Olympiad problems

PEN Q Problems, 4

Tags: algebra , polynomial , inequalities , complex analysis , complex numbers , Polynomials

A prime $p$ has decimal digits $p_{n}p_{n-1} \cdots p_0$ with $p_{n}>1$. Show that the polynomial $p_{n}x^{n} + p_{n-1}x^{n-1}+\cdots+ p_{1}x + p_0$ cannot be represented as a product of two nonconstant polynomials with integer coefficients

2015 Dutch IMO TST, 2

Tags: Polynomials , algebra , polynomial

Determine all polynomials P(x) with real coefficients such that [(x + 1)P(x − 1) − (x − 1)P(x)] is a constant polynomial.

Kvant 2024, M2799

Tags: algebra , polynomial , Polynomials , algebra proposed

Let $n$ be a positive integer. Ilya and Sasha both choose a pair of different polynomials of degree $n$ with real coefficients. Lenya knows $n$, his goal is to find out whether Ilya and Sasha have the same pair of polynomials. Lenya selects a set of $k$ real numbers $x_1<x_2<\dots<x_k$ and reports these numbers. Then Ilya fills out a $2 \times k$ table: For each $i=1,2,\dots,k$ he writes a pair of numbers $P(x_i),Q(x_i)$ (in any of the two possible orders) intwo the two cells of the $i$-th column, where $P$ and $Q$ are his polynomials. Sasha fills out a similar table. What is the minimal $k$ such that Lenya can surely achieve the goal by looking at the tables? [i]Proposed by L. Shatunov[/i]

PEN Q Problems, 5

Tags: algebra , polynomial , induction , Polynomials

(Eisentein's Criterion) Let $f(x)=a_{n}x^{n} +\cdots +a_{1}x+a_{0}$ be a nonconstant polynomial with integer coefficients. If there is a prime $p$ such that $p$ divides each of $a_{0}$, $a_{1}$, $\cdots$,$a_{n-1}$ but $p$ does not divide $a_{n}$ and $p^2$ does not divide $a_{0}$, then $f(x)$ is irreducible in $\mathbb{Q}[x]$.

2011 Ukraine Team Selection Test, 11

Tags: Polynomials , polynomial , algebra , inequalities , Integer Polynomial

Let $ P (x) $ and $ Q (x) $ be polynomials with real coefficients such that $ P (0)> 0 $ and all coefficients of the polynomial $ S (x) = P (x) \cdot Q (x) $ are integers. Prove that for any positive $ x $ the inequality holds: $$S ({{x} ^ {2}}) - {{S} ^ {2}} (x) \le \frac {1} {4} ({{P} ^ {2}} ({{ x} ^ {3}}) + Q ({{x} ^ {3}})). $$

1987 IMO Longlists, 39

Tags: algebra , polynomial , algebra unsolved , Polynomials

Let $A$ be a set of polynomials with real coefficients and let them satisfy the following conditions: [b](i)[/b] if $f \in A$ and $\deg( f ) \leq 1$, then $f(x) = x - 1$; [b](ii)[/b] if $f \in A$ and $\deg( f ) \geq 2$, then either there exists $g \in A$ such that $f(x) = x^{2+\deg(g)} + xg(x) -1$ or there exist $g, h \in A$ such that $f(x) = x^{1+\deg(g)}g(x) + h(x)$; [b](iii)[/b] for every $g, h \in A$, both $x^{2+\deg(g)} + xg(x) -1$ and $x^{1+\deg(g)}g(x) + h(x)$ belong to $A.$ Let $R_n(f)$ be the remainder of the Euclidean division of the polynomial $f(x)$ by $x^n$. Prove that for all $f \in A$ and for all natural numbers $n \geq 1$ we have $R_n(f)(1) \leq 0$, and that if $R_n(f)(1) = 0$ then $R_n(f) \in A$.

2017 China Team Selection Test, 4

Tags: Tstst , Polynomials , algebra , polynomial , China

Find out all the integer pairs $(m,n)$ such that there exist two monic polynomials $P(x)$ and $Q(x)$ ,with $\deg{P}=m$ and $\deg{Q}=n$,satisfy that $$P(Q(t))\not=Q(P(t))$$ holds for any real number $t$.

PEN Q Problems, 11

Tags: algebra , polynomial , Polynomials

Show that the polynomial $x^{8} +98 x^{4}+1$ can be expressed as the product of two nonconstant polynomials with integer coefficients.

1971 IMO Longlists, 16

Tags: algebra , system of equations , Vieta , Polynomials , IMO Shortlist

Knowing that the system \[x + y + z = 3,\]\[x^3 + y^3 + z^3 = 15,\]\[x^4 + y^4 + z^4 = 35,\] has a real solution $x, y, z$ for which $x^2 + y^2 + z^2 < 10$, find the value of $x^5 + y^5 + z^5$ for that solution.

2023 India Regional Mathematical Olympiad, 3

Tags: algebra , Polynomials , polynomial

Let $f(x)$ be a polynomial with real coefficients of degree 2. Suppose that for some pairwise distinct real numbers , $a,b,c$ we have:\\ \[f(a)=bc , f(b)=ac, f(c)=ab\] Dertermine $f(a+b+c)$ in terms of $a,b,c$.

