Olympiad problems

2024 Israel TST, P3

Let $n$ be a positive integer and $p$ be a prime number of the form $8k+5$. A polynomial $Q$ of degree at most $2023$ and nonnegative integer coefficients less than or equal to $n$ will be called "cool" if \[p\mid Q(2)\cdot Q(3) \cdot \ldots \cdot Q(p-2)-1.\] Prove that the number of cool polynomials is even.

PEN G Problems, 16

Tags: irrational number , algebra , polynomial , integration , inequalities

For each integer $n \ge 1$, prove that there is a polynomial $P_{n}(x)$ with rational coefficients such that $x^{4n}(1-x)^{4n}=(1+x)^{2}P_{n}(x)+(-1)^{n}4^{n}$. Define the rational number $a_{n}$ by \[a_{n}= \frac{(-1)^{n-1}}{4^{n-1}}\int_{0}^{1}P_{n}(x) \; dx,\; n=1,2, \cdots.\] Prove that $a_{n}$ satisfies the inequality \[\left\vert \pi-a_{n}\right\vert < \frac{1}{4^{5n-1}}, \; n=1,2, \cdots.\]

2005 District Olympiad, 1

Tags: inequalities , algebra , polynomial , rearrangement inequality

a) Prove that if $x,y>0$ then \[ \frac x{y^2} + \frac y{x^2} \geq \frac 1x + \frac 1y. \] b) Prove that if $a,b,c$ are positive real numbers, then \[ \frac {a+b}{c^2} + \frac {b+c}{a^2} + \frac {c+a}{b^2} \geq 2 \left( \frac 1a + \frac 1b + \frac 1c \right). \]

2019 Korea - Final Round, 5

Tags: algebra , polynomial

Find all pairs $(p,q)$ such that the equation $$x^4+2px^2+qx+p^2-36=0$$ has exactly $4$ integer roots(counting multiplicity).

1991 Irish Math Olympiad, 5

Tags: polynomial , algebra

Find all polynomials $f(x) = x^{n} + a_{1}x^{n-1} + \cdots + a_{n}$ with the following properties (a) all the coefficients $a_{1}, a_{2}, ..., a_{n}$ belong to the set $\{ -1, 1 \}$; and (b) all the roots of the equation $f(x)=0$ are real.

2013 Kosovo National Mathematical Olympiad, 2

Tags: algebra , polynomial , calculus , integration

Find all integer $n$ such that $n-5$ divide $n^2+n-27$.

1976 Chisinau City MO, 121

Tags: algebra , odd , polynomial , integer

Prove that the polynomial $P (x)$ with integer coefficients, taking odd values for $x = 0$ and $x= 1$, has no integer roots.

2012 IMO Shortlist, N5

Tags: algebra , number theory , prime number , polynomial

For a nonnegative integer $n$ define $\operatorname{rad}(n)=1$ if $n=0$ or $n=1$, and $\operatorname{rad}(n)=p_1p_2\cdots p_k$ where $p_1<p_2<\cdots <p_k$ are all prime factors of $n$. Find all polynomials $f(x)$ with nonnegative integer coefficients such that $\operatorname{rad}(f(n))$ divides $\operatorname{rad}(f(n^{\operatorname{rad}(n)}))$ for every nonnegative integer $n$.

2020 Thailand TST, 4

Tags: algebra , polynomial

Let $n$ be a positive integer and let $P$ be the set of monic polynomials of degree $n$ with complex coefficients. Find the value of \[ \min_{p \in P} \left \{ \max_{|z| = 1} |p(z)| \right \} \]

PEN A Problems, 22

Tags: polynomial , binomial theorem , divisibility theory , algebra

Prove that the number \[\sum_{k=0}^{n}\binom{2n+1}{2k+1}2^{3k}\] is not divisible by $5$ for any integer $n\geq 0$.

2013 ISI Entrance Examination, 6

Tags: polynomial , limit , algebra

Let $p(x)$ and $q(x)$ be two polynomials, both of which have their sum of coefficients equal to $s.$ Let $p,q$ satisfy $p(x)^3-q(x)^3=p(x^3)-q(x^3).$ Show that (i) There exists an integer $a\geq1$ and a polynomial $r(x)$ with $r(1)\neq0$ such that \[p(x)-q(x)=(x-1)^ar(x).\] (ii) Show that $s^2=3^{a-1},$ where $a$ is described as above.

2025 Bulgarian Spring Mathematical Competition, 11.4

Tags: algebra , polynomial

We call two non-constant polynomials [i]friendly[/i] if each of them has only real roots, and every root of one polynomial is also a root of the other. For two friendly polynomials $ P(x), Q(x) $ and a constant $ C \in \mathbb{R}, C \neq 0 $, it is given that $ P(x) + C $ and $ Q(x) + C $ are also friendly polynomials. Prove that $ P(x) \equiv Q(x) $.

