Olympiad problems

2003 Iran MO (3rd Round), 29

Let $ c\in\mathbb C$ and $ A_c \equal{} \{p\in \mathbb C[z]|p(z^2 \plus{} c) \equal{} p(z)^2 \plus{} c\}$. a) Prove that for each $ c\in C$, $ A_c$ is infinite. b) Prove that if $ p\in A_1$, and $ p(z_0) \equal{} 0$, then $ |z_0| < 1.7$. c) Prove that each element of $ A_c$ is odd or even. Let $ f_c \equal{} z^2 \plus{} c\in \mathbb C[z]$. We see easily that $ B_c: \equal{} \{z,f_c(z),f_c(f_c(z)),\dots\}$ is a subset of $ A_c$. Prove that in the following cases $ A_c \equal{} B_c$. d) $ |c| > 2$. e) $ c\in \mathbb Q\backslash\mathbb Z$. f) $ c$ is a non-algebraic number g) $ c$ is a real number and $ c\not\in [ \minus{} 2,\frac14]$.

2005 Kazakhstan National Olympiad, 4

Tags: polynomial , algebra

Find all polynomials $ P(x)$ with real coefficients such that for every positive integer $ n$ there exists a rational $ r$ with $ P(r)=n$.

2018 Canada National Olympiad, 4

Tags: algebra , number theory , polynomial

Find all polynomials $p(x)$ with real coefficients that have the following property: there exists a polynomial $q(x)$ with real coefficients such that $$p(1) + p(2) + p(3) +\dots + p(n) = p(n)q(n)$$ for all positive integers $n$.

2007 Estonia Math Open Senior Contests, 4

Tags: algebra , polynomial , number theory

The Fibonacci sequence is determined by conditions $ F_0 \equal{} 0, F1 \equal{} 1$, and $ F_k\equal{}F_{k\minus{}1}\plus{}F_{k\minus{}2}$ for all $ k \ge 2$. Let $ n$ be a positive integer and let $ P(x) \equal{} a_mx^m \plus{}. . .\plus{} a_1x\plus{} a_0$ be a polynomial that satisfies the following two conditions: (1) $ P(F_n) \equal{} F_{n}^{2}$ ; (2) $ P(F_k) \equal{} P(F_{k\minus{}1}) \plus{} P(F_{k\minus{}2}$ for all $ k \ge 2$. Find the sum of the coefficients of P.

2006 Bulgaria Team Selection Test, 3

Tags: algebra , polynomial , combinatorics

[b]Problem 6.[/b] Let $p>2$ be prime. Find the number of the subsets $B$ of the set $A=\{1,2,\ldots,p-1\}$ such that, the sum of the elements of $B$ is divisible by $p.$ [i] Ivan Landgev[/i]

2001 All-Russian Olympiad, 1

Tags: algebra , polynomial , quadratic , quadratic formula

The polynomial $ P(x)\equal{}x^3\plus{}ax^2\plus{}bx\plus{}d$ has three distinct real roots. The polynomial $ P(Q(x))$, where $ Q(x)\equal{}x^2\plus{}x\plus{}2001$, has no real roots. Prove that $ P(2001)>\frac{1}{64}$.

2018 Saint Petersburg Mathematical Olympiad, 4

Tags: algebra , polynomial , abstract algebra

$f(x)$ is polynomial with integer coefficients, with module not exceeded $5*10^6$. $f(x)=nx$ has integer root for $n=1,2,...,20$. Prove that $f(0)=0$

2019 Ukraine Team Selection Test, 2

Tags: algebra , polynomial , divisible

Polynomial $p(x)$ with real coefficients, which is different from the constant, has the following property: [i] for any naturals $n$ and $k$ the $\frac{p(n+1)p(n+2)...p(n+k)}{p(1)p(2)...p(k)}$ is an integer.[/i] Prove that this polynomial is divisible by $x$.

2018 Brazil Team Selection Test, 1

Tags: algebra , polynomial , ineqstd

Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that $$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$ If $M>1$, prove that the polynomial $$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$ has no positive roots.

2010 IberoAmerican Olympiad For University Students, 5

Tags: matrix , algebra , polynomial , inequalities , linear algebra

Let $A,B$ be matrices of dimension $2010\times2010$ which commute and have real entries, such that $A^{2010}=B^{2010}=I$, where $I$ is the identity matrix. Prove that if $\operatorname{tr}(AB)=2010$, then $\operatorname{tr}(A)=\operatorname{tr}(B)$.

2003 IMC, 6

Tags: algebra , polynomial , real analysis

Let $ p=\sum\limits_{k=0}^n a_kX^k\in R[X] $ a polynomial such that all his roots lie in the half plane $ \{z\in C| Re(z)<0 \}. $ Prove that $ a_ka_{k+3}<a_{k+1}a_{k+2}, $ for every k=0,1,2...,n-3.