2014 Poland - Second Round, 3.

Tags: algebra , polynomial , Polynomials

For each positive integer $n$, determine the smallest possible value of the polynomial $$ W_n(x)=x^{2n}+2x^{2n-1}+3x^{2n-2}+\ldots + (2n-1)x^2+2nx. $$

2014 IMO Shortlist, A5

Tags: IMO Shortlist , algebra , Functional inequality , Polynomials

Consider all polynomials $P(x)$ with real coefficients that have the following property: for any two real numbers $x$ and $y$ one has \[|y^2-P(x)|\le 2|x|\quad\text{if and only if}\quad |x^2-P(y)|\le 2|y|.\] Determine all possible values of $P(0)$. [i]Proposed by Belgium[/i]

1997 Romania National Olympiad, 3

Tags: algebra , superior algebra , Field theory , Polynomials

Let $K$ be a finite field, $n \ge 2$ an integer, $f \in K[X]$ an irreducible polynomial of degree $n,$ and $g$ the product of all the nonconstant polynomials in $K[X]$ of degree at most $n-1.$ Prove that $f$ divides $g-1.$

2021 Turkey MO (2nd round), 2

Tags: Polynomials

If a polynomial with real coefficients of degree $d$ has at least $d$ coefficients equal to $1$ and has $d$ real roots, what is the maximum possible value of $d$? (Note: The roots of the polynomial do not have to be different from each other.)

1985 Bulgaria National Olympiad, Problem 1

Tags: number theory , Polynomials , algebra

Let $f(x)$ be a non-constant polynomial with integer coefficients and $n,k$ be natural numbers. Show that there exist $n$ consecutive natural numbers $a,a+1,\ldots,a+n-1$ such that the numbers $f(a),f(a+1),\ldots,f(a+n-1)$ all have at least $k$ prime factors. (We say that the number $p_1^{\alpha_1}\cdots p_s^{\alpha_s}$ has $\alpha_1+\ldots+\alpha_s$ prime factors.)

2014 BMT Spring, 4

Tags: algebra , Polynomials

The function $f(x)=x^5-20x^4+ax^3+bx^2+cx+24$ has the interesting property that its roots can be arranged to form an arithmetic sequence. Determine $f(8)$.

2012 Irish Math Olympiad, 3

Tags: Polynomials , algebra , polynomial , Combo

Find, with proof, all polynomials $f$ such that $f$ has nonnegative integer coefficients, $f$($1$) = $8$ and $f$($2$) = $2012$.

2016 IFYM, Sozopol, 6

Tags: algebra , Polynomials

Find all polynomials $P\in \mathbb{Q}[x]$, which satisfy the following equation: $P^2 (n)+\frac{1}{4}=P(n^2+\frac{1}{4})$ for $\forall$ $n\in \mathbb{N}$.

2023 IFYM, Sozopol, 8

Tags: algebra , Polynomials

Do there exist a natural number $n$ and real numbers $a_0, a_1, \dots, a_n$, each equal to $1$ or $-1$, such that the polynomial $a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0$ is divisible by the polynomial $x^{2023} - 2x^{2022} + c$, where: \\ (a) $c = 1$ \\ (b) $c = -1$? [i] (For polynomials $P(x)$ and $Q(x)$ with real coefficients, we say that $P(x)$ is divisible by $Q(x)$ if there exists a polynomial $R(x)$ with real coefficients such that $P(x) = Q(x)R(x)$.)[/i]

2021 Nigerian MO Round 3, Problem 5

Tags: calculus , derivative , degree , Polynomials , algebra , polynomial , functions

Let $f(x)=\frac{P(x)}{Q(x)}$, where $P(x), Q(x)$ are two non-constant polynomials with no common zeros and $P(0)=P(1)=0$. Suppose $f(x)f\left(\frac{1}{x}\right)=f(x)+f\left(\frac{1}{x}\right)$ for infinitely many values of $x$. a) Show that $\text{deg}(P)<\text{deg}(Q)$. b) Show that $P'(1)=2Q'(1)-\text{deg}(Q)\cdot Q(1)$. Here, $P'(x)$ denotes the derivative of $P(x)$ as usual.

2001 China Team Selection Test, 1

Tags: algebra , Polynomials , polynomial

Let $p(x)$ be a polynomial with real coefficients such that $p(0)=p(n)$. Prove that there are at least $n$ pairs of real numbers $(x,y)$ where $p(x)=p(y)$ and $y-x$ is a positive integer

2019 IFYM, Sozopol, 4

Tags: number theory , Polynomials

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 $n\geq 0$?

2014 Contests, 3.

Tags: algebra , polynomial , Polynomials

For each positive integer $n$, determine the smallest possible value of the polynomial $$ W_n(x)=x^{2n}+2x^{2n-1}+3x^{2n-2}+\ldots + (2n-1)x^2+2nx. $$

2023 International Zhautykov Olympiad, 3

Tags: number theory , Polynomials

Let $a_1, a_2, \cdots, a_k$ be natural numbers. Let $S(n)$ be the number of solutions in nonnegative integers to $a_1x_1 + a_2x_2 + \cdots + a_kx_k = n$. Suppose $S(n) \neq 0$ for all big enough $n$. Show that for all sufficiently large $n$, we have $S(n+1) < 2S(n)$.

2019 AIME Problems, 8

Tags: AIME , AMC , AIME II , function , 2019 AIME II , 2019 AMC , Polynomials

The polynomial $f(z)=az^{2018}+bz^{2017}+cz^{2016}$ has real coefficients not exceeding $2019$, and $f(\tfrac{1+\sqrt{3}i}{2})=2015+2019\sqrt{3}i$. Find the remainder when $f(1)$ is divided by $1000$.