Found problems: 3597
2023 Olimphíada, 4
We say that a prime $p$ is $\textit{philé}$ if there is a polynomial $P$ of non-negative integer coefficients smaller than $p$ and with degree $3$, that is, $P(x) = ax^3 + bx^2 + cx + d$ where $a, b, c, d < p$, such that $$\{P(n) | 1 \leq n \leq p\}$$ is a complete residue system modulo $p$. Find all $\textit{philé}$ primes.
Note: A set $A$ is a complete residue system modulo $p$ if for every integer $k$, with $0 \leq k \leq p - 1$, there exists an element $a \in A$ such that $$p | a-k.$$
Russian TST 2019, P1
Let $t\in (1,2)$. Show that there exists a polynomial $P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ with the coefficients in $\{1,-1\}$ such that $\left|P(t)-2019\right| \leqslant 1.$
[i]Proposed by N. Safaei (Iran)[/i]
2025 India STEMS Category C, 5
Let $P \in \mathbb{R}[x]$. Suppose that the multiset of real roots (where roots are counted with multiplicity) of $P(x)-x$ and $P^3(x)-x$ are distinct. Prove that for all $n\in \mathbb{N}$, $P^n(x)-x$ has at least $\sigma(n)-2$ distinct real roots.
(Here $P^n(x):=P(P^{n-1}(x))$ with $P^1(x) = P(x)$, and $\sigma(n)$ is the sum of all positive divisors of $n$).
[i]Proposed by Malay Mahajan[/i]
1989 Federal Competition For Advanced Students, P2, 6
Determine all functions $ f: \mathbb{N}_0 \rightarrow \mathbb{N}_0$ such that $ f(f(n))\plus{}f(n)\equal{}2n\plus{}6$ for all $ n \in \mathbb{N}_0$.
2014 Putnam, 6
Let $n$ be a positive integer. What is the largest $k$ for which there exist $n\times n$ matrices $M_1,\dots,M_k$ and $N_1,\dots,N_k$ with real entries such that for all $i$ and $j,$ the matrix product $M_iN_j$ has a zero entry somewhere on its diagonal if and only if $i\ne j?$
1994 Moldova Team Selection Test, 1
Let $P(X)=X^n+a_1X^{n-1}+\ldots+a_n$ be a plynomial with real roots $x_1. x_2,\ldots,x_n$. Denote $E_k=x_1^k+x_2^k+\ldots+x_n^k, \forall k\in\mathbb{N}$. There exists an $m\in\mathbb{N}$ such that $E_m=E_{m+1}=E_{m+2}=1$. Find $\max\{P(-2),P(2)\}$.
2005 ISI B.Stat Entrance Exam, 2
Let
\[f(x)=\int_0^1 |t-x|t \, dt\]
for all real $x$. Sketch the graph of $f(x)$. What is the minimum value of $f(x)$?
2021 South Africa National Olympiad, 5
Determine all polynomials $a(x)$, $b(x)$, $c(x)$, $d(x)$ with real coefficients satisfying the simultaneous equations
\begin{align*}
b(x) c(x) + a(x) d(x) & = 0 \\
a(x) c(x) + (1 - x^2) b(x) d(x) & = x + 1.
\end{align*}
2008 All-Russian Olympiad, 1
Numbers $ a,b,c$ are such that the equation $ x^3 \plus{} ax^2 \plus{} bx \plus{} c$ has three real roots.Prove that if $ \minus{} 2\leq a \plus{} b \plus{} c\leq 0$,then at least one of these roots belongs to the segment $ [0,2]$
2013 IMO Shortlist, A6
Let $m \neq 0 $ be an integer. Find all polynomials $P(x) $ with real coefficients such that
\[ (x^3 - mx^2 +1 ) P(x+1) + (x^3+mx^2+1) P(x-1) =2(x^3 - mx +1 ) P(x) \]
for all real number $x$.
2020 Tournament Of Towns, 4
We say that a nonconstant polynomial $p(x)$ with real coefficients is split into two squares if it is represented as $a(x) +b(x)$ where $a(x)$ and $b(x)$ are squares of polynomials with real coefficients. Is there such a polynomial $p(x)$ that it may be split into two squares:
a) in exactly one way;
b) in exactly two ways?
Note: two splittings that differ only in the order of summands are considered to be the same.
