Found problems: 15925
2020 AIME Problems, 14
Let $P(x)$ be a quadratic polynomial with complex coefficients whose $x^2$ coefficient is $1$. Suppose the equation $P(P(x))=0$ has four distinct solutions, $x=3,4,a,b$. Find the sum of all possible values of $(a+b)^2$.
1993 USAMO, 1
For each integer $\, n \geq 2, \,$ determine, with proof, which of the two positive real numbers $\, a \,$ and $\, b \,$ satisfying \[ a^n = a + 1, \hspace{.3in} b^{2n} = b + 3a \] is larger.
2013 AMC 12/AHSME, 17
Let $a,b,$ and $c$ be real numbers such that \begin{align*}
a+b+c &= 2, \text{ and} \\
a^2+b^2+c^2&= 12
\end{align*}
What is the difference between the maximum and minimum possible values of $c$?
$ \textbf{(A)}\ 2\qquad\textbf{(B)}\ \frac{10}{3}\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ \frac{16}{3}\qquad\textbf{(E)}\ \frac{20}{3} $
2022 JBMO Shortlist, A1
Find all pairs of positive integers $(a, b)$ such that $$11ab \le a^3 - b^3 \le 12ab.$$
Kvant 2024, M2812
On the coordinate plane, at some points with integer coordinates, there is a pebble (a finite number of pebbles). It is allowed to make the following move: select a pair of pebbles, take some vector $\vec{a}$ with integer coordinates and then move one of the selected pebbles to vector $\vec{a}$, and the other to the opposite vector $-\vec{a}$; it is forbidden that there should be more than one pebble at one point. Is it always possible to achieve a situation in which all the pebbles lie on the same straight line in a few moves?
[i] K. Ivanov [/i]
2011 Iran MO (3rd Round), 3
We define the polynomial $f(x)$ in $\mathbb R[x]$ as follows:
$f(x)=x^n+a_{n-2}x^{n-2}+a_{n-3}x^{n-3}+.....+a_1x+a_0$
Prove that there exists an $i$ in the set $\{1,....,n\}$ such that we have
$|f(i)|\ge \frac{n!}{\dbinom{n}{i}}$.
[i]proposed by Mohammadmahdi Yazdi[/i]
2008 Princeton University Math Competition, A5/B8
Let $H_k =\Sigma_{i=1}^k \frac{1}{i}$ for all positive integers $k$. Find an closed-form expression for $\Sigma_{i=1}^k H_i$ in terms of $n$ and $H_n$.
2006 All-Russian Olympiad Regional Round, 10.5
Prove that for every $x$ such that $\sin x \ne 0$, there is such natural $n$, which $$ | \sin nx| \ge \frac{\sqrt3}{2}.$$
1992 Poland - First Round, 4
Determine all functions $f: R \longrightarrow R$ such that
$f(x+y)-f(x-y)=f(x)*f(y)$ for $x,y \in R$
1998 Belarus Team Selection Test, 2
For any sequence of real numbers $(a_n), n \in N$, define a new sequence $(b_n)$ as $b_n =a_{n+2}+sa_{n+1}+ta_{n}$, where $s,t$ are given real numbers.
Find all ordered pairs $(s,t)$ satisfying the following property: any sequence $(a_n)$ converges as soon as the sequence $(b_n)$ converges.
2018 Iran MO (3rd Round), 3
A)Let $x,y$ be two complex numbers on the unit circle so that:
$\frac{\pi }{3} \le \arg (x)-\arg (y) \le \frac{5 \pi }{3}$
Prove that for any $z \in \mathbb{C}$ we have:
$|z|+|z-x|+|z-y| \ge |zx-y|$
B)Let $x,y$ be two complex numbers so that:
$\frac{\pi }{3} \le \arg (x)-\arg (y) \le \frac{2 \pi }{3}$
Prove that for any $z \in \mathbb{C}$ we have:
$|z|+|z-y|+|z-x| \ge | \frac{\sqrt{3}}{2} x +(y-\frac{x}{2})i|$
2002 China Team Selection Test, 2
Given an integer $k$. $f(n)$ is defined on negative integer set and its values are integers. $f(n)$ satisfies \[ f(n)f(n+1)=(f(n)+n-k)^2, \] for $n=-2,-3,\cdots$. Find an expression of $f(n)$.
2002 Vietnam National Olympiad, 3
For a positive integer $ n$, consider the equation $ \frac{1}{x\minus{}1}\plus{}\frac{1}{4x\minus{}1}\plus{}\cdots\plus{}\frac{1}{k^2x\minus{}1}\plus{}\cdots\plus{}\frac{1}{n^2x\minus{}1}\equal{}\frac{1}{2}$.
(a) Prove that, for every $ n$, this equation has a unique root greater than $ 1$, which is denoted by $ x_n$.
(b) Prove that the limit of sequence $ (x_n)$ is $ 4$ as $ n$ approaches infinity.
