Found problems: 3597
2001 China Team Selection Test, 3
For a given natural number $k > 1$, find all functions $f:\mathbb{R} \to \mathbb{R}$ such that for all $x, y \in \mathbb{R}$, $f[x^k + f(y)] = y +[f(x)]^k$.
1980 Polish MO Finals, 4
Show that for every polynomial $W$ in three variables there exist polynomials $U$ and $V$ such that:
$$W(x,y,z) = U(x,y,z)+V(x,y,z),$$
$$U(x,y,z) = U(y,x,z),$$
$$V(x,y,z) = -V(x,z,y).$$
1994 China Team Selection Test, 2
Given distinct prime numbers $p$ and $q$ and a natural number $n \geq 3$, find all $a \in \mathbb{Z}$ such that the polynomial $f(x) = x^n + ax^{n-1} + pq$ can be factored into 2 integral polynomials of degree at least 1.
2006 Stanford Mathematics Tournament, 10
Find the smallest positive $m$ for which there are at least 11 even and 11 odd positive integers $n$ so that $\tfrac{n^3+m}{n+2}$ is an integer.
1992 AIME Problems, 8
For any sequence of real numbers $A=(a_1,a_2,a_3,\ldots)$, define $\Delta A$ to be the sequence $(a_2-a_1,a_3-a_2,a_4-a_3,\ldots)$, whose $n^\text{th}$ term is $a_{n+1}-a_n$. Suppose that all of the terms of the sequence $\Delta(\Delta A)$ are $1$, and that $a_{19}=a_{92}=0$. Find $a_1$.
2016 Azerbaijan Team Selection Test, 2
Find all polynomials $P(x)$ with real coefficents, such that for all $x,y,z$ satisfying $x+y+z=0$, the equation below is true: \[P(x+y)^3+P(y+z)^3+P(z+x)^3=3P((x+y)(y+z)(z+x))\]
2004 China National Olympiad, 2
For a given positive integer $n\ge 2$, suppose positive integers $a_i$ where $1\le i\le n$ satisfy $a_1<a_2<\ldots <a_n$ and $\sum_{i=1}^n \frac{1}{a_i}\le 1$. Prove that, for any real number $x$, the following inequality holds
\[\left(\sum_{i=1}^n\frac{1}{a_i^2+x^2}\right)^2\le\frac{1}{2}\cdot\frac{1}{a_1(a_1-1)+x^2} \]
[i]Li Shenghong[/i]
1988 IMO, 1
Show that the solution set of the inequality
\[ \sum^{70}_{k \equal{} 1} \frac {k}{x \minus{} k} \geq \frac {5}{4}
\]
is a union of disjoint intervals, the sum of whose length is 1988.
2022 South Africa National Olympiad, 6
Show that there are infinitely many polynomials P with real coefficients such that if x, y, and z are real numbers such that $x^2+y^2+z^2+2xyz=1$, then
$$P\left(x\right)^2+P\left(y\right)^2+P\left(z\right)^2+2P\left(x\right)P\left(y\right)P\left(z\right) = 1$$
2007 Irish Math Olympiad, 1
Let $ r,s,$ and $ t$ be the roots of the cubic polynomial: $ p(x)\equal{}x^3\minus{}2007x\plus{}2002.$
Determine the value of: $ \frac{r\minus{}1}{r\plus{}1}\plus{}\frac{s\minus{}1}{s\plus{}1}\plus{}\frac{t\minus{}1}{t\plus{}1}$.
2023 AMC 12/AHSME, 14
For how many ordered pairs $(a,b)$ of integers does the polynomial $x^3+ax^2+bx+6$ have $3$ distinct integer roots?
$\textbf{(A)}\ 5 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 4$
PEN L Problems, 13
The sequence $\{x_{n}\}_{n \ge 1}$ is defined by \[x_{1}=x_{2}=1, \; x_{n+2}= 14x_{n+1}-x_{n}-4.\] Prove that $x_{n}$ is always a perfect square.
PEN A Problems, 22
Prove that the number \[\sum_{k=0}^{n}\binom{2n+1}{2k+1}2^{3k}\] is not divisible by $5$ for any integer $n\geq 0$.
2022 Bulgarian Autumn Math Competition, Problem 9.1
Given is the equation:
\[x^2+mx+2022=0\]
a) Find all the values of the parameter $m$, such that the two solutions of the equation $x_1, x_2$ are $\textbf{natural}$ numbers
b)Find all the values of the parameter $m$, such that the two solutions of the equation $x_1, x_2$ are $\textbf{integer}$ numbers
2021 All-Russian Olympiad, 6
Given is a polynomial $P(x)$ of degree $n>1$ with real coefficients. The equation $P(P(P(x)))=P(x)$ has $n^3$ distinct real roots. Prove that these roots could be split into two groups with equal arithmetic mean.
