Olympiad problems

2014 Dutch IMO TST, 5

Tags: polynomial , calculus , integration , ratio , arithmetic sequence , algebra

Let $P(x)$ be a polynomial of degree $n \le 10$ with integral coefficients such that for every $k \in \{1, 2, \dots, 10\}$ there is an integer $m$ with $P(m) = k$. Furthermore, it is given that $|P(10) - P(0)| < 1000$. Prove that for every integer $k$ there is an integer $m$ such that $P(m) = k.$

2016 USA TSTST, 1

Tags: polynomial

Let $A = A(x,y)$ and $B = B(x,y)$ be two-variable polynomials with real coefficients. Suppose that $A(x,y)/B(x,y)$ is a polynomial in $x$ for infinitely many values of $y$, and a polynomial in $y$ for infinitely many values of $x$. Prove that $B$ divides $A$, meaning there exists a third polynomial $C$ with real coefficients such that $A = B \cdot C$. [i]Proposed by Victor Wang[/i]

2023 Iran MO (2nd Round), P5

Tags: algebra , polynomial

5. We call $(P_n)_{n\in \mathbb{N}}$ an arithmetic sequence with common difference $Q(x)$ if $\forall n: P_{n+1} = P_n + Q$ $\newline$ We have an arithmetic sequence with a common difference $Q(x)$ and the first term $P(x)$ such that $P,Q$ are monic polynomials with integer coefficients and don't share an integer root. Each term of the sequence has at least one integer root. Prove that: $\newline$ a) $P(x)$ is divisible by $Q(x)$ $\newline$ b) $\text{deg}(\frac{P(x)}{Q(x)}) = 1$

1976 Kurschak Competition, 3

Tags: algebra , polynomial , trinomial

Prove that if the quadratic $x^2 +ax+b$ is always positive (for all real $x$) then it can be written as the quotient of two polynomials whose coefficients are all positive.

1974 IMO Longlists, 39

Tags: polynomial , complex number , algebra

Let $n$ be a positive integer, $n \geq 2$, and consider the polynomial equation \[x^n - x^{n-2} - x + 2 = 0.\] For each $n,$ determine all complex numbers $x$ that satisfy the equation and have modulus $|x| = 1.$

1985 Miklós Schweitzer, 5

Tags: number theory , relatively prime , algebra , polynomial , absolute value

Let $F(x,y)$ and $G(x,y)$ be relatively prime homogeneous polynomials of degree at least one having integer coefficients. Prove that there exists a number $c$ depending only on the degrees and the maximum of the absolute values of the coefficients of $F$ and $G$ such that $F(x,y)\neq G(x,y)$ for any integers $x$ and $y$ that are relatively prime and satisfy $\max \{ |x|,|y|\} > c$. [K. Gyory]

1962 All Russian Mathematical Olympiad, 024

Tags: polynomial , number theory , algebra

Given $x,y,z$, three different integers. Prove that $$(x-y)^5+(y-z)^5+(z-x)^5$$ is divisible by $$5(x-y)(y-z)(z-x)$$

1985 Traian Lălescu, 1.2

Tags: algebra , polynomial

Prove that all real roots of the polynomial $$ P=X^{1985}-X^{1984}+1983\cdot X^{1983}+1994\cdot X^{992} -884064 $$ are positive.

2018 Vietnam Team Selection Test, 3

Tags: polynomial , algebra

For every positive integer $n\ge 3$, let $\phi_n$ be the set of all positive integers less than and coprime to $n$. Consider the polynomial: $$P_n(x)=\sum_{k\in\phi_n} {x^{k-1}}.$$ a. Prove that $P_n(x)=(x^{r_n}+1)Q_n(x)$ for some positive integer $r_n$ and polynomial $Q_n(x)\in\mathbb{Z}[x]$ (not necessary non-constant polynomial). b. Find all $n$ such that $P_n(x)$ is irreducible over $\mathbb{Z}[x]$.

2004 Regional Olympiad - Republic of Srpska, 1

Tags: polynomial , algebra

Define the sequence $(a_n)_{n\geq 1}$ by $a_1=1$, $a_2=p$ and \[a_{n+1}=pa_n-a_{n-1} \textrm { for all } n>1.\] Prove that for $n>1$ the polynomial $x^n-a_nx+a_{n-1}$ is divisible by $x^2-px+1$. Using this result, solve the equation \[x^4-56x+15=0.\]

2014 Online Math Open Problems, 27

Tags: function , modular arithmetic , algebra , polynomial , induction , quadratic

Let $p = 2^{16}+1$ be a prime, and let $S$ be the set of positive integers not divisible by $p$. Let $f: S \to \{0, 1, 2, ..., p-1\}$ be a function satisfying \[ f(x)f(y) \equiv f(xy)+f(xy^{p-2}) \pmod{p} \quad\text{and}\quad f(x+p) = f(x) \] for all $x,y \in S$. Let $N$ be the product of all possible nonzero values of $f(81)$. Find the remainder when when $N$ is divided by $p$. [i]Proposed by Yang Liu and Ryan Alweiss[/i]

2014 Contests, 1

Tags: binomial coefficient , inequalities , algebra , polynomial , binomial theorem

Let $a$, $b$, $c$ be real numbers greater than or equal to $1$. Prove that \[ \min \left(\frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )\leq abc. \]

2016 Saint Petersburg Mathematical Olympiad, 7

Tags: algebra , polynomial

A polynomial $P$ with real coefficients is called [i]great,[/i] if for some integer $a>1$ and for all integers $x$, there exists an integer $z$ such that $aP(x)=P(z)$. Find all [i]great[/i] polynomials. [i]Proposed by A. Golovanov[/i]

Kvant 2022, M2700

Tags: algebra , polynomial

What is the maximal possible number of roots on the interval (0,1) for a polynomial of degree 2022 with integer coefficients and with the leading coefficient equal to 1?