2001 China Team Selection Test, 3
Let $F = \max_{1 \leq x \leq 3} |x^3 - ax^2 - bx - c|$. When $a$, $b$, $c$ run over all the real numbers, find the smallest possible value of $F$.
2002 Romania National Olympiad, 4
Let $I\subseteq \mathbb{R}$ be an interval and $f:I\rightarrow\mathbb{R}$ a function such that:
\[|f(x)-f(y)|\le |x-y|,\quad\text{for all}\ x,y\in I. \]
Show that $f$ is monotonic on $I$ if and only if, for any $x,y\in I$, either $f(x)\le f\left(\frac{x+y}{2}\right)\le f(y)$ or $f(y)\le f\left(\frac{x+y}{2}\right)\le f(x)$.
2021 German National Olympiad, 3
For a fixed $k$ with $4 \le k \le 9$ consider the set of all positive integers with $k$ decimal digits such that each of the digits from $1$ to $k$ occurs exactly once.
Show that it is possible to partition this set into two disjoint subsets such that the sum of the cubes of the numbers in the first set is equal to the sum of the cubes in the second set.
2020 China Team Selection Test, 3
For a non-empty finite set $A$ of positive integers, let $\text{lcm}(A)$ denote the least common multiple of elements in $A$, and let $d(A)$ denote the number of prime factors of $\text{lcm}(A)$ (counting multiplicity). Given a finite set $S$ of positive integers, and $$f_S(x)=\sum_{\emptyset \neq A \subset S} \frac{(-1)^{|A|} x^{d(A)}}{\text{lcm}(A)}.$$
Prove that, if $0 \le x \le 2$, then $-1 \le f_S(x) \le 0$.
1998 IMC, 2
$S$ ist the set of all cubic polynomials $f$ with $|f(\pm 1)| \leq 1$ and $|f(\pm \frac{1}{2})| \leq 1$. Find $\sup_{f \in S} \max_{-1 \leq x \leq 1} |f''(x)|$ and all members of $f$ which give equality.
2005 Czech And Slovak Olympiad III A, 1
Consider all arithmetical sequences of real numbers $(x_i)^{\infty}=1$ and $(y_i)^{\infty} =1$ with the common first term, such that for some $k > 1, x_{k-1}y_{k-1} = 42, x_ky_k = 30$, and $x_{k+1}y_{k+1} = 16$. Find all such pairs of sequences with the maximum possible $k$.
2000 IMO Shortlist, 2
Let $ a, b, c$ be positive integers satisfying the conditions $ b > 2a$ and $ c > 2b.$ Show that there exists a real number $ \lambda$ with the property that all the three numbers $ \lambda a, \lambda b, \lambda c$ have their fractional parts lying in the interval $ \left(\frac {1}{3}, \frac {2}{3} \right].$
2023 SG Originals, Q6
$\mathbb{Z}[x]$ represents the set of all polynomials with integer coefficients. Find all functions $f:\mathbb{Z}[x]\rightarrow \mathbb{Z}[x]$ such that for any 2 polynomials $P,Q$ with integer coefficients and integer $r$, the following statement is true. \[P(r)\mid Q(r) \iff f(P)(r)\mid f(Q)(r).\]
(We define $a|b$ if and only if $b=za$ for some integer $z$. In particular, $0|0$.)
[i]Proposed by the4seasons.[/i]
2000 Romania National Olympiad, 1
For the real numbers $a, b, c, d$, the following inequalities hold:
$$a + b + c \le 3d, \,\,\, b + c + d \le 3a, \,\,\,c + d + a \le 3b, \,\,\,d + a + b\le 3c.$$
Compare the numbers $a, b, c, d$.
2009 Tuymaada Olympiad, 4
The sum of several non-negative numbers is not greater than 200, while the sum of their squares is not less than 2500. Prove that among them there are four numbers whose sum is not less than 50.
[i]Proposed by A. Khabrov[/i]
2018 Rio de Janeiro Mathematical Olympiad, 4
Find every real values that $a$ can assume such that
$$\begin{cases}
x^3 + y^2 + z^2 = a\\
x^2 + y^3 + z^2 = a\\
x^2 + y^2 + z^3 = a
\end{cases}$$
has a solution with $x, y, z$ distinct real numbers.
1992 AMC 12/AHSME, 29
An "unfair" coin has a $2/3$ probability of turning up heads. If this coin is tossed $50$ times, what is the probability that the total number of heads is even?
$ \textbf{(A)}\ 25\left(\frac{2}{3}\right)^{50}\qquad\textbf{(B)}\ \frac{1}{2}\left(1 - \frac{1}{3^{50}}\right)\qquad\textbf{(C)}\ \frac{1}{2}\qquad\textbf{(D)}\ \frac{1}{2}\left(1 + \frac{1}{3^{50}}\right)\qquad\textbf{(E)}\ \frac{2}{3} $