Found problems: 3597
1970 IMO Longlists, 12
Let $\{x_i\}, 1\le i\le 6$ be a given set of six integers, none of which are divisible by $7$.
$(a)$ Prove that at least one of the expressions of the form $x_1\pm x_2\pm x_3\pm x_4\pm x_5\pm x_6$ is divisible by $7$, where the $\pm$ signs are independent of each other.
$(b)$ Generalize the result to every prime number.
2017 NIMO Summer Contest, 9
Let $P$ be a cubic monic polynomial with roots $a$, $b$, and $c$. If $P(1)=91$ and $P(-1)=-121$, compute the maximum possible value of \[\dfrac{ab+bc+ca}{abc+a+b+c}.\]
[i]Proposed by David Altizio[/i]
2017 Danube Mathematical Olympiad, 1
Find all polynomials $P(x)$ with integer coefficients such that $a^2+b^2-c^2$ divides $P(a)+P(b)-P(c)$, for all integers $a,b,c$.
2012 All-Russian Olympiad, 4
Initially there are $n+1$ monomials on the blackboard: $1,x,x^2, \ldots, x^n $. Every minute each of $k$ boys simultaneously write on the blackboard the sum of some two polynomials that were written before. After $m$ minutes among others there are the polynomials $S_1=1+x,S_2=1+x+x^2,S_3=1+x+x^2+x^3,\ldots ,S_n=1+x+x^2+ \ldots +x^n$ on the blackboard. Prove that $ m\geq \frac{2n}{k+1} $.
2010 AMC 10, 21
The polynomial $ x^3\minus{}ax^2\plus{}bx\minus{}2010$ has three positive integer zeros. What is the smallest possible value of $ a$?
$ \textbf{(A)}\ 78 \qquad
\textbf{(B)}\ 88 \qquad
\textbf{(C)}\ 98 \qquad
\textbf{(D)}\ 108 \qquad
\textbf{(E)}\ 118$
2008 India National Olympiad, 6
Let $ P(x)$ be a polynomial with integer coefficients. Prove that there exist two polynomials $ Q(x)$ and $ R(x)$, again with integer coefficients, such that
[b](i)[/b] $ P(x) \cdot Q(x)$ is a polynomial in $ x^2$ , and
[b](ii)[/b] $ P(x) \cdot R(x)$ is a polynomial in $ x^3$.
2009 China Team Selection Test, 2
Find all integers $ n\ge 2$ having the following property: for any $ k$ integers $ a_{1},a_{2},\cdots,a_{k}$ which aren't congruent to each other (modulo $ n$), there exists an integer polynomial $ f(x)$ such that congruence equation $ f(x)\equiv 0 (mod n)$ exactly has $ k$ roots $ x\equiv a_{1},a_{2},\cdots,a_{k} (mod n).$
1975 IMO Shortlist, 10
Determine the polynomials P of two variables so that:
[b]a.)[/b] for any real numbers $t,x,y$ we have $P(tx,ty) = t^n P(x,y)$ where $n$ is a positive integer, the same for all $t,x,y;$
[b]b.)[/b] for any real numbers $a,b,c$ we have $P(a + b,c) + P(b + c,a) + P(c + a,b) = 0;$
[b]c.)[/b] $P(1,0) =1.$
2005 MOP Homework, 1
Let $n$ be a natural number and $f_1$, $f_2$, ..., $f_n$ be polynomials with integers coeffcients. Show that there exists a polynomial $g(x)$ which can be factored (with at least two terms of degree at least $1$) over the integers such that $f_i(x)+g(x)$ cannot be factored (with at least two terms of degree at least $1$) over the integers for every $i$.
2010 Contests, 3
Find all two-variable polynomials $p(x,y)$ such that for each $a,b,c\in\mathbb R$:
\[p(ab,c^2+1)+p(bc,a^2+1)+p(ca,b^2+1)=0\]
2020 Silk Road, 3
A polynomial $ Q (x) = k_n x ^ n + k_ {n-1} x ^ {n-1} + \ldots + k_1 x + k_0 $ with real coefficients is called [i]powerful[/i] if the equality $ | k_0 | = | k_1 | + | k_2 | + \ldots + | k_ {n-1} | + | k_n | $, and [i]non-increasing[/i] , if $ k_0 \geq k_1 \geq \ldots \geq k_ {n-1} \geq k_n $.
Let for the polynomial $ P (x) = a_d x ^ d + a_ {d-1} x ^ {d-1} + \ldots + a_1 x + a_0 $ with nonzero real coefficients, where $ a_d> 0 $, the polynomial $ P (x) (x-1) ^ t (x + 1) ^ s $ is [i]powerful[/i] for some non-negative integers $ s $ and $ t $ ($ s + t> 0 $). Prove that at least one of the polynomials $ P (x) $ and $ (- 1) ^ d P (-x) $ is [i]nonincreasing[/i].
2010 AMC 12/AHSME, 21
The graph of $ y \equal{} x^6 \minus{} 10x^5 \plus{} 29x^4 \minus{} 4x^3 \plus{} ax^2$ lies above the line $ y \equal{} bx \plus{} c$ except at three values of $ x$, where the graph and the line intersect. What is the largest of those values?
