Found problems: 1132
2002 Moldova National Olympiad, 2
Let $ a,b,c\geq 0$ such that $ a\plus{}b\plus{}c\equal{}1$. Prove that:
$ a^2\plus{}b^2\plus{}c^2\geq 4(ab\plus{}bc\plus{}ca)\minus{}1$
2013 Finnish National High School Mathematics Competition, 5
Find all integer triples $(m,p,q)$ satisfying \[2^mp^2+1=q^5\] where $m>0$ and both $p$ and $q$ are prime numbers.
2005 IberoAmerican, 1
Determine all triples of real numbers $(a,b,c)$ such that \begin{eqnarray*} xyz &=& 8 \\ x^2y + y^2z + z^2x &=& 73 \\ x(y-z)^2 + y(z-x)^2 + z(x-y)^2 &=& 98 . \end{eqnarray*}
2010 IMC, 4
Let $a,b$ be two integers and suppose that $n$ is a positive integer for which the set $\mathbb{Z} \backslash \{ax^n + by^n \mid x,y \in \mathbb{Z}\}$ is finite. Prove that $n=1$.
2012 NIMO Problems, 9
Let $f(x) = x^2 - 2x$. A set of real numbers $S$ is [i]valid[/i] if it satisfies the following:
$\bullet$ If $x \in S$, then $f(x) \in S$.
$\bullet$ If $x \in S$ and $\underbrace{f(f(\dots f}_{k\ f\text{'s}}(x)\dots )) = x$ for some integer $k$, then $f(x) = x$.
Compute the number of 7-element valid sets.
[i]Proposed by Lewis Chen[/i]
2006 Stanford Mathematics Tournament, 5
There exist two positive numbers $ x$ such that $ \sin(\arccos(\tan(\arcsin x)))\equal{}x$. Find the product of the two possible $ x$.
2001 Saint Petersburg Mathematical Olympiad, 10.1
Quadratic trinomials $f$ and $g$ with integer coefficients obtain only positive values and the inequality $\dfrac{f(x)}{g(x)}\geq
\sqrt{2}$ is true $\forall x\in\mathbb{R}$. Prove that $\dfrac{f(x)}{g(x)}>\sqrt{2}$ is true $\forall x\in\mathbb{R}$
[I]Proposed by A. Khrabrov[/i]
2010 Princeton University Math Competition, 6
Define $\displaystyle{f(x) = x + \sqrt{x + \sqrt{x + \sqrt{x + \sqrt{x + \ldots}}}}}$. Find the smallest integer $x$ such that $f(x)\ge50\sqrt{x}$.
(Edit: The official question asked for the "smallest integer"; the intended question was the "smallest positive integer".)
2004 Italy TST, 3
Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all $m,n\in\mathbb{N}$,
\[(2^m+1)f(n)f(2^mn)=2^mf(n)^2+f(2^mn)^2+(2^m-1)^2n. \]
2016 SDMO (High School), 1
Quadratic equation $ x^2\plus{}ax\plus{}b\plus{}1\equal{}0$ have 2 positive integer roots, for integers $ a,b$. Show that $ a^2\plus{}b^2$ is not a prime.
1989 AMC 12/AHSME, 28
Find the sum of the roots of $\tan^2x-9\tan x+1=0$ that are between $x=0$ and $x=2\pi$ radians.
$ \textbf{(A)}\ \frac{\pi}{2} \qquad\textbf{(B)}\ \pi \qquad\textbf{(C)}\ \frac{3\pi}{2} \qquad\textbf{(D)}\ 3\pi \qquad\textbf{(E)}\ 4\pi $
2009 Hong Kong TST, 5
Let $ a,b,c$ be the three sides of a triangle. Determine all possible values of $ \frac {a^2 \plus{} b^2 \plus{} c^2}{ab \plus{} bc \plus{} ca}$
2002 USA Team Selection Test, 6
Find in explicit form all ordered pairs of positive integers $(m, n)$ such that $mn-1$ divides $m^2 + n^2$.
PEN A Problems, 18
Let $m$ and $n$ be natural numbers and let $mn+1$ be divisible by $24$. Show that $m+n$ is divisible by $24$.
