Found problems: 1132
2010 Stanford Mathematics Tournament, 1
Compute
\[\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}}}\]
2000 Putnam, 2
Prove that there exist infinitely many integers $n$ such that $n$, $n+1$, $n+2$ are each the sum of the squares of two integers. [Example: $0=0^2+0^2$, $1=0^2+1^2$, $2=1^2+1^2$.]
2011 N.N. Mihăileanu Individual, 1
Let be a quadratic polynom that has the property that the modulus of the sum between the leading and the free coefficient is smaller than the modulus of the middle coefficient. Prove that this polynom admits two distinct real roots, one belonging to the interval $ (-1,1) , $ and the other belonging outside of the interval $ (-1,1). $
2006 JBMO ShortLists, 5