Found problems: 3597
2010 Harvard-MIT Mathematics Tournament, 1
Suppose that $p(x)$ is a polynomial and that $p(x)-p^\prime (x)=x^2+2x+1$. Compute $p(5)$.
2003 China Team Selection Test, 2
Can we find positive reals $a_1, a_2, \dots, a_{2002}$ such that for any positive integer $k$, with $1 \leq k \leq 2002$, every complex root $z$ of the following polynomial $f(x)$ satisfies the condition $|\text{Im } z| \leq |\text{Re } z|$,
\[f(x)=a_{k+2001}x^{2001}+a_{k+2000}x^{2000}+ \cdots + a_{k+1}x+a_k,\] where $a_{2002+i}=a_i$, for $i=1,2, \dots, 2001$.
2013 ELMO Shortlist, 3
Find all $f:\mathbb{R}\to\mathbb{R}$ such that for all $x,y\in\mathbb{R}$, $f(x)+f(y) = f(x+y)$ and $f(x^{2013}) = f(x)^{2013}$.
[i]Proposed by Calvin Deng[/i]
2014 Israel National Olympiad, 5
Let $p$ be a polynomial with integer coefficients satisfying $p(16)=36,p(14)=16,p(5)=25$. Determine all possible values of $p(10)$.
1991 Hungary-Israel Binational, 1
Suppose $f(x)$ is a polynomial with integer coefficients such that $f(0) = 11$ and $f(x_1) = f(x_2) = ... = f(x_n) = 2002$ for some distinct integers $x_1, x_2, . . . , x_n$. Find the largest possible value of $n$.
2022 Indonesia MO, 2
Let $P(x)$ be a polynomial with integer coefficient such that $P(1) = 10$ and $P(-1) = 22$.
(a) Give an example of $P(x)$ such that $P(x) = 0$ has an integer root.
(b) Suppose that $P(0) = 4$, prove that $P(x) = 0$ does not have an integer root.
2015 Switzerland Team Selection Test, 2
Let $a$, $b$, $c$ be real numbers greater than or equal to $1$. Prove that
\[ \min \left(\frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )\leq abc. \]
2010 Paenza, 2
A polynomial $f$ with integer coefficients is written on the blackboard. The teacher is a mathematician who has $3$ kids: Andrew, Beth and Charles. Andrew, who is $7$, is the youngest, and Charles is the oldest. When evaluating the polynomial on his kids' ages he obtains:
[list]$f(7) = 77$
$f(b) = 85$, where $b$ is Beth's age,
$f(c) = 0$, where $c$ is Charles' age.[/list]
How old is each child?
2023 Chile Classification NMO Seniors, 1
The function $f(x) = ax + b$ satisfies the following equalities:
\begin{align*}
f(f(f(1))) &= 2023, \\
f(f(f(0))) &= 1996.
\end{align*}
Find the value of $a$.
2007 IMC, 5
For each positive integer $ k$, find the smallest number $ n_{k}$ for which there exist real $ n_{k}\times n_{k}$ matrices $ A_{1}, A_{2}, \ldots, A_{k}$ such that all of the following conditions hold:
(1) $ A_{1}^{2}= A_{2}^{2}= \ldots = A_{k}^{2}= 0$,
(2) $ A_{i}A_{j}= A_{j}A_{i}$ for all $ 1 \le i, j \le k$, and
(3) $ A_{1}A_{2}\ldots A_{k}\ne 0$.
1959 AMC 12/AHSME, 27
Which one of the following is [i] not [/i] true for the equation \[ix^2-x+2i=0,\] where $i=\sqrt{-1}$?
$ \textbf{(A)}\ \text{The sum of the roots is 2} \qquad$
$\textbf{(B)}\ \text{The discriminant is 9}\qquad$
$\textbf{(C)}\ \text{The roots are imaginary}\qquad$
$\textbf{(D)}\ \text{The roots can be found using the quadratic formula}\qquad$
$\textbf{(E)}\ \text{The roots can be found by factoring, using imaginary numbers} $
2012 Purple Comet Problems, 5
Find the sum of the squares of the values $x$ that satisfy $\frac{1}{x} + \frac{2}{x+3}+\frac{3}{x+6} = 1$.
