Found problems: 3597
2014 USA Team Selection Test, 3
For a prime $p$, a subset $S$ of residues modulo $p$ is called a [i]sum-free multiplicative subgroup[/i] of $\mathbb F_p$ if
$\bullet$ there is a nonzero residue $\alpha$ modulo $p$ such that $S = \left\{ 1, \alpha^1, \alpha^2, \dots \right\}$ (all considered mod $p$), and
$\bullet$ there are no $a,b,c \in S$ (not necessarily distinct) such that $a+b \equiv c \pmod p$.
Prove that for every integer $N$, there is a prime $p$ and a sum-free multiplicative subgroup $S$ of $\mathbb F_p$ such that $\left\lvert S \right\rvert \ge N$.
[i]Proposed by Noga Alon and Jean Bourgain[/i]
1990 Bulgaria National Olympiad, Problem 3
Let $n=p_1p_2\cdots p_s$, where $p_1,\ldots,p_s$ are distinct odd prime numbers.
(a) Prove that the expression
$$F_n(x)=\prod\left(x^{\frac n{p_{i_1}\cdots p_{i_k}}}-1\right)^{(-1)^k},$$where the product goes over all subsets $\{p_{i_1},\ldots,p_{i_k}\}$ or $\{p_1,\ldots,p_s\}$ (including itself and the empty set), can be written as a polynomial in $x$ with integer coefficients.
(b) Prove that if $p$ is a prime divisor of $F_n(2)$, then either $p\mid n$ or $n\mid p-1$.
1981 IMO Shortlist, 13
Let $P$ be a polynomial of degree $n$ satisfying
\[P(k) = \binom{n+1}{k}^{-1} \qquad \text{ for } k = 0, 1, . . ., n.\]
Determine $P(n + 1).$
2023 Tuymaada Olympiad, 5
A small ship sails on an infinite coordinate sea. At the moment $t$ the ship is at the point with coordinates $(f(t), g(t))$, where $f$ and $g$ are two polynomials of third degree. Yesterday at $14:00$ the ship was at the same point as at $13:00$, and at $20:00$, it was at the same point as at $19:00$. Prove that the ship sails along a straight line.
PEN F Problems, 8
Find all polynomials $W$ with real coefficients possessing the following property: if $x+y$ is a rational number, then $W(x)+W(y)$ is rational.
2007 Princeton University Math Competition, 10
Find the real root of $x^5+5x^3+5x-1$. Hint: Let $x = u+k/u$.
2009 Indonesia MO, 1
Find all positive integers $ n\in\{1,2,3,\ldots,2009\}$ such that
\[ 4n^6 \plus{} n^3 \plus{} 5\]
is divisible by $ 7$.
1990 Vietnam National Olympiad, 2
Suppose $ f(x)\equal{}a_0x^n\plus{}a_1x^{n\minus{}1}\plus{}\ldots\plus{}a_{n\minus{}1}x\plus{}a_n$ ($ a_0\neq 0$) is a polynomial with real coefficients satisfying $ f(x)f(2x^2) \equal{} f(2x^3 \plus{} x)$ for all $ x \in\mathbb{R}$. Prove that $ f(x)$ has no real roots.
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$.
2010 Brazil National Olympiad, 2
Let $P(x)$ be a polynomial with real coefficients. Prove that there exist positive integers $n$ and $k$ such that $k$ has $n$ digits and more than $P(n)$ positive divisors.
1980 IMO Longlists, 12
Find all pairs of solutions $(x,y)$:
\[ x^3 + x^2y + xy^2 + y^3 = 8(x^2 + xy + y^2 + 1). \]
2013 IFYM, Sozopol, 4
Find all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$.
2016 Miklós Schweitzer, 3
Prove that for any polynomial $P$ with real coefficients, and for any positive integer $n$, there exists a polynomial $Q$ with real coefficients such that $P(x)^2 +Q(x)^2$ is divisible by $(1+x^2)^n$.
2019 Tournament Of Towns, 1
The polynomial P(x,y) is such that for every integer n >= 0 each of the polynomials P(x,n) and P(n,y) either is a constant zero or has a degree not greater than n. Is it possible that P(x,x) has an odd degree?
