Found problems: 3597
2007 Harvard-MIT Mathematics Tournament, 2
Determine the real number $a$ having the property that $f(a)=a$ is a relative minimum of $f(x)=x^4-x^3-x^2+ax+1$.
2020 Serbian Mathematical Olympiad, Problem 1
Find all monic polynomials $P(x)$ such that the polynomial $P(x)^2-1$ is divisible by the polynomial $P(x+1)$.
Russian TST 2021, P3
Given an integer $n \geqslant 3$ the polynomial $f(x_1, \ldots, x_n)$ with integer coefficients is called [i]good[/i] if $f(0,\ldots, 0) = 0$ and \[f(x_1, \ldots, x_n)=f(x_{\pi_1}, \ldots, x_{\pi_n}),\]for any permutation of $\pi$ of the numbers $1,\ldots, n$. Denote by $\mathcal{J}$ the set of polynomials of the form \[p_1q_1+\cdots+p_mq_m,\]where $m$ is a positive integer and $q_1,\ldots , q_m$ are polynomials with integer coefficients, and $p_1,\ldots , p_m$ are good polynomials. Find the smallest natural number $D{}$ such that each monomial of degree $D{}$ lies in the set $\mathcal{J}$.
2007 All-Russian Olympiad Regional Round, 9.1
Pete chooses $ 1004$ monic quadratic polynomial $ f_{1},\cdots,f_{1004}$, such that each integer from $ 0$ to $ 2007$ is a root of at least one of them. Vasya considers all equations of the form $ f_{i}\equal{}f_{j}(i\not \equal{}j)$ and computes their roots; for each such root , Pete has to pay to Vasya $ 1$ ruble . Find the least possible value of Vasya's income.
2002 IMO Shortlist, 3
Let $P$ be a cubic polynomial given by $P(x)=ax^3+bx^2+cx+d$, where $a,b,c,d$ are integers and $a\ne0$. Suppose that $xP(x)=yP(y)$ for infinitely many pairs $x,y$ of integers with $x\ne y$. Prove that the equation $P(x)=0$ has an integer root.
2017 Taiwan TST Round 1, 5
Let $n$ be an odd number larger than 1, and $f(x)$ is a polynomial with degree $n$ such that $f(k)=2^k$ for $k=0,1,\cdots,n$. Prove that there is only finite integer $x$ such that $f(x)$ is the power of two.
2014 IMAR Test, 3
Let $f$ be a primitive polynomial with integral coefficients (their highest common factor is $1$ ) such that $f$ is irreducible in $\mathbb{Q}[X]$ , and $f(X^2)$ is reducible in $\mathbb{Q}[X]$ . Show that $f= \pm(u^2-Xv^2)$ for some polynomials $u$ and $v$ with integral coefficients.
1978 IMO Shortlist, 15
Let $p$ be a prime and $A = \{a_1, \ldots , a_{p-1} \}$ an arbitrary subset of the set of natural numbers such that none of its elements is divisible by $p$. Let us define a mapping $f$ from $\mathcal P(A)$ (the set of all subsets of $A$) to the set $P = \{0, 1, \ldots, p - 1\}$ in the following way:
$(i)$ if $B = \{a_{i_{1}}, \ldots , a_{i_{k}} \} \subset A$ and $\sum_{j=1}^k a_{i_{j}} \equiv n \pmod p$, then $f(B) = n,$
$(ii)$ $f(\emptyset) = 0$, $\emptyset$ being the empty set.
Prove that for each $n \in P$ there exists $B \subset A$ such that $f(B) = n.$
2007 China Team Selection Test, 3
Consider a $ 7\times 7$ numbers table $ a_{ij} \equal{} (i^2 \plus{} j)(i \plus{} j^2), 1\le i,j\le 7.$ When we add arbitrarily each term of an arithmetical progression consisting of $ 7$ integers to corresponding to term of certain row (or column) in turn, call it an operation. Determine whether such that each row of numbers table is an arithmetical progression, after a finite number of operations.
1961 AMC 12/AHSME, 22
If $3x^3-9x^2+kx-12$ is divisible by $x-3$, then it is also divisible by:
${{ \textbf{(A)}\ 3x^2-x+4 \qquad\textbf{(B)}\ 3x^2-4 \qquad\textbf{(C)}\ 3x^2+4 \qquad\textbf{(D)}\ 3x-4 }\qquad\textbf{(E)}\ 3x+4 } $
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 $P(x_0)<Q(x_0)$.