1975 Vietnam National Olympiad, 1
The roots of the equation $x^3 - x + 1 = 0$ are $a, b, c$. Find $a^8 + b^8 + c^8$.
1971 IMO Longlists, 31
Determine whether there exist distinct real numbers $a, b, c, t$ for which:
[i](i)[/i] the equation $ax^2 + btx + c = 0$ has two distinct real roots $x_1, x_2,$
[i](ii)[/i] the equation $bx^2 + ctx + a = 0$ has two distinct real roots $x_2, x_3,$
[i](iii)[/i] the equation $cx^2 + atx + b = 0$ has two distinct real roots $x_3, x_1.$
1997 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 5
Determine $ m > 0$ so that $ x^4 \minus{} (3m\plus{}2)x^2 \plus{} m^2 \equal{} 0$ has four real solutions forming an arithmetic series: i.e., that the solutions may be written $ a, a\plus{}b, a\plus{}2b,$ and $ a\plus{}3b$ for suitable $ a$ and $ b$.
A. 1
B. 3
C. 7
D. 12
E. None of these
2009 Stanford Mathematics Tournament, 9
Find the shortest distance between the point $(6,12)$ and the parabola given by the equation $x=\frac{y^2}{2}$
2005 IberoAmerican Olympiad For University Students, 7
Prove that for any integers $n,p$, $0<n\leq p$, all the roots of the polynomial below are real:
\[P_{n,p}(x)=\sum_{j=0}^n {p\choose j}{p\choose {n-j}}x^j\]
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?
STEMS 2021 Math Cat B, Q4
Let $n$ be a fixed positive integer.
- Show that there exist real polynomials $p_1, p_2, p_3, \cdots, p_k \in \mathbb{R}[x_1, \cdots, x_n]$ such that
\[(x_1 + x_2 + \cdots + x_n)^2 + p_1(x_1, \cdots, x_n)^2 + p_2(x_1, \cdots, x_n)^2 + \cdots + p_k(x_1, \cdots, x_n)^2 = n(x_1^2 + x_2^2 + \cdots + x_n^2)\]
- Find the least natural number $k$, depending on $n$, such that the above polynomials $p_1, p_2, \cdots, p_k$ exist.
2008 Thailand Mathematical Olympiad, 5
Let $P(x)$ be a polynomial of degree $2008$ with the following property: all roots of $P$ are real, and for all real $a$, if $P(a) = 0$ then $P(a+ 1) = 1$. Prove that P must have a repeated root.
2000 Manhattan Mathematical Olympiad, 1
Prove there exists no polynomial $f(x)$, with integer coefficients, such that $f(7) = 11$ and $f(11) = 13$.
1963 All Russian Mathematical Olympiad, 038
Find such real $p, q, a, b$, that for all $x$ an equality is held: $$(2x-1)^{20} - (ax+b)^{20} = (x^2+px+q)^{10}$$
2009 Brazil Team Selection Test, 3
Let $P(x) = x^4 + ax^3 + bx^2 + cx + d$ be a monic polynomial of degree $4$. It is known that all the roots of $P$ are real, distinct and belong to the interval $[-1, 1]$.
(a) Prove that $P(x) > -4$ for all real $x$.
(b) Find the highest value of the real constant $k$ such that $P(x) > k$ for every real $x$ and for every polynomial $P(x)$ satisfying the given conditions.
1997 AIME Problems, 15
The sides of rectangle $ABCD$ have lengths 10 and 11. An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD.$ The maximum possible area of such a triangle can be written in the form $p\sqrt{q}-r,$ where $p, q,$ and $r$ are positive integers, and $q$ is not divisible by the square of any prime number. Find $p+q+r.$
2025 Romania EGMO TST, P1
find all real coefficient polynomial $ P(x)$ such that $ P(x)P(x\plus{}1)\equal{}P(x^2\plus{}x\plus{}1)$ for all $ x$