Sergey Markelov
1963 Miklós Schweitzer, 4
Call a polynomial positive reducible if it can be written as a product of two nonconstant polynomials with positive real coefficients. Let $ f(x)$ be a polynomial with $ f(0)\not\equal{}0$ such that $ f(x^n)$ is positive reducible for some natural number $ n$. Prove that $ f(x)$ itself is positive reducible. [L. Redei]
1976 IMO Longlists, 43
Prove that if for a polynomial $P(x, y)$, we have
\[P(x - 1, y - 2x + 1) = P(x, y),\]
then there exists a polynomial $\Phi(x)$ with $P(x, y) = \Phi(y - x^2).$
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 $m,n$.
2010 AMC 12/AHSME, 24
The set of real numbers $ x$ for which
\[ \frac{1}{x\minus{}2009}\plus{}\frac{1}{x\minus{}2010}\plus{}\frac{1}{x\minus{}2011}\ge 1\]
is the union of intervals of the form $ a<x\le b$. What is the sum of the lengths of these intervals?
$ \textbf{(A)}\ \frac{1003}{335} \qquad \textbf{(B)}\ \frac{1004}{335} \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ \frac{403}{134} \qquad \textbf{(E)}\ \frac{202}{67}$
2024/2025 TOURNAMENT OF TOWNS, P2
Two polynomials with real coefficients have the leading coefficients equal to 1 . Each polynomial has an odd degree that is equal to the number of its distinct real roots. The product of the values of the first polynomial at the roots of the second polynomial is equal to 2024. Find the product of the values of the second polynomial at the roots of the first one.
Sergey Yanzhinov
2021 IMO Shortlist, N8
Find all positive integers $n$ for which there exists a polynomial $P(x) \in \mathbb{Z}[x]$ such that for every positive integer $m\geq 1$, the numbers $P^m(1), \ldots, P^m(n)$ leave exactly $\lceil n/2^m\rceil$ distinct remainders when divided by $n$. (Here, $P^m$ means $P$ applied $m$ times.)
[i]Proposed by Carl Schildkraut, USA[/i]
2012 IMO Shortlist, A7
We say that a function $f:\mathbb{R}^k \rightarrow \mathbb{R}$ is a metapolynomial if, for some positive integers $m$ and $n$, it can be represented in the form
\[f(x_1,\cdots , x_k )=\max_{i=1,\cdots , m} \min_{j=1,\cdots , n}P_{i,j}(x_1,\cdots , x_k),\]
where $P_{i,j}$ are multivariate polynomials. Prove that the product of two metapolynomials is also a metapolynomial.
2003 Austrian-Polish Competition, 1
Find all real polynomials $p(x) $ such that $p(x-1)p(x+1)= p(x^2-1)$.
2019 CMIMC, 6
Let $a, b$ and $c$ be the distinct solutions to the equation $x^3-2x^2+3x-4=0$. Find the value of
$$\frac{1}{a(b^2+c^2-a^2)}+\frac{1}{b(c^2+a^2-b^2)}+\frac{1}{c(a^2+b^2-c^2)}.$$
2010 India IMO Training Camp, 2
Two polynomials $P(x)=x^4+ax^3+bx^2+cx+d$ and $Q(x)=x^2+px+q$ have real coefficients, and $I$ is an interval on the real line of length greater than $2$. Suppose $P(x)$ and $Q(x)$ take negative values on $I$, and they take non-negative values outside $I$. Prove that there exists a real number $x_0$ such that $P(x_0)<Q(x_0)$.
2023 IMAR Test, P4
Let $n{}$ be a non-negative integer and consider the standard power expansion of the following polynomial \[\sum_{k=0}^n\binom{n}{k}^2(X+1)^{2k}(X-1)^{2(n-k)}=\sum_{k=0}^{2n}a_kX^k.\]The coefficients $a_{2k+1}$ all vanish since the polynomial is invariant under the change $X\mapsto -X.$ Prove that the coefficients $a_{2k}$ are all positive.
2016 Ukraine Team Selection Test, 10
Let $a_1,\ldots, a_n$ be real numbers. Define polynomials $f,g$ by $$f(x)=\sum_{k=1}^n a_kx^k,\ g(x)=\sum_{k=1}^n \frac{a_k}{2^k-1}x^k.$$ Assume that $g(2016)=0$. Prove that $f(x)$ has a root in $(0;2016)$.
2016 IMO Shortlist, N1
For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$
[i]Proposed by Warut Suksompong, Thailand[/i]
VMEO III 2006, 11.1
Given a polynomial $P(x)=x^4+x^3+3x^2-6x+1$. Calculate $P(\alpha^2+\alpha+1)$ where \[ \alpha=\sqrt[3]{\frac{1+\sqrt{5}}{2}}+\sqrt[3]{\frac{1-\sqrt{5}}{2}} \]