2000 Spain Mathematical Olympiad, 1

Tags: polynomial , parameterization , quadratic , algebra

Consider the polynomials \[P(x) = x^4 + ax^3 + bx^2 + cx + 1 \quad \text{and} \quad Q(x) = x^4 + cx^3 + bx^2 + ax + 1.\] Find the conditions on the parameters $a, b, $c with $a\neq c$ for which $P(x)$ and $Q(x)$ have two common roots and, in such cases, solve the equations $P(x) = 0$ and $Q(x) = 0.$

2021 Iran Team Selection Test, 4

Tags: combi , number theory , polynomial , prime number , function , algebra

Assume $\Omega(n),\omega(n)$ be the biggest and smallest prime factors of $n$ respectively . Alireza and Amin decided to play a game. First Alireza chooses $1400$ polynomials with integer coefficients. Now Amin chooses $700$ of them, the set of polynomials of Alireza and Amin are $B,A$ respectively . Amin wins if for all $n$ we have : $$\max_{P \in A}(\Omega(P(n))) \ge \min_{P \in B}(\omega(P(n)))$$ Who has the winning strategy. Proposed by [i]Alireza Haghi[/i]

1972 Polish MO Finals, 3

Tags: inequalities , algebra , polynomial

Prove that there is a polynomial $P(x)$ with integer coefficients such that for all $x$ in the interval $\left[ \frac{1}{10} , \frac{9}{10}\right]$ we have $$\left|P(x) -\frac12 \right| < \frac{ 1}{1000 }.$$

2012 Tuymaada Olympiad, 2

Tags: polynomial , quadratic , inequalities , sum of roots , algebra

Let $P(x)$ be a real quadratic trinomial, so that for all $x\in \mathbb{R}$ the inequality $P(x^3+x)\geq P(x^2+1)$ holds. Find the sum of the roots of $P(x)$. [i]Proposed by A. Golovanov, M. Ivanov, K. Kokhas[/i]

1990 Dutch Mathematical Olympiad, 3

Tags: polynomial , induction , binomial theorem , algebra

A polynomial $ f(x)\equal{}ax^4\plus{}bx^3\plus{}cx^2\plus{}dx$ with $ a,b,c,d>0$ is such that $ f(x)$ is an integer for $ x \in \{ \minus{}2,\minus{}1,0,1,2 \}$ and $ f(1)\equal{}1$ and $ f(5)\equal{}70$. $ (a)$ Show that $ a\equal{}\frac{1}{24}, b\equal{}\frac{1}{4},c\equal{}\frac{11}{24},d\equal{}\frac{1}{4}$. $ (b)$ Prove that $ f(x)$ is an integer for all $ x \in \mathbb{Z}$.

2018 Iran Team Selection Test, 3

Tags: polynomial , integer polynomial , number theory

$n>1$ and distinct positive integers $a_1,a_2,\ldots,a_{n+1}$ are given. Does there exist a polynomial $p(x)\in\Bbb{Z}[x]$ of degree $\le n$ that satisfies the following conditions? a. $\forall_{1\le i < j\le n+1}: \gcd(p(a_i),p(a_j))>1 $ b. $\forall_{1\le i < j < k\le n+1}: \gcd(p(a_i),p(a_j),p(a_k))=1 $ [i]Proposed by Mojtaba Zare[/i]

2024 Princeton University Math Competition, A1 / B3

Tags: algebra , polynomial

Consider polynomial $f(x)=ax^3+bx^2+cx+d$ where $a, b, c, d$ are nonnegative integers satisfying $ab+bc+cd+ad=20$. Find the sum of all distinct possible values of $f(1)$.

1997 Romania Team Selection Test, 4

Tags: polynomial , number theory , greatest common divisor , modular arithmetic , algebra

Let $n\ge 2$ be an integer and let $P(X)=X^n+a_{n-1}X^{n-1}+\ldots +a_1X+1$ be a polynomial with positive integer coefficients. Suppose that $a_k=a_{n-k}$ for all $k\in 1,2,\ldots,n-1$. Prove that there exist infinitely many pairs of positive integers $x,y$ such that $x|P(y)$ and $y|P(x)$. [i]Remus Nicoara[/i]

1969 IMO Longlists, 24

Tags: algebra , polynomial , number theory , divisibility

$(GBR 1)$ The polynomial $P(x) = a_0x^k + a_1x^{k-1} + \cdots + a_k$, where $a_0,\cdots, a_k$ are integers, is said to be divisible by an integer $m$ if $P(x)$ is a multiple of $m$ for every integral value of $x$. Show that if $P(x)$ is divisible by $m$, then $a_0 \cdot k!$ is a multiple of $m$. Also prove that if $a, k,m$ are positive integers such that $ak!$ is a multiple of $m$, then a polynomial $P(x)$ with leading term $ax^k$can be found that is divisible by $m.$

2016 Saudi Arabia GMO TST, 2

Tags: algebra , polynomial

Let $a, b$ be given two real number with $a \ne 0$. Find all polynomials $P$ with real coefficients such that $x P(x - a) = (x - b)P(x)$ for all $x\in R$

2019 Simurgh, 4

Tags: algebra , polynomial

Assume that every root of polynomial $P(x) = x^d - a_1x^{d-1} + ... + (-1)^{d-k}a_d$ is in $[0,1]$. Show that for every $k = 1,2,...,d$ the following inequality holds: $ a_k - a_{k+1} + ... + (-1)^{d-k}a_d \geq 0 $