$ \textbf{(A)}\ 4 \qquad
\textbf{(B)}\ 5 \qquad
\textbf{(C)}\ 6 \qquad
\textbf{(D)}\ 7 \qquad
\textbf{(E)}\ 8$
2002 Tournament Of Towns, 1
Show that if the last digit of the number $x^2+xy+y^2$ is $0$ (where $x,y\in\mathbb{N}$ ) then last two digits are zero.
2010 Tournament Of Towns, 4
Assume that $P(x)$ is a polynomial with integer non negative coefficients, different from constant. Baron Munchausen claims that he can restore $P(x)$ provided he knows the values of $P(2)$ and $P(P(2))$ only. Is the baron's claim valid?
1991 India National Olympiad, 7
Solve the following system for real $x,y,z$
\[ \{ \begin{array}{ccc} x+ y -z & =& 4 \\ x^2 - y^2 + z^2 & = & -4 \\ xyz & =& 6. \end{array} \]
2000 China Team Selection Test, 1
Let $F$ be the set of all polynomials $\Gamma$ such that all the coefficients of $\Gamma (x)$ are integers and $\Gamma (x) = 1$ has integer roots. Given a positive intger $k$, find the smallest integer $m(k) > 1$ such that there exist $\Gamma \in F$ for which $\Gamma (x) = m(k)$ has exactly $k$ distinct integer roots.
2021 Belarusian National Olympiad, 10.4
Quadratic polynomials $P(x)$ and $Q(x)$ with leading coefficients $1$, both of which have real roots, are called friendly if for all $t \in [0,1]$ quadratic polynomial $tP(x)+(1-t)Q(x)$ also has real roots.
a) Provide an example of quadratic polynomials $P(x)$ and $Q(x)$ with leading coefficients $1$ and which have real roots, that are not friendly.
b) Prove that for any two quadratic polynomials $P(x)$ and $Q(x)$ with leading coefficients $1$ that have real roots, there is a quadratic polynomial $R(x)$ which has a leading coefficient $1$ and which is friendly to both $P$ and $Q$
2012 Online Math Open Problems, 30
Let $P(x)$ denote the polynomial
\[3\sum_{k=0}^{9}x^k + 2\sum_{k=10}^{1209}x^k + \sum_{k=1210}^{146409}x^k.\]Find the smallest positive integer $n$ for which there exist polynomials $f,g$ with integer coefficients satisfying $x^n - 1 = (x^{16} + 1)P(x) f(x) + 11\cdot g(x)$.
[i]Victor Wang.[/i]
1983 Federal Competition For Advanced Students, P2, 2
Let $ x_1,x_2,x_3$ be the roots of: $ x^3\minus{}6x^2\plus{}ax\plus{}a\equal{}0$. Find all real numbers $ a$ for which $ (x_1\minus{}1)^3\plus{}(x_2\minus{}1)^3\plus{}(x_3\minus{}1)^3\equal{}0$. Also, for each such $ a$, determine the corresponding values of $ x_1,x_2,$ and $ x_3$.
2011 China Team Selection Test, 2
Let $\ell$ be a positive integer, and let $m,n$ be positive integers with $m\geq n$, such that $A_1,A_2,\cdots,A_m,B_1,\cdots,B_m$ are $m+n$ pairwise distinct subsets of the set $\{1,2,\cdots,\ell\}$. It is known that $A_i\Delta B_j$ are pairwise distinct, $1\leq i\leq m, 1\leq j\leq n$, and runs over all nonempty subsets of $\{1,2,\cdots,\ell\}$. Find all possible values of $m,n$.
2007 AMC 12/AHSME, 18
The polynomial $ f(x) \equal{} x^{4} \plus{} ax^{3} \plus{} bx^{2} \plus{} cx \plus{} d$ has real coefficients, and $ f(2i) \equal{} f(2 \plus{} i) \equal{} 0.$ What is $ a \plus{} b \plus{} c \plus{} d?$
$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 16$
PEN E Problems, 16
Prove that for any prime $p$ in the interval $\left]n, \frac{4n}{3}\right]$, $p$ divides \[\sum^{n}_{j=0}{{n}\choose{j}}^{4}.\]
2017 Miklós Schweitzer, 5
For every non-constant polynomial $p$, let $H_p=\big\{z\in \mathbb{C} \, \big| \, |p(z)|=1\big\}$. Prove that if $H_p=H_q$ for some polynomials $p,q$, then there exists a polynomial $r$ such that $p=r^m$ and $q=\xi\cdot r^n$ for some positive integers $m,n$ and constant $|\xi|=1$.
India EGMO 2022 TST, 6
Suppose $P(x)$ is a non-constant polynomial with real coefficients, and even degree. Bob writes the polynomial $P(x)$ on a board. At every step, if the polynomial on the board is $f(x)$, he can replace it with
1. $f(x)+c$ for a real number $c$, or
2. the polynomial $P(f(x))$.
Can he always find a finite sequence of steps so the final polynomial on the board has exactly $2020$ real roots? What about $2021$?
[i]~Sutanay Bhattacharya[/i]
1982 Spain Mathematical Olympiad, 4
Determine a polynomial of non-negative real coefficients that satisfies the following two conditions:
$$p(0) = 0, p(|z|) \le x^4 + y^4,$$
being $|z|$ the module of the complex number $z = x + iy$ .