Found problems: 3597
2025 Vietnam Team Selection Test, 6
For each prime $p$ of the form $4k+3$ with $k \in \mathbb{Z}^+$, consider the polynomial $$Q(x)=px^{2p} - x^{2p-1} + p^2x^{\frac{3p+1}{2}} - px^{p+1} +2(p^2+1)x^p -px^{p-1}+ p^2 x^{\frac{p-1}{2}} -x + p.$$ Determine all ordered pairs of polynomials $f, g$ with integer coefficients such that $Q(x)=f(x)g(x)$.
2014 Contests, 2
find all polynomials with integer coefficients that $P(\mathbb{Z})= ${$p(a):a\in \mathbb{Z}$} has a Geometric progression.
2007 Romania National Olympiad, 2
Let $f: \mathbb{R}\to\mathbb{R}$ be a continuous function, and $a<b$ be two points in the image of $f$ (that is, there exists $x,y$ such that $f(x)=a$ and $f(y)=b$).
Show that there is an interval $I$ such that $f(I)=[a,b]$.
1969 IMO Shortlist, 54
$(POL 3)$ Given a polynomial $f(x)$ with integer coefficients whose value is divisible by $3$ for three integers $k, k + 1,$ and $k + 2$. Prove that $f(m)$ is divisible by $3$ for all integers $m.$
2025 Ukraine National Mathematical Olympiad, 11.5
Initially, two constant polynomials are written on the board: \(0\) and \(1\). At each step, it is allowed to add \(1\) to one of the polynomials and to multiply another one by the polynomial \(45x + 2025\). Can the polynomials become equal at some point?
[i]Proposed by Oleksii Masalitin[/i]
2006 Purple Comet Problems, 11
Consider the polynomials \begin{align*}P(x) &= (x + \sqrt{2})(x^2 - 2x + 2)\\Q(x) &= (x - \sqrt{2})(x^2 + 2x + 2)\\R(x) &= (x^2 + 2)(x^8 + 16).\end{align*} Find the coefficient of $x^4$ in $P(x)\cdot Q(x)\cdot R(x)$.
2000 Vietnam National Olympiad, 1
For every integer $ n \ge 3$ and any given angle $ \alpha$ with $ 0 < \alpha < \pi$, let $ P_n(x) \equal{} x^n \sin\alpha \minus{} x \sin n\alpha \plus{} \sin(n \minus{} 1)\alpha$.
(a) Prove that there is a unique polynomial of the form $ f(x) \equal{} x^2 \plus{} ax \plus{} b$ which divides $ P_n(x)$ for every $ n \ge 3$.
(b) Prove that there is no polynomial $ g(x) \equal{} x \plus{} c$ which divides $ P_n(x)$ for every $ n \ge 3$.
1962 All Russian Mathematical Olympiad, 016
Prove that there are no integers $a,b,c,d$ such that the polynomial $ax^3+bx^2+cx+d$ equals $1$ at $x=19$, and equals $2$ at $x=62$.
2003 Tournament Of Towns, 2
$P(x)$ is a polynomial with real coefficients such that $P(a_1) = 0, P(a_{i+1}) = a_i$ ($i = 1, 2,\ldots$) where $\{a_i\}_{i=1,2,\ldots}$ is an infinite sequence of distinct natural numbers. Determine the possible values of degree of $P(x)$.
1983 Putnam, B6
Let $ k$ be a positive integer, let $ m\equal{}2^k\plus{}1$, and let $ r\neq 1$ be a complex root of $ z^m\minus{}1\equal{}0$. Prove that there exist polynomials $ P(z)$ and $ Q(z)$ with integer coefficients such that $ (P(r))^2\plus{}(Q(r))^2\equal{}\minus{}1$.
1970 Miklós Schweitzer, 8
Let $ \pi_n(x)$ be a polynomial of degree not exceeding $ n$ with real coefficients such that \[ |\pi_n(x)| \leq \sqrt{1\minus{}x^2}
\;\textrm{for}\ \;\minus{}1\leq x \leq 1 \ .\] Then \[ |\pi'_n(x)| \leq 2(n\minus{}1).\]
[i]P. Turan[/i]
1969 IMO Shortlist, 28
$(GBR 5)$ Let us define $u_0 = 0, u_1 = 1$ and for $n\ge 0, u_{n+2} = au_{n+1}+bu_n, a$ and $b$ being positive integers. Express $u_n$ as a polynomial in $a$ and $b.$ Prove the result. Given that $b$ is prime, prove that $b$ divides $a(u_b -1).$
2013 Poland - Second Round, 5
Let $W(x)$ be a polynomial of integer coefficients such that for any pair of different rational number $r_1$, $r_2$ dependence $W(r_1) \neq W(r_2)$ is true. Decide, whether the assuptions imply that for any pair of different real numbers $t_1$, $t_2$ dependence $W(t_1) \neq W(t_2)$ is true.