2020 Iran Team Selection Test, 3
We call a number $n$ [i]interesting [/i]if for each permutation $\sigma$ of $1,2,\ldots,n$ there exist polynomials $P_1,P_2,\ldots ,P_n$ and $\epsilon > 0$ such that:
$i)$ $P_1(0)=P_2(0)=\ldots =P_n(0)$
$ii)$ $P_1(x)>P_2(x)>\ldots >P_n(x)$ for $-\epsilon<x<0$
$iii)$ $P_{\sigma (1)} (x)>P_{\sigma (2)}(x)> \ldots >P_{\sigma (n)} (x) $ for $0<x<\epsilon$
Find all [i]interesting [/i]$n$.
[i]Proposed by Mojtaba Zare Bidaki[/i]
2023 IFYM, Sozopol, 5
Is it true that for any polynomial $P(x)$ with real coefficients of degree $2023$, there exists a natural number $n$ such that the equation $P(x) = n^{-100}$ has no rational root?
2000 Turkey Team Selection Test, 1
$(a)$ Prove that for every positive integer $n$, the number of ordered pairs $(x, y)$ of integers satisfying $x^2-xy+y^2 = n$ is divisible by $3.$
$(b)$ Find all ordered pairs of integers satisfying $x^2-xy+y^2=727.$
2023 Ukraine National Mathematical Olympiad, 11.5
Let's call a polynomial [i]mixed[/i] if it has both positive and negative coefficients ($0$ isn't considered positive or negative). Is the product of two mixed polynomials always mixed?
[i]Proposed by Vadym Koval[/i]
2012 IberoAmerican, 3
Let $n$ to be a positive integer. Given a set $\{ a_1, a_2, \ldots, a_n \} $ of integers, where $a_i \in \{ 0, 1, 2, 3, \ldots, 2^n -1 \},$ $\forall i$, we associate to each of its subsets the sum of its elements; particularly, the empty subset has sum of its elements equal to $0$. If all of these sums have different remainders when divided by $2^n$, we say that $\{ a_1, a_2, \ldots, a_n \} $ is [i]$n$-complete[/i].
For each $n$, find the number of [i]$n$-complete[/i] sets.
2002 Tuymaada Olympiad, 3
Is there a quadratic trinomial with integer coefficients, such that all values which are natural to be natural powers of two?
2013 Saudi Arabia BMO TST, 5
Let $k$ be a real number such that the product of real roots of the equation $$X^4 + 2X^3 + (2 + 2k)X^2 + (1 + 2k)X + 2k = 0$$ is $-2013$. Find the sum of the squares of these real roots.
2007 USAMO, 5
Prove that for every nonnegative integer $n$, the number $7^{7^{n}}+1$ is the product of at least $2n+3$ (not necessarily distinct) primes.
2023 BMT, 26
For positive integers $i$ and $N$, let $k_{N,i}$ be the $i$th smallest positive integer such that the polynomial $\frac{x^2}{2023} + \frac{N_x}{7} - k_{N,i}$ has integer roots. Compute the minimum positive integer $N$ satisfying the condition $\frac{k_{N,2023}}{k_{N,1000}}< 3$.
Submit your answer as a positive integer $E$. If the correct answer is $A$, your score for this question will be
$\max \left( 0, 25 \min \left( \frac{A}{E} , \frac{E}{A}\right)^{\frac32}\right)$, rounded to the nearest integer.
2006 Balkan MO, 2
Let $ABC$ be a triangle and $m$ a line which intersects the sides $AB$ and $AC$ at interior points $D$ and $F$, respectively, and intersects the line $BC$ at a point $E$ such that $C$ lies between $B$ and $E$. The parallel lines from the points $A$, $B$, $C$ to the line $m$ intersect the circumcircle of triangle $ABC$ at the points $A_1$, $B_1$ and $C_1$, respectively (apart from $A$, $B$, $C$). Prove that the lines $A_1E$ , $B_1F$ and $C_1D$ pass through the same point.
[i]Greece[/i]
2000 Spain Mathematical Olympiad, 1
Consider the polynomials
\[P(x) = x^4 + ax^3 + bx^2 + cx + 1 \quad \text{and} \quad Q(x) = x^4 + cx^3 + bx^2 + ax + 1.\]
Find the conditions on the parameters $a, b, $c with $a\neq c$ for which $P(x)$ and $Q(x)$ have two common roots and, in such cases, solve the equations $P(x) = 0$ and $Q(x) = 0.$