1969 IMO Longlists, 14
$(CZS 3)$ Let $a$ and $b$ be two positive real numbers. If $x$ is a real solution of the equation $x^2 + px + q = 0$ with real coefficients $p$ and $q$ such that $|p| \le a, |q| \le b,$ prove that $|x| \le \frac{1}{2}(a +\sqrt{a^2 + 4b})$ Conversely, if $x$ satisfies the above inequality, prove that there exist real numbers $p$ and
$q$ with $|p|\le a, |q|\le b$ such that $x$ is one of the roots of the equation $x^2+px+ q = 0.$
1982 Vietnam National Olympiad, 1
Determine a quadric polynomial with intergral coefficients whose roots are $\cos 72^{\circ}$ and $\cos 144^{\circ}.$
2016 Chile TST IMO, 4
Let \( f \) and \( g \) be two nonzero polynomials with integer coefficients such that \( \deg(f) > \deg(g) \). Suppose that for infinitely many prime numbers \( p \), the polynomial \( pf + g \) has a rational root. Prove that \( f \) has a rational root.
Clarification: A rational root of a polynomial \( f \) is a number \( q \in \mathbb{Q} \) such that \( f(q) = 0 \).
2016 Tournament Of Towns, 5
On a blackboard, several polynomials of degree $37$ are written, each of them has the leading coefficient equal to $1$. Initially all coefficients of each polynomial are non-negative. By one move it is allowed to erase any pair of polynomials $f, g$ and replace it by another pair of polynomials $f_1, g_1$ of degree $37$ with the leading coefficients equal to $1$ such that either $f_1+g_1 = f+g$ or $f_1g_1 = fg$. Prove that it is impossible that after some move each polynomial
on the blackboard has $37$ distinct positive roots. [i](8 points)[/i]
[i]Alexandr Kuznetsov[/i]
2017 Korea USCM, 4
For a real coefficient cubic polynomial $f(x)=ax^3+bx^2+cx+d$, denote three roots of the equation $f(x)=0$ by $\alpha,\beta,\gamma$. Prove that the three roots $\alpha,\beta,\gamma$ are distinct real numbers iff the real symmetric matrix
$$\begin{pmatrix} 3 & p_1 & p_2 \\ p_1 & p_2 & p_3 \\ p_2 & p_3 & p_4 \end{pmatrix},\quad p_i = \alpha^i + \beta^i + \gamma^i$$
is positive definite.
2023 NMTC Junior, P3
Let $a_i (i=1,2,3,4,5,6)$ are reals. The polynomial
$f(x)=a_1+a_2x+a_3x^2+a_4x^3+a_5x^4+a_6a^5+7x^6-4x^7+x^8$ can be factorized into linear factors $x-x_i$ where
$i \in {1,2,3,...,8}$.
Find the possible values of $a_1$.
1994 China National Olympiad, 5
For arbitrary natural number $n$, prove that $\sum^n_{k=0}C^k_n2^kC^{[(n-k)/2]}_{n-k}=C^n_{2n+1}$, where $C^0_0=1$ and $[\dfrac{n-k}{2}]$ denotes the integer part of $\dfrac{n-k}{2}$.
2008 International Zhautykov Olympiad, 2
A polynomial $ P(x)$ with integer coefficients is called good,if it can be represented as a sum of cubes of several polynomials (in variable $ x$) with integer coefficients.For example,the polynomials $ x^3 \minus{} 1$ and $ 9x^3 \minus{} 3x^2 \plus{} 3x \plus{} 7 \equal{} (x \minus{} 1)^3 \plus{} (2x)^3 \plus{} 2^3$ are good.
a)Is the polynomial $ P(x) \equal{} 3x \plus{} 3x^7$ good?
b)Is the polynomial $ P(x) \equal{} 3x \plus{} 3x^7 \plus{} 3x^{2008}$ good?
Justify your answers.
2016 Balkan MO, 3
Find all monic polynomials $f$ with integer coefficients satisfying the following condition: there exists a positive integer $N$ such that $p$ divides $2(f(p)!)+1$ for every prime $p>N$ for which $f(p)$ is a positive integer.
[i]Note: A monic polynomial has a leading coefficient equal to 1.[/i]
[i](Greece - Panagiotis Lolas and Silouanos Brazitikos)[/i]
2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 2
Justify your answer whether $A=\left(
\begin{array}{ccc}
-4 & -1& -1 \\
1 & -2& 1 \\
0 & 0& -3
\end{array}
\right)$ is similar to $B=\left(
\begin{array}{ccc}
-2 & 1& 0 \\
-1 & -4& 1 \\
0 & 0& -3
\end{array}
\right),\ A,\ B\in{M(\mathbb{C})}$ or not.