Russian TST 2019, P3
Let $m,n\geq 2$ be integers. Let $f(x_1,\dots, x_n)$ be a polynomial with real coefficients such that $$f(x_1,\dots, x_n)=\left\lfloor \frac{x_1+\dots + x_n}{m} \right\rfloor\text{ for every } x_1,\dots, x_n\in \{0,1,\dots, m-1\}.$$ Prove that the total degree of $f$ is at least $n$.
2009 Today's Calculation Of Integral, 426
Consider the polynomial $ f(x) \equal{} ax^2 \plus{} bx \plus{} c$, with degree less than or equal to 2.
When $ f$ varies with subject to the constrain $ f(0) \equal{} 0,\ f(2) \equal{} 2$, find the minimum value of $ S\equal{}\int_0^2 |f'(x)|\ dx$.
2009 Math Prize For Girls Problems, 20
Let $ y_0$ be chosen randomly from $ \{0, 50\}$, let $ y_1$ be chosen randomly from $ \{40, 60, 80\}$, let $ y_2$ be chosen randomly from $ \{10, 40, 70, 80\}$, and let $ y_3$ be chosen randomly from $ \{10, 30, 40, 70, 90\}$. (In each choice, the possible outcomes are equally likely to occur.) Let $ P$ be the unique polynomial of degree less than or equal to $ 3$ such that $ P(0) \equal{} y_0$, $ P(1) \equal{} y_1$, $ P(2) \equal{} y_2$, and $ P(3) \equal{} y_3$. What is the expected value of $ P(4)$?
2005 Greece National Olympiad, 1
Find the polynomial $P(x)$ with real coefficients such that $P(2)=12$ and $P(x^2)=x^2(x^2+1)P(x)$ for each $x\in\mathbb{R}$.
2007 All-Russian Olympiad, 5
Two numbers are written on each vertex of a convex $100$-gon. Prove that it is possible to remove a number from each vertex so that the remaining numbers on any two adjacent vertices are different.
[i]F. Petrov [/i]
2010 IMC, 5
Suppose that for a function $f: \mathbb{R}\to \mathbb{R}$ and real numbers $a<b$ one has $f(x)=0$ for all $x\in (a,b).$ Prove that $f(x)=0$ for all $x\in \mathbb{R}$ if
\[\sum^{p-1}_{k=0}f\left(y+\frac{k}{p}\right)=0\]
for every prime number $p$ and every real number $y.$
2022 Indonesia TST, N
Let $n$ be a natural number, with the prime factorisation
\[ n = p_1^{e_1} p_2^{e_2} \cdots p_r^{e_r} \] where $p_1, \ldots, p_r$ are distinct primes, and $e_i$ is a natural number. Define
\[ rad(n) = p_1p_2 \cdots p_r \] to be the product of all distinct prime factors of $n$. Determine all polynomials $P(x)$ with rational coefficients such that there exists infinitely many naturals $n$ satisfying $P(n) = rad(n)$.
2013 South africa National Olympiad, 4
Determine all pairs of polynomials $f$ and $g$ with real coefficients such that \[ x^2 \cdot g(x) = f(g(x)). \]
IV Soros Olympiad 1997 - 98 (Russia), 10.8
Let $a$ be the root of the equation $x^3-x-1=0$. Find an equation of the third degree with integer coefficients whose root is $a^3$.
2015 JBMO Shortlist, A2
If $x^3-3\sqrt3 x^2 +9x - 3\sqrt3 -64=0$ find the value of $x^6-8x^5+13x^4-5x^3+49x^2-137x+2015$ .
2009 CHKMO, 1
Let $ f(x) \equal{} c_m x^m \plus{} c_{m\minus{}1} x^{m\minus{}1} \plus{}...\plus{} c_1 x \plus{} c_0$, where each $ c_i$ is a non-zero integer. Define a sequence $ \{ a_n \}$ by $ a_1 \equal{} 0$ and $ a_{n\plus{}1} \equal{} f(a_n)$ for all positive integers $ n$.
(a) Let $ i$ and $ j$ be positive integers with $ i<j$. Show that $ a_{j\plus{}1} \minus{} a_j$ is a multiple of $ a_{i\plus{}1} \minus{} a_i$.
(b) Show that $ a_{2008} \neq 0$
2016 CCA Math Bonanza, T8
As $a$, $b$ and $c$ range over [i]all[/i] real numbers, let $m$ be the smallest possible value of $$2\left(a+b+c\right)^2+\left(ab-4\right)^2+\left(bc-4\right)^2+\left(ca-4\right)^2$$ and $n$ be the number of ordered triplets $\left(a,b,c\right)$ such that the above quantity is minimized. Compute $m+n$.
[i]2016 CCA Math Bonanza Team #8[/i]