Found problems: 3597
1982 IMO Shortlist, 7
Let $p(x)$ be a cubic polynomial with integer coefficients with leading coefficient $1$ and with one of its roots equal to the product of the other two. Show that $2p(-1)$ is a multiple of $p(1)+p(-1)-2(1+p(0)).$
III Soros Olympiad 1996 - 97 (Russia), 11.8
Find any polynomial with integer coefficients, the smallest value of which on the entire line is equal to :
a) $-\sqrt2$
b) $\sqrt2$
2021 Brazil Team Selection Test, 3
A magician intends to perform the following trick. She announces a positive integer $n$, along with $2n$ real numbers $x_1 < \dots < x_{2n}$, to the audience. A member of the audience then secretly chooses a polynomial $P(x)$ of degree $n$ with real coefficients, computes the $2n$ values $P(x_1), \dots , P(x_{2n})$, and writes down these $2n$ values on the blackboard in non-decreasing order. After that the magician announces the secret polynomial to the audience. Can the magician find a strategy to perform such a trick?
2010 India Regional Mathematical Olympiad, 2
Let $P_1(x) = ax^2 - bx - c$, $P_2(x) = bx^2 - cx - a$, $P_3(x) = cx^2 - ax - b$ be three quadratic polynomials. Suppose there exists a real number $\alpha$ such that $P_1(\alpha) = P_2(\alpha) = P_3(\alpha)$. Prove that $a = b = c$.
2015 Junior Balkan Team Selection Tests - Romania, 1
Define the set $M_q=\{x \in \mathbb{Q} \mid x^3-2015x=q \}$ , where $q$ is an arbitrary rational number.
[b]a)[/b] Show that there exists values for $q$ such that the set is null as well as values for which it has exactly one element.
[b]b)[/b] Determine all the possible values for the cardinality of $M_q$
2016 Iran MO (3rd Round), 2
We call a function $g$ [i]special [/i] if $g(x)=a^{f(x)}$ (for all $x$) where $a$ is a positive integer and $f$ is polynomial with integer coefficients such that $f(n)>0$ for all positive integers $n$.
A function is called an [i]exponential polynomial[/i] if it is obtained from the product or sum of special functions. For instance, $2^{x}3^{x^{2}+x-1}+5^{2x}$ is an exponential polynomial.
Prove that there does not exist a non-zero exponential polynomial $f(x)$ and a non-constant polynomial $P(x)$ with integer coefficients such that
$$P(n)|f(n)$$
for all positive integers $n$.
1988 Romania Team Selection Test, 10
Let $ p > 2$ be a prime number. Find the least positive number $ a$ which can be represented as
\[ a \equal{} (X \minus{} 1)f(X) \plus{} (X^{p \minus{} 1} \plus{} X^{p \minus{} 2} \plus{} \cdots \plus{} X \plus{} 1)g(X),
\]
where $ f(X)$ and $ g(X)$ are integer polynomials.
[i]Mircea Becheanu[/i].
1996 Moldova Team Selection Test, 5
Find all polynomials $P(X)$ of fourth degree with real coefficients that verify the properties:
[b]a)[/b] $P(-x)=P(x), \forall x\in\mathbb{R};$
[b]b)[/b] $P(x)\geq0, \forall x\in\mathbb{R};$
[b]c)[/b] $P(0)=1;$
[b]d)[/b] $P(X)$ has exactly two local minimums $x_1$ and $x_2$ such that $|x_1-x_2|=2.$
1940 Putnam, B5
Suppose that the rational numbers $a, b$ and $c$ are the roots of the equation $x^3+ax^2 + bx + c = 0$. Find all such rational numbers $a, b$ and $c$. Justify your answer
2012 Vietnam National Olympiad, 2
Let $\langle a_n\rangle $ and $ \langle b_n\rangle$ be two arithmetic sequences of numbers, and let $m$ be an integer greater than $2.$ Define $P_k(x)=x^2+a_kx+b_k,\ k=1,2,\cdots, m.$ Prove that if the quadratic expressions $P_1(x), P_m(x)$ do not have any real roots, then all the remaining polynomials also don't have real roots.
2022 IMO Shortlist, A7
For a positive integer $n$ we denote by $s(n)$ the sum of the digits of $n$. Let $P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be a polynomial, where $n \geqslant 2$ and $a_i$ is a positive integer for all $0 \leqslant i \leqslant n-1$. Could it be the case that, for all positive integers $k$, $s(k)$ and $s(P(k))$ have the same parity?
