c^3$

1945 Moscow Mathematical Olympiad, 091

Tags: polynomial , factoring , algebra

a) Divide $a^{128} - b^{128}$ by $(a + b)(a^2 + b^2)(a^4 + b^4)(a^8 + b^8)(a^{16} + b^{16})(a^{32} + b^{32})(a^{64} + b^{64}) $. b) Divide $a^{2^k} - b^{2^k}$ by $(a + b)(a^2 + b^2)(a^4 + b^4) ... (a^{2^{k-1}} + b^{2^{k-1}})$

1949-56 Chisinau City MO, 11

Tags: factoring , algebra

Factor the polynomial $x^3+x^2z+xyz+y^2z-y^3$.

1972 Czech and Slovak Olympiad III A, 4

Tags: number theory , composite , factoring

Show that there are infinitely many positive integers $a$ such that the number $n^4+a$ is composite for every positive integer $n.$ Give 5 (different) numbers $a$ with the mentioned property.

1940 Moscow Mathematical Olympiad, 054

Tags: algebra , factoring , factor

Factor $(b - c)^3 + (c - a)^3 + (a - b)^3$.

2021 Kyiv City MO Round 1, 7.3

Tags: number theory , factoring

Petryk factored the number $10^6 = 1000000$ as a product of $7$ distinct positive integers. Among all such factorings, find the one in which the largest of these $7$ factors is the smallest possible. [i]Proposed by Bogdan Rublov[/i]