PEN H Problems, 9

Tags: diophantine equation , algebra , polynomial , vieta , absolute value

Determine all integers $a$ for which the equation \[x^{2}+axy+y^{2}=1\] has infinitely many distinct integer solutions $x, \;y$.

2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 2

Tags: linear algebra , matrix , algebra , polynomial

Justify your answer whether $A=\left( \begin{array}{ccc} -4 & -1& -1 \\ 1 & -2& 1 \\ 0 & 0& -3 \end{array} \right)$ is similar to $B=\left( \begin{array}{ccc} -2 & 1& 0 \\ -1 & -4& 1 \\ 0 & 0& -3 \end{array} \right),\ A,\ B\in{M(\mathbb{C})}$ or not.

2012 Harvard-MIT Mathematics Tournament, 10

Tags: hmmt , algebra , polynomial , function

Suppose that there are $16$ variables $\{a_{i,j}\}_{0\leq i,j\leq 3}$, each of which may be $0$ or $1$. For how many settings of the variables $a_{i,j}$ do there exist positive reals $c_{i,j}$ such that the polynomial \[f(x,y)=\sum_{0\leq i,j\leq 3}a_{i,j}c_{i,j}x^iy^j\] $(x,y\in\mathbb{R})$ is bounded below?

2022 Grand Duchy of Lithuania, 1

Tags: algebra , polynomial

Given a polynomial with integer coefficients $$P(x) = x^{20} + a_{19}x^{19} +... + a_1x + a_0,$$ having $20$ different real roots. Determine the maximum number of roots such a polynomial $P$ can have in the interval $(99, 100)$.

2012 Dutch BxMO/EGMO TST, 1

Tags: trinomial , quadratic , polynomial , root , algebra

Do there exist quadratic polynomials $P(x)$ and $Q(x)$ with real coeffcients such that the polynomial $P(Q(x))$ has precisely the zeros $x = 2, x = 3, x =5$ and $x = 7$?

1982 All Soviet Union Mathematical Olympiad, 347

Tags: polynomial , algebra

Can you find three polynomials $P,Q,R$ of three variables $x,y,z$, providing the condition: a)$P(x-y+z)^3 + Q(y-z-1)^3 +R(z-2x+1)^3 = 1$ b)$P(x-y+z)^3 + Q(y-z-1)^3 +R(z-x+1)^3 = 1$ for all $x,y,z$?

2013 Tournament of Towns, 2

Tags: polynomial , algebra

Find all positive integers $n$ for which the following statement holds: For any two polynomials $P(x)$ and $Q(x)$ of degree $n$ there exist monomials $ax^k$ and $bx^{ell}, 0 \le k,\ ell \le n$, such that the graphs of $P(x) + ax^k$ and $Q(x) + bx^{ell}$ have no common points.

1993 Moldova Team Selection Test, 1

Tags: algebra , polynomial

Prove that $\frac{5^{125}-1}{5^{25}-1}$ is a composite number.

1973 IMO, 3

Tags: algebra , polynomial , root , minimum value , coefficient

Determine the minimum value of $a^{2} + b^{2}$ when $(a,b)$ traverses all the pairs of real numbers for which the equation \[ x^{4} + ax^{3} + bx^{2} + ax + 1 = 0 \] has at least one real root.

1960 Polish MO Finals, 4

Tags: algebra , polynomial

Prove that if the equation $$x^4 + ax + b = 0$$ has two equal roots, then $$\left( \frac{a}{4} \right)^4 =\left( \frac{b}{3} \right)^3.$$

2006 Iran Team Selection Test, 4

Tags: algebra , polynomial , divisibility tests , number theory

Let $n$ be a fixed natural number. Find all $n$ tuples of natural pairwise distinct and coprime numbers like $a_1,a_2,\ldots,a_n$ such that for $1\leq i\leq n$ we have \[ a_1+a_2+\ldots+a_n|a_1^i+a_2^i+\ldots+a_n^i \]

2024 Azerbaijan Senior NMO, 4

Tags: algebra , polynomial

Let $P(x)$ be a polynomial with the coefficients being $0$ or $1$ and degree $2023$. If $P(0)=1$, then prove that every real root of this polynomial is less than $\frac{1-\sqrt{5}}{2}$.

2017 Saudi Arabia JBMO TST, 1

Tags: polynomial , algebra

Given a polynomial $f(x) = x^4+ax^3+bx^2+cx$. It is known that each of the equations $f(x) = 1$ and $f(x) = 2$ has four real roots (not necessarily distinct). Prove that if the roots of the first equation satisfy the equality $x_1 + x_2 = x_3 + x_4$, then the same equation holds for the roots of the second equation