2004 239 Open Mathematical Olympiad, 1
Given non-constant linear functions $p_1(x), p_2(x), \dots p_n(x)$. Prove that at least $n-2$ of polynomials $p_1p_2\dots p_{n-1}+p_n, p_1p_2\dots p_{n-2} p_n + p_{n-1},\dots p_2p_3\dots p_n+p_1$ have a real root.
2014 France Team Selection Test, 3
Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.
1984 IMO Longlists, 12
Let $n$ be a positive integer and $a_1, a_2, \dots , a_{2n}$ mutually distinct integers. Find all integers $x$ satisfying
\[(x - a_1) \cdot (x - a_2) \cdots (x - a_{2n}) = (-1)^n(n!)^2.\]
2006 District Olympiad, 1
Let $x>0$ be a real number and $A$ a square $2\times 2$ matrix with real entries such that $\det {(A^2+xI_2 )} = 0$.
Prove that $\det{ (A^2+A+xI_2) } = x$.
2015 Taiwan TST Round 2, 1
Let $f(x)=\sum_{i=0}^{n}a_ix^i$ and $g(x)=\sum_{i=0}^{n}b_ix^i$, where $a_n$,$b_n$ can be zero.
Called $f(x)\ge g(x)$ if exist $r$ such that $\forall i>r,a_i=b_i,a_r>b_r$ or $f(x)=g(x)$.
Prove that: if the leading coefficients of $f$ and $g$ are positive, then $f(f(x))+g(g(x))\ge f(g(x))+g(f(x))$
1985 Polish MO Finals, 5
$p(x,y)$ is a polynomial such that $p(cos t, sin t) = 0$ for all real $t$.
Show that there is a polynomial $q(x,y)$ such that $p(x,y) = (x^2 + y^2 - 1) q(x,y)$.
Kvant 2023, M2754
Given are reals $a, b$. Prove that at least one of the equations $x^4-2b^3x+a^4=0$ and $x^4-2a^3x+b^4=0$ has a real root.
Proposed by N. Agakhanov
2011 USA Team Selection Test, 5
Let $c_n$ be a sequence which is defined recursively as follows: $c_0 = 1$, $c_{2n+1} = c_n$ for $n \geq 0$, and $c_{2n} = c_n + c_{n-2^e}$ for $n > 0$ where $e$ is the maximal nonnegative integer such that $2^e$ divides $n$. Prove that
\[\sum_{i=0}^{2^n-1} c_i = \frac{1}{n+2} {2n+2 \choose n+1}.\]
2004 Serbia Team Selection Test, 3
Let $P(x)$ be a polynomial of degree $n$ whose roots are $i-1, i-2,\cdot\cdot\cdot, i-n$ (where $i^2=-1$), and let $R(x)$ and $S(x)$ be the polynomials with real coefficients such that $P(x)=R(x)+iS(x)$. Show that the polynomial $R$ has $n$ real roots. (R. Stanojevic)
2003 India IMO Training Camp, 6
A zig-zag in the plane consists of two parallel half-lines connected by a line segment. Find $z_n$, the maximum number of regions into which $n$ zig-zags can divide the plane. For example, $z_1=2,z_2=12$(see the diagram). Of these $z_n$ regions how many are bounded? [The zig-zags can be as narrow as you please.] Express your answers as polynomials in $n$ of degree not exceeding $2$.
[asy]
draw((30,0)--(-70,0), Arrow);
draw((30,0)--(-20,-40));
draw((-20,-40)--(80,-40), Arrow);
draw((0,-60)--(-40,20), dashed, Arrow);
draw((0,-60)--(0,15), dashed);
draw((0,15)--(40,-65),dashed, Arrow);
[/asy]
1998 All-Russian Olympiad, 1
The angle formed by the rays $y=x$ and $y=2x$ ($x \ge 0$) cuts off two arcs from a given parabola $y=x^2+px+q$. Prove that the projection of one arc onto the $x$-axis is shorter by $1$ than that of the second arc.
1992 IMO Longlists, 71
Let $P_1(x, y)$ and $P_2(x, y)$ be two relatively prime polynomials with complex coefficients. Let $Q(x, y)$ and $R(x, y)$ be polynomials with complex coefficients and each of degree not exceeding $d$. Prove that there exist two integers $A_1, A_2$ not simultaneously zero with $|A_i| \leq d + 1 \ (i = 1, 2)$ and such that the polynomial $A_1P_1(x, y) + A_2P_2(x, y)$ is coprime to $Q(x, y)$ and $R(x, y).$
2015 CCA Math Bonanza, I12
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]
2010 Stanford Mathematics Tournament, 8
Let $P(x)$ be a polynomial of degree $n$ such that $P(x)=3^k$ for $0\le k \le n$. Find $P(n+1)$