Found problems: 3597
2011 IMO Shortlist, 2
Let $A_1A_2A_3A_4$ be a non-cyclic quadrilateral. Let $O_1$ and $r_1$ be the circumcentre and the circumradius of the triangle $A_2A_3A_4$. Define $O_2,O_3,O_4$ and $r_2,r_3,r_4$ in a similar way. Prove that
\[\frac{1}{O_1A_1^2-r_1^2}+\frac{1}{O_2A_2^2-r_2^2}+\frac{1}{O_3A_3^2-r_3^2}+\frac{1}{O_4A_4^2-r_4^2}=0.\]
[i]Proposed by Alexey Gladkich, Israel[/i]
2008 District Olympiad, 4
Let be a finite field $ K. $ Say that two polynoms $ f,g $ from $ K[X] $ are [i]neighbours,[/i] if the have the same degree and they differ by exactly one coefficient.
[b]a)[/b] Show that all the neighbours of $ 1+X^2 $ from $ \mathbb{Z}_3[X] $ are reducible in $ \mathbb{Z}_3[X] . $
[b]b)[/b] If $ |K|\ge 4, $ show that any polynomial of degree $ |K|-1 $ from $ K[X] $ has a neighbour from $ K[X] $ that is reducible in $ K[X] , $ and also has a neighbour that doesn´t have any root in $ K. $
2009 China Western Mathematical Olympiad, 1
Let $M$ be the set of the real numbers except for finitely many elements. Prove that for every positive integer $n$ there exists a polynomial $f(x)$ with $\deg f = n$, such that all the coefficients and the $n$ real roots of $f$ are all in $M$.
PEN E Problems, 11
In 1772 Euler discovered the curious fact that $n^2 +n+41$ is prime when $n$ is any of $0,1,2, \cdots, 39$. Show that there exist $40$ consecutive integer values of $n$ for which this polynomial is not prime.
2010 IFYM, Sozopol, 3
Let $n\ge 2$ be an even integer and $a,b$ real numbers such that $b^n=3a+1$. Show that the polynomial $P(X)=(X^2+X+1)^n-X^n-a$ is divisible by $Q(X)=X^3+X^2+X+b$ if and only if $b=1$.
2022 International Zhautykov Olympiad, 5
A polynomial $f(x)$ with real coefficients of degree greater than $1$ is given. Prove that there are infinitely many positive integers which cannot be represented in the form \[f(n+1)+f(n+2)+\cdots+f(n+k)\]
where $n$ and $k$ are positive integers.
1980 Miklós Schweitzer, 7
Let $ n \geq 2$ be a natural number and $ p(x)$ a real polynomial of degree at most $ n$ for which \[ \max _{ \minus{}1 \leq x \leq 1} |p(x)| \leq 1, \; p(\minus{}1)\equal{}p(1)\equal{}0 \ .\] Prove that then \[ |p'(x)| \leq \frac{n \cos \frac{\pi}{2n}}{\sqrt{1\minus{}x^2 \cos^2 \frac{\pi}{2n}}} \;\;\;\;\; \left( \minus{}\frac{1}{\cos \frac{\pi}{2n}} < x < \frac{1}{\cos \frac{\pi}{2n}} \\\\\ \right).\]
[i]J. Szabados[/i]
2012 USAMTS Problems, 5
An ordered quadruple $(y_1,y_2,y_3,y_4)$ is $\textbf{quadratic}$ if there exist real numbers $a$, $b$, and $c$ such that \[y_n=an^2+bn+c\] for $n=1,2,3,4$.
Prove that if $16$ numbers are placed in a $4\times 4$ grid such that all four rows are quadratic and the first three columns are also quadratic then the fourth column must also be quadratic.
[i](We say that a row is quadratic if its entries, in order, are quadratic. We say the same for a column.)[/i]
[asy]
size(100);
defaultpen(linewidth(0.8));
for(int i=0;i<=4;i=i+1)
draw((i,0)--(i,4));
for(int i=0;i<=4;i=i+1)
draw((0,i)--(4,i));
[/asy]
2020-2021 Winter SDPC, #4
Find all polynomials $P(x)$ with integer coefficients such that for all positive integers $n$, we have that $P(n)$ is not zero and $\frac{P(\overline{nn})}{P(n)}$ is an integer, where $\overline{nn}$ is the integer obtained upon concatenating $n$ with itself.
