Olympiad problems

2008 Postal Coaching, 2

Find all polynomials $P$ with integer coefficients such that wherever $a, b \in N$ and $a+b$ is a square we have $P(a) + P(b)$ is also a square.

2006 All-Russian Olympiad Regional Round, 10.4

Tags: trinomial , polynomial , algebra

Given $n > 1$ monic square trinomials $x^2 - a_1x + b_1$,$...$, $x^2-a_nx + b_n$, and all $2n$ numbers are $a_1$,$...$, $a_n$, $b_1$,$...$, $b_n$ are different. Can it happen that each of the numbers $a_1$,$...$, $a_n$, $b_1$,$...$, $b_n is the root of one of these trinomials?

2019 IFYM, Sozopol, 4

Tags: polynomial , number theory

Is it true that for $\forall$ prime number $p$, there exist non-constant polynomials $P$ and $Q$ with $P,Q\in \mathbb{Z} [x]$ for which the remainder modulo $p$ of the coefficient in front of $x^n$ in the product $PQ$ is 1 for $n=0$ and $n=4$; $p-2$ for $n=2$ and is 0 for all other $n\geq 0$?

2013 Costa Rica - Final Round, 5

Tags: algebra , polynomial , divisible

Determine the number of polynomials of degree $5$ with different coefficients in the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$ such that they are divisible by $x^2-x + 1$. Justify your answer.

2004 AIME Problems, 13

Tags: polynomial , algebra , relatively prime , geometric series , number theory

The polynomial \[P(x)=(1+x+x^2+\cdots+x^{17})^2-x^{17}\] has 34 complex roots of the form $z_k=r_k[\cos(2\pi a_k)+i\sin(2\pi a_k)], k=1, 2, 3,\ldots, 34$, with $0<a_1\le a_2\le a_3\le\cdots\le a_{34}<1$ and $r_k>0$. Given that $a_1+a_2+a_3+a_4+a_5=m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.

1996 Taiwan National Olympiad, 6

Tags: polynomial , algebra , number theory

Let $q_{0},q_{1},...$ be a sequence of integers such that a) for any $m>n$ we have $m-n\mid q_{m}-q_{n}$, and b) $|q_{n}|\leq n^{10}, \ \forall n\geq 0$. Prove there exists a polynomial $Q$ such that $q_{n}=Q(n), \ \forall n\geq 0$.

PEN B Problems, 5

Tags: polynomial , induction , symmetry , modular arithmetic , vieta , search , algebra

Let $p$ be an odd prime. If $g_{1}, \cdots, g_{\phi(p-1)}$ are the primitive roots $\pmod{p}$ in the range $1<g \le p-1$, prove that \[\sum_{i=1}^{\phi(p-1)}g_{i}\equiv \mu(p-1) \pmod{p}.\]

2024 Belarusian National Olympiad, 10.5

Tags: polynomial , algebra

Let $n$ be a positive integer. On the blackboard all quadratic polynomials with positive integer coefficients, that do not exceed $n$, without real roots are written Find all $n$ for which the number of written polynomials is even [i]A. Voidelevich[/i]

1999 Switzerland Team Selection Test, 9

Tags: polynomial , algebra , number theory

Suppose that $P(x)$ is a polynomial with degree $10$ and integer coefficients. Prove that, there is an infinite arithmetic progression (open to bothside) not contain value of $P(k)$ with $k\in\mathbb{Z}$

2017 Taiwan TST Round 2, 1

Tags: number theory , functional equation , sum of digits , polynomial

For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$ [i]Proposed by Warut Suksompong, Thailand[/i]

1987 Federal Competition For Advanced Students, P2, 6

Tags: algebra , sum of roots , polynomial

Determine all polynomials $ P_n(x)\equal{}x^n\plus{}a_1 x^{n\minus{}1}\plus{}...\plus{}a_{n\minus{}1} x\plus{}a_n$ with integer coefficients whose $ n$ zeros are precisely the numbers $ a_1,...,a_n$ (counted with their respective multiplicities).

2016 Thailand Mathematical Olympiad, 7

Tags: absolute value , algebra , polynomial

Given $P(x)=a_{2016}x^{2016}+a_{2015}x^{2015}+...+a_1x+a_0$ be a polynomial with real coefficients and $a_{2016} \neq 0$ satisfies $|a_1+a_3+...+a_{2015}| > |a_0+a_2+...+a_{2016}|$ Prove that $P(x)$ has an odd number of complex roots with absolute value less than $1$ (count multiple roots also) edited: complex roots

2013 APMO, 4

Tags: combinatorics , algebra , polynomial

Let $a$ and $b$ be positive integers, and let $A$ and $B$ be finite sets of integers satisfying (i) $A$ and $B$ are disjoint; (ii) if an integer $i$ belongs to either to $A$ or to $B$, then either $i+a$ belongs to $A$ or $i-b$ belongs to $B$. Prove that $a\left\lvert A \right\rvert = b \left\lvert B \right\rvert$. (Here $\left\lvert X \right\rvert$ denotes the number of elements in the set $X$.)