1988 AMC 12/AHSME, 15
If $a$ and $b$ are integers such that $x^2 - x - 1$ is a factor of $ax^3 + bx^2 + 1$, then $b$ is
$ \textbf{(A)}\ -2\qquad\textbf{(B)}\ -1\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 1\qquad\textbf{(E)}\ 2 $
2017 Purple Comet Problems, 6
For some constant $k$ the polynomial $p(x) = 3x^2 + kx + 117$ has the property that $p(1) = p(10)$. Evaluate $p(20)$.
2011 All-Russian Olympiad, 2
In the notebooks of Peter and Nick, two numbers are written. Initially, these two numbers are 1 and 2 for Peter and 3 and 4 for Nick. Once a minute, Peter writes a quadratic trinomial $f(x)$, the roots of which are the two numbers in his notebook, while Nick writes a quadratic trinomial $g(x)$ the roots of which are the numbers in [i]his[/i] notebook. If the equation $f(x)=g(x)$ has two distinct roots, one of the two boys replaces the numbers in his notebook by those two roots. Otherwise, nothing happens. If Peter once made one of his numbers 5, what did the other one of his numbers become?
PEN J Problems, 6
Show that if $m$ and $n$ are relatively prime positive integers, then $\phi( 5^m -1) \neq 5^{n}-1$.
2011 Austria Beginners' Competition, 2
Let $p$ and $q$ be real numbers. The quadratic equation $$x^2 + px + q = 0$$
has the real solutions $x_1$ and $x_2$. In addition, the following two conditions apply:
(i) The numbers $x_1$ and $x_2$ differ from each other by exactly $ 1$.
(ii) The numbers $p$ and $q$ differ from each other by exactly $ 1$.
Show that then $p$, $q$, $x_1$ and $x_2$ are integers.
(G. Kirchner, University of Innsbruck)
1988 IMO Shortlist, 22
Let $ p$ be the product of two consecutive integers greater than 2. Show that there are no integers $ x_1, x_2, \ldots, x_p$ satisfying the equation
\[ \sum^p_{i \equal{} 1} x^2_i \minus{} \frac {4}{4 \cdot p \plus{} 1} \left( \sum^p_{i \equal{} 1} x_i \right)^2 \equal{} 1
\]
[b]OR[/b]
Show that there are only two values of $ p$ for which there are integers $ x_1, x_2, \ldots, x_p$ satisfying
\[ \sum^p_{i \equal{} 1} x^2_i \minus{} \frac {4}{4 \cdot p \plus{} 1} \left( \sum^p_{i \equal{} 1} x_i \right)^2 \equal{} 1
\]
2004 India National Olympiad, 2
$p > 3$ is a prime. Find all integers $a$, $b$, such that $a^2 + 3ab + 2p(a+b) + p^2 = 0$.
2011 NIMO Summer Contest, 14
In circle $\theta_1$ with radius $1$, circles $\phi_1, \phi_2, \dots, \phi_8$, with equal radii, are drawn such that for $1 \le i \le 8$, $\phi_i$ is tangent to $\omega_1$, $\phi_{i-1}$, and $\phi_{i+1}$, where $\phi_0 = \phi_8$ and $\phi_1 = \phi_9$. There exists a circle $\omega_2$ such that $\omega_1 \neq \omega_2$ and $\omega_2$ is tangent to $\phi_i$ for $1 \le i \le 8$. The radius of $\omega_2$ can be expressed in the form $a - b\sqrt{c} -d\sqrt{e - \sqrt{f}} + g \sqrt{h - j \sqrt{k}}$ such that $a, b, \dots, k$ are positive integers and the numbers $e, f, k, \gcd(h, j)$ are squarefree. What is $a+b+c+d+e+f+g+h+j+k$.
[i]Proposed by Eugene Chen
[/i]
1969 Canada National Olympiad, 7
Show that there are no integers $a,b,c$ for which $a^2+b^2-8c=6$.
2002 Tournament Of Towns, 3
Show that if the last digit of the number $x^2+xy+y^2$ is $0$ (where $x,y\in\mathbb{N}$ ) then last two digits are zero.
2012 ELMO Shortlist, 6
Prove that if $a$ and $b$ are positive integers and $ab>1$, then
\[\left\lfloor\frac{(a-b)^2-1}{ab}\right\rfloor=\left\lfloor\frac{(a-b)^2-1}{ab-1}\right\rfloor.\]Here $\lfloor x\rfloor$ denotes the greatest integer not exceeding $x$.
[i]Calvin Deng.[/i]