This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 3597

2014 Contests, 3
Tags: complex analysis , algebra , polynomial
Let $p,q\in \mathbb{R}[x]$ such that $p(z)q(\overline{z})$ is always a real number for every complex number $z$. Prove that $p(x)=kq(x)$ for some constant $k \in \mathbb{R}$ or $q(x)=0$. [i]Proposed by Mohammad Ahmadi[/i]

2005 Alexandru Myller, 2
Tags: superior algebra , matrix , linear algebra , algebra , polynomial
Let $A\in M_4(\mathbb R)$ be an invertible matrix s.t. $\det(A+^tA)=5\det A$ and $\det (A-^tA)=\det A$. Prove that for every complex root $\omega$ of order 5 of unitity (i.e. $\omega^5=1,\omega\not\in\mathbb R$) the following relation holds $\det(\omega A+^tA)=0$. [i]Dan Popescu[/i]

2015 CCA Math Bonanza, I12
Tags: algebra , polynomial
Positive integers $x,y,z$ satisfy $x^3+xy+x^2+xz+y+z=301$. Compute $y+z-x$. [i]2015 CCA Math Bonanza Individual Round #12[/i]

2007 IMC, 3
Tags: quadratic , analytic geometry , linear algebra , algebra , matrix , polynomial
Call a polynomial $ P(x_{1}, \ldots, x_{k})$ [i]good[/i] if there exist $ 2\times 2$ real matrices $ A_{1}, \ldots, A_{k}$ such that $ P(x_{1}, \ldots, x_{k}) = \det \left(\sum_{i=1}^{k}x_{i}A_{i}\right).$ Find all values of $ k$ for which all homogeneous polynomials with $ k$ variables of degree 2 are good. (A polynomial is homogeneous if each term has the same total degree.)

2023 Princeton University Math Competition, B2
Tags: polynomial , algebra
Let $f$ be a polynomial with degree at most $n-1$. Show that $$ \sum_{k=0}^n\left(\begin{array}{l} n \\ k \end{array}\right)(-1)^k f(k)=0 $$

2002 China Team Selection Test, 3
Tags: algebra , polynomial
Let \[ f(x_1,x_2,x_3) = -2 \cdot (x_1^3+x_2^3+x_3^3) + 3 \cdot (x_1^2(x_2+x_3) + x_2^2 \cdot (x_1+x_3) + x_3^2 \cdot ( x_1+x_2 ) - 12x_1x_2x_3. \] For any reals $r,s,t$, we denote \[ g(r,s,t)=\max_{t\leq x_3\leq t+2} |f(r,r+2,x_3)+s|. \] Find the minimum value of $g(r,s,t)$.

2012 ELMO Shortlist, 6
Tags: vieta , quadratic , number theory , floor function , algebra , inequalities , polynomial
Prove that if $a$ and $b$ are positive integers and $ab>1$, then \[\left\lfloor\frac{(a-b)^2-1}{ab}\right\rfloor=\left\lfloor\frac{(a-b)^2-1}{ab-1}\right\rfloor.\]Here $\lfloor x\rfloor$ denotes the greatest integer not exceeding $x$. [i]Calvin Deng.[/i]

2009 Miklós Schweitzer, 4
Tags: polynomial , superior algebra , algebra
Prove that the polynomial \[ f(x) \equal{} \frac {x^n \plus{} x^m \minus{} 2}{x^{\gcd(m,n)} \minus{} 1}\] is irreducible over $ \mathbb{Q}$ for all integers $ n > m > 0$.

2013 ELMO Shortlist, 2
Tags: geometry , algebra , 3d geometry , polynomial , number theory , modular arithmetic
For what polynomials $P(n)$ with integer coefficients can a positive integer be assigned to every lattice point in $\mathbb{R}^3$ so that for every integer $n \ge 1$, the sum of the $n^3$ integers assigned to any $n \times n \times n$ grid of lattice points is divisible by $P(n)$? [i]Proposed by Andre Arslan[/i]

2022 Israel TST, 2
Tags: polynomial , algebra , function
Let $f: \mathbb{Z}^2\to \mathbb{R}$ be a function. It is known that for any integer $C$ the four functions of $x$ \[f(x,C), f(C,x), f(x,x+C), f(x, C-x)\] are polynomials of degree at most $100$. Prove that $f$ is equal to a polynomial in two variables and find its maximal possible degree. [i]Remark: The degree of a bivariate polynomial $P(x,y)$ is defined as the maximal value of $i+j$ over all monomials $x^iy^j$ appearing in $P$ with a non-zero coefficient.[/i]

1997 Federal Competition For Advanced Students, Part 2, 3
Tags: algebra , polynomial
For every natural number $n$, find all polynomials $x^2+ax+b$, where $a^2 \geq 4b$, that divide $x^{2n} + ax^n + b$.

2017 Middle European Mathematical Olympiad, 1
Tags: polynomial , algebra
Determine all pairs of polynomials $(P, Q)$ with real coefficients satisfying $$P(x + Q(y)) = Q(x + P(y))$$ for all real numbers $x$ and $y$.