2017 IMO, 6
An ordered pair $(x, y)$ of integers is a primitive point if the greatest common divisor of $x$ and $y$ is $1$. Given a finite set $S$ of primitive points, prove that there exist a positive integer $n$ and integers $a_0, a_1, \ldots , a_n$ such that, for each $(x, y)$ in $S$, we have:
$$a_0x^n + a_1x^{n-1} y + a_2x^{n-2}y^2 + \cdots + a_{n-1}xy^{n-1} + a_ny^n = 1.$$
[i]Proposed by John Berman, United States[/i]
1987 Poland - Second Round, 4
Determine all pairs of real numbers $ a, b $ for which the polynomials $ x^4 + 2ax^2 + 4bx + a^2 $ and $ x^3 + ax - b $ have two different common real roots.
2016 Kosovo National Mathematical Olympiad, 2
Sum of all coefficients of polynomial $P(x)$ is equal with $2$ . Also the sum of coefficients which are at odd exponential in $x^k$ are equal to sum of coefficients which are at even exponential in $x^k$ . Find the residue of polynomial $P(x)$ when it is divide by $x^2-1$ .
1956 Putnam, A4
Suppose that the $n$ times differentiable real function $f(x)$ has at least $n+1$ distinct zeros in the closed interval $[a,b]$ and that the polynomial $P(z)=z^n +c_{n-1}z^{n-1}+\ldots+c_1 x +c_0$ has only real zeroes. Show that
$f^{(n)}(x)+ c_{n-1} f^{(n-1)}(x) +\ldots +c_1 f'(x)+ c_0 f(x)$ has at least one zero in $[a,b]$, where $f^{(n)}$ denotes the $n$-th derivative of $f.$
2017 IMO Shortlist, N7
An ordered pair $(x, y)$ of integers is a primitive point if the greatest common divisor of $x$ and $y$ is $1$. Given a finite set $S$ of primitive points, prove that there exist a positive integer $n$ and integers $a_0, a_1, \ldots , a_n$ such that, for each $(x, y)$ in $S$, we have:
$$a_0x^n + a_1x^{n-1} y + a_2x^{n-2}y^2 + \cdots + a_{n-1}xy^{n-1} + a_ny^n = 1.$$
[i]Proposed by John Berman, United States[/i]
1984 All Soviet Union Mathematical Olympiad, 383
The teacher wrote on a blackboard: $$x^2 + 10x + 20$$ Then all the pupils in the class came up in turn and either decreased or increased by $1$ either the free coefficient or the coefficient at $x$, but not both. Finally they have obtained: $$x^2 + 20x + 10$$ Is it true that some time during the process there was written the square polynomial with the integer roots?
2001 Moldova National Olympiad, Problem 5
Let $a,b,c,d$ be real numbers. Prove that the set $M=\left\{ax^3+bx^2+cx+d|x\in\mathbb R\right\}$ contains no irrational numbers if and only if $a=b=c=0$ and $d$ is rational.
1980 IMO, 1
Let $p(x)$ be a polynomial with integer coefficients such that $p(0)=p(1)=1$. We define the sequence $a_0, a_1, a_2, \ldots, a_n, \ldots$ that starts with an arbitrary nonzero integer $a_0$ and satisfies $a_{n+1}=p(a_n)$ for all $n \in \mathbb N\cup \{0\}$. Prove that $\gcd(a_i,a_j)=1$ for all $i,j \in \mathbb N \cup \{0\}$.
2013 Balkan MO Shortlist, A4
Find all positive integers $n$ such that there exist non-constant polynomials with integer coefficients $f_1(x),...,f_n(x)$ (not necessarily distinct) and $g(x)$ such that $$1 + \prod_{k=1}^{n}\left(f^2_k(x)-1\right)=(x^2+2013)^2g^2(x)$$
2018 Peru IMO TST, 2
Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that
$$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$
If $M>1$, prove that the polynomial
$$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$
has no positive roots.
1986 AIME Problems, 1
What is the sum of the solutions to the equation $\sqrt[4]x =\displaystyle \frac{12}{7-\sqrt[4]x}$?
1955 Moscow Mathematical Olympiad, 316
Prove that if $\frac{p}{q}$ is an irreducible rational number that serves as a root of the polynomial $f(x) = a_0x^n + a_1x^{n-1} + ... + a_n$ with integer coefficients, then $p - kq$ is a divisor of $f(k)$ for any integer $k$.
1965 Swedish Mathematical Competition, 5
Let $S$ be the set of all real polynomials $f(x) = ax^3 + bx^2 + cx + d$ such that $|f(x)| \le 1$ for all $ -1 \le x \le 1$. Show that the set of possible $|a|$ for $f$ in $S$ is bounded above and find the smallest upper bound.
1997 Pre-Preparation Course Examination, 1
Let $n$ be a positive integer. Prove that there exist polynomials$f(x)$and $g(x$) with integer coefficients such that
\[f(x)\left(x + 1 \right)^{2^n}+ g(x) \left(x^{2^n}+ 1 \right) = 2.\]