1974 Putnam, A6

Tags: algebra , polynomial

Given $n$, let $k = k(n)$ be the minimal degree of any monic integral polynomial $$f(x)=x^k + a_{k-1}x^{k-1}+\ldots+a_0$$ such that the value of $f(x)$ is exactly divisible by $n$ for every integer $x.$ Find the relationship between $n$ and $k(n)$. In particular, find the value of $k(n)$ corresponding to $n = 10^6.$

2005 Brazil Undergrad MO, 1

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

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.

1981 Canada National Olympiad, 4

Tags: polynomial , function , algebra

$P(x),Q(x)$ are two polynomials such that $P(x)=Q(x)$ has no real solution, and $P(Q(x))\equiv Q(P(x))\forall x\in\mathbb{R}$. Prove that $P(P(x))=Q(Q(x))$ has no real solution.

2013 Hitotsubashi University Entrance Examination, 5

Tags: probability , function , algebra , polynomial , geometric series , probability and stats

Throw a die $n$ times, let $a_k$ be a number shown on the die in the $k$-th place. Define $s_n$ by $s_n=\sum_{k=1}^n 10^{n-k}a_k$. (1) Find the probability such that $s_n$ is divisible by 4. (2) Find the probability such that $s_n$ is divisible by 6. (3) Find the probability such that $s_n$ is divisible by 7. Last Edited Thanks, jmerry & JBL

1997 Moldova Team Selection Test, 5

Tags: polynomial , algebra

Let $P(x)\in\mathbb{Z}[x]$ with deg $P=2015$. Let $Q(x)=(P(x))^2-9$. Prove that: the number of distinct roots of $Q(x)$ can not bigger than $2015$

2017 Princeton University Math Competition, B2

Tags: algebra , polynomial

Let $a_1(x), a_2(x)$, and $a_3(x)$ be three polynomials with integer coefficients such that every polynomial with integer coefficients can be written in the form $p_1(x)a_1(x) + p_2(x)a_2(x) + p_3(x)a_3(x)$ for some polynomials $p_1(x), p_2(x), p_3(x)$ with integer coefficients. Show that every polynomial is of the form $p_1(x)a_1(x)^2 + p_2(x)a_2(x)^2 + p_3(x)a_3(x)^2$ for some polynomials $p_1(x), p_2(x), p_3(x)$ with integer coefficients.

2023 Bangladesh Mathematical Olympiad, P10

Tags: algebra , polynomial , inequalities

Let all possible $2023$-degree real polynomials: $P(x)=x^{2023}+a_1x^{2022}+a_2x^{2021}+\cdots+a_{2022}x+a_{2023}$, where $P(0)+P(1)=0$, and the polynomial has 2023 real roots $r_1, r_2,\cdots r_{2023}$ [not necessarily distinct] so that $0\leq r_1,r_2,\cdots r_{2023}\leq1$. What is the maximum value of $r_1 \cdot r_2 \cdots r_{2023}?$

2021 Durer Math Competition Finals, 8

Tags: polynomial , algebra

John found all real numbers $p$ such that in the polynomial $g(x) = (x -1)^2(p + 2x)^2$ , the quadratic term has coefficient $2021$. What is the sum of all of these values $p$?

2014 IMAC Arhimede, 4

Tags: polynomial , algebra , sum of powers , sum , rational

Let $n$ be a natural number and let $P (t) = 1 + t + t^2 + ... + t^{2n}$. If $x \in R$ such that $P (x)$ and $P (x^2)$ are rational numbers, prove that $x$ is rational number.

2004 Iran MO (3rd Round), 30

Tags: algebra , polynomial , number theory

Find all polynomials $ p\in\mathbb Z[x]$ such that $ (m,n)\equal{}1\Rightarrow (p(m),p(n))\equal{}1$

2008 Baltic Way, 1

Tags: polynomial , search , algebra

Determine all polynomials $p(x)$ with real coefficients such that $p((x+1)^3)=(p(x)+1)^3$ and $p(0)=0$.

2011 VJIMC, Problem 1

Tags: number theory , polynomial

(a) Is there a polynomial $P(x)$ with real coefficients such that $P\left(\frac1k\right)=\frac{k+2}k$ for all positive integers $k$? (b) Is there a polynomial $P(x)$ with real coefficients such that $P\left(\frac1k\right)=\frac1{2k+1}$ for all positive integers $k$?

1974 USAMO, 1

Tags: algebra , polynomial

Let $ a,b,$ and $ c$ denote three distinct integers, and let $ P$ denote a polynomial having integer coefficients. Show that it is impossible that $ P(a) \equal{} b, P(b) \equal{} c,$ and $ P(c) \equal{} a$.

2011 Putnam, A4

Tags: linear algebra , matrix , modular arithmetic , vector , polynomial , putnam matrices

For which positive integers $n$ is there an $n\times n$ matrix with integer entries such that every dot product of a row with itself is even, while every dot product of two different rows is odd?