Russian TST 2019, P1

Tags: algebra , polynomial

Let $t\in (1,2)$. Show that there exists a polynomial $P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ with the coefficients in $\{1,-1\}$ such that $\left|P(t)-2019\right| \leqslant 1.$ [i]Proposed by N. Safaei (Iran)[/i]

1989 Greece National Olympiad, 3

Tags: algebra , inequalities , polynomial

If $a\ge 0$ prove that $a^4+ a^3-10 a^2+9 a+4>0$.

2012 AMC 12/AHSME, 20

Tags: algebra , polynomial

Consider the polynomial \[P(x)=\prod_{k=0}^{10}(x^{2^k}+2^k)=(x+1)(x^2+2)(x^4+4)\cdots(x^{1024}+1024).\] The coefficient of $x^{2012}$ is equal to $2^a$. What is $a$? $ \textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 24 $

1968 Poland - Second Round, 1

Tags: algebra , integer polynomial , polynomial

Prove that if a polynomial with integer coefficients takes a value equal to $1$ in absolute value at three different integer points, then it has no integer zeros.

2005 Brazil Undergrad MO, 1

Tags: function , algebra , polynomial , linear algebra , matrix

Determine the number of possible values for the determinant of $A$, given that $A$ is a $n\times n$ matrix with real entries such that $A^3 - A^2 - 3A + 2I = 0$, where $I$ is the identity and $0$ is the all-zero matrix.

2020 Centroamerican and Caribbean Math Olympiad, 5

Tags: inequalities , algebra , inequality , n-variable inequality , polynomial

Let $P(x)$ be a polynomial with real non-negative coefficients. Let $k$ be a positive integer and $x_1, x_2, \dots, x_k$ positive real numbers such that $x_1x_2\cdots x_k=1$. Prove that $$P(x_1)+P(x_2)+\cdots+P(x_k)\geq kP(1).$$

2001 India IMO Training Camp, 1

Tags: polynomial , complex number , algebra , number theory

Complex numbers $\alpha$ , $\beta$ , $\gamma$ have the property that $\alpha^k +\beta^k +\gamma^k$ is an integer for every natural number $k$. Prove that the polynomial \[(x-\alpha)(x-\beta )(x-\gamma )\] has integer coefficients.

1999 Federal Competition For Advanced Students, Part 2, 2

Tags: algebra , polynomial

Given a real number $A$ and an integer $n$ with $2 \leq n \leq 19$, find all polynomials $P(x)$ with real coefficients such that $P(P(P(x))) = Ax^n +19x+99$.

1991 Spain Mathematical Olympiad, 3

Tags: triangle inequality , polynomial , algebra

What condition must be satisfied by the coefficients $u,v,w$ if the roots of the polynomial $x^3 -ux^2+vx-w$ are the sides of a triangle

2005 China Team Selection Test, 3

Tags: polynomial , inequalities , triangle inequality , induction , algebra , trigonometry

Let $n$ be a positive integer, and $a_j$, for $j=1,2,\ldots,n$ are complex numbers. Suppose $I$ is an arbitrary nonempty subset of $\{1,2,\ldots,n\}$, the inequality $\left|-1+ \prod_{j\in I} (1+a_j) \right| \leq \frac 12$ always holds. Prove that $\sum_{j=1}^n |a_j| \leq 3$.

1990 Tournament Of Towns, (257) 1

Tags: algebra , polynomial

Prove that for all natural $n$ there exists a polynomial $P(x)$ divisible by $(x-1)^n$ such that its degree is not greater than $2^n$ and each of its coefficients is equal to $1$, $0$ or $-1$. (D. Fomin, Leningrad)

2003 Alexandru Myller, 1

Tags: algebra , polynomial

[b]1)[/b] Show that there exist quadratic polynoms $ P\in\mathbb{R}[X] $ whose composition with themselves have $ 1,2 $ and $ 3 $ as their fixed points. [b]2)[/b] Prove that the polynoms referred to at [b]1)[/b] are not integer. [i]Gheorghe Iurea[/i]