2009 Iran MO (2nd Round), 1
Tags: polynomial , quadratic , algebra
Let $ p(x) $ be a quadratic polynomial for which : \[ |p(x)| \leq 1 \qquad \forall x \in \{-1,0,1\} \] Prove that: \[ \ |p(x)|\leq\frac{5}{4} \qquad \forall x \in [-1,1]\]

1996 Flanders Math Olympiad, 4
Tags: polynomial , algebra
Consider a real poylnomial $p(x)=a_nx^n+...+a_1x+a_0$. (a) If $\deg(p(x))>2$ prove that $\deg(p(x)) = 2 + deg(p(x+1)+p(x-1)-2p(x))$. (b) Let $p(x)$ a polynomial for which there are real constants $r,s$ so that for all real $x$ we have \[ p(x+1)+p(x-1)-rp(x)-s=0 \]Prove $\deg(p(x))\le 2$. (c) Show, in (b) that $s=0$ implies $a_2=0$.

2008 IMC, 1
Tags: ratio , algebra , modular arithmetic , polynomial , relatively prime , number theory
Let $ n, k$ be positive integers and suppose that the polynomial $ x^{2k}\minus{}x^k\plus{}1$ divides $ x^{2n}\plus{}x^n\plus{}1$. Prove that $ x^{2k}\plus{}x^k\plus{}1$ divides $ x^{2n}\plus{}x^n\plus{}1$.

2022 Vietnam National Olympiad, 1
Tags: polynomial , algebra
Consider 2 non-constant polynomials $P(x),Q(x)$, with nonnegative coefficients. The coefficients of $P(x)$ is not larger than $2021$ and $Q(x)$ has at least one coefficient larger than $2021$. Assume that $P(2022)=Q(2022)$ and $P(x),Q(x)$ has a root $\frac p q \ne 0 (p,q\in \mathbb Z,(p,q)=1)$. Prove that $|p|+n|q|\le Q(n)-P(n)$ for all $n=1,2,...,2021$

2024 Canada National Olympiad, 3
Tags: polynomial , algebra
Let $N{}$ be the number of positive integers with $10$ digits $\overline{d_9d_8\cdots d_0}$ in base $10$ (where $0\le d_i\le9$ for all $i$ and $d_9>0$) such that the polynomial \[d_9x^9+d_8x^8+\cdots+d_1x+d_0\] is irreducible in $\Bbb Q$. Prove that $N$ is even. (A polynomial is irreducible in $\Bbb Q$ if it cannot be factored into two non-constant polynomials with rational coefficients.)

2006 China Team Selection Test, 2
Tags: algebra , polynomial , function
The function $f(n)$ satisfies $f(0)=0$, $f(n)=n-f \left( f(n-1) \right)$, $n=1,2,3 \cdots$. Find all polynomials $g(x)$ with real coefficient such that \[ f(n)= [ g(n) ], \qquad n=0,1,2 \cdots \] Where $[ g(n) ]$ denote the greatest integer that does not exceed $g(n)$.

2022 JHMT HS, 1
Tags: polynomial , algebra
If three of the roots of the quartic polynomial $f(x) = x^4 + ax^3 + bx^2 + cx + d$ are $0$, $2$, and $4$, and the sum of $a$, $b$, and $c$ is at most $12$, then find the largest possible value of $f(1)$.

1953 Moscow Mathematical Olympiad, 241
Tags: algebra , factor , polynomial
Prove that the polynomial $x^{200} y^{200} +1$ cannot be represented in the form $f(x)g(y)$, where $f$ and $g$ are polynomials of only $x$ and $y$, respectively.

III Soros Olympiad 1996 - 97 (Russia), 11.8
Tags: polynomial , algebra
Find any polynomial with integer coefficients, the smallest value of which on the entire line is equal to : a) $-\sqrt2$ b) $\sqrt2$

1989 Bulgaria National Olympiad, Problem 3
Tags: polynomial , irreducibility , algebra
Let $p$ be a real number and $f(x)=x^p-x+p$. Prove that: (a) Every root $\alpha$ of $f(x)$ satisfies $|\alpha|<p^{\frac1{p-1}}$; (b) If $p$ is a prime number, then $f(x)$ cannot be written as the product of two non-constant polynomials with integer coefficients.

2008 Hanoi Open Mathematics Competitions, 3
Tags: coefficient , combinatorics , polynomial , algebra
Find the coefficient of $x$ in the expansion of $(1 + x)(1 - 2x)(1 + 3x)(1 - 4x) ...(1 - 2008x)$.

2020 South East Mathematical Olympiad, 3
Tags: polynomial , algebra
Given a polynomial $f(x)=x^{2020}+\sum_{i=0}^{2019} c_ix^i$, where $c_i \in \{ -1,0,1 \}$. Denote $N$ the number of positive integer roots of $f(x)=0$ (counting multiplicity). If $f(x)=0$ has no negative integer roots, find the maximum of $N$.

2015 AMC 12/AHSME, 18
Tags: function , quadratic , vieta , sfft , polynomial , algebra
The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of all possible values of $a$? $\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$