1998 IMO, 4
Determine all pairs $(x,y)$ of positive integers such that $x^{2}y+x+y$ is divisible by $xy^{2}+y+7$.
2020 Tuymaada Olympiad, 2
All non-zero coefficients of the polynomial $f(x)$ equal $1$, while the sum of the coefficients is $20$. Is it possible that thirteen coefficients of $f^2(x)$ equal $9$?
[i](S. Ivanov, K. Kokhas)[/i]
1953 Polish MO Finals, 1
Test whether equation $$\frac{1}{x - a} + \frac{1}{x - b} + \frac{1}{x - c} = 0,$$ where $ a $, $ b $, $ c $ denote the given real numbers, has real roots.
2024 Tuymaada Olympiad, 7
Given are two polynomial $f$ and $g$ of degree $100$ with real coefficients. For each positive integer $n$ there is an integer $k$ such that
\[\frac{f(k)}{g(k)}=\frac{n+1}{n}.\]
Prove that $f$ and $g$ have a common non-constant factor.
2017 Harvard-MIT Mathematics Tournament, 6
A polynomial $P$ of degree $2015$ satisfies the equation $P(n)=\frac{1}{n^2}$ for $n=1, 2, \dots, 2016$. Find $\lfloor 2017P(2017)\rfloor$.
2023 SAFEST Olympiad, 6
Find all polynomials $P(x)$ with integer coefficients, such that for all positive integers $m, n$, $$m+n \mid P^{(m)}(n)-P^{(n)}(m).$$
[i]Proposed by Navid Safaei, Iran[/i]
2013 Princeton University Math Competition, 16
Is $\cos 1^\circ$ rational? Prove.
2021 Stars of Mathematics, 2
Let $n{}$ be a positive integer. Show that there exists a polynomial $f{}$ of degree $n{}$ with integral coefficients such that \[f^2=(x^2-1)g^2+1,\] where $g{}$ is a polynomial with integral coefficients.
2016 Kyrgyzstan National Olympiad, 5
Given two monic polynomials $P(x)$ and $Q(x)$ with degrees 2016.
$P(x)=Q(x)$ has no real root. [b]Prove that P(x)=Q(x+1) has at least one real root.[/b]
VI Soros Olympiad 1999 - 2000 (Russia), 10.6
A natural number $n$ is given. Find the longest interval of a real line such that for numbers taken arbitrarily from it $a_0$, $a_1$, $a_2$, $...$, $a_{2n-1}$ the polynomial $x^{2n}+a_{2n-1}x^{2n-1}+...+a_1x + a_0$ has no roots on the entire real axis. (The left and right ends of the interval do not belong to the interval.)
2010 Irish Math Olympiad, 5
Find all polynomials $f(x)=x^3+bx^2+cx+d$, where $b,c,d,$ are real numbers, such that $f(x^2-2)=-f(-x)f(x)$.
2018 Mathematical Talent Reward Programme, SAQ: P 5
[list=1]
[*] Prove that, the sequence of remainders obtained when the Fibonacci numbers are divided by $n$ is periodic, where $n$ is a natural number.
[*] There exists no such non-constant polynomial with integer coefficients such that for every Fibonacci number $n,$ $ P(n)$ is a prime.
[/list]
1995 China Team Selection Test, 3
Prove that the interval $\lbrack 0,1 \rbrack$ can be split into black and white intervals for any quadratic polynomial $P(x)$, such that the sum of weights of the black intervals is equal to the sum of weights of the white intervals. (Define the weight of the interval $\lbrack a,b \rbrack$ as $P(b) - P(a)$.)
Does the same result hold with a degree 3 or degree 5 polynomial?
2008 District Olympiad, 2
Determine $ x$ irrational so that $ x^2\plus{}2x$ and $ x^3\minus{}6x$ are both rational.
2006 IberoAmerican Olympiad For University Students, 3
Let $p_1(x)=p(x)=4x^3-3x$ and $p_{n+1}(x)=p(p_n(x))$ for each positive integer $n$. Also, let $A(n)$ be the set of all the real roots of the equation $p_n(x)=x$.
Prove that $A(n)\subseteq A(2n)$ and that the product of the elements of $A(n)$ is the average of the elements of $A(2n)$.
PEN Q Problems, 1
Suppose $p(x) \in \mathbb{Z}[x]$ and $P(a)P(b)=-(a-b)^2$ for some distinct $a, b \in \mathbb{Z}$. Prove that $P(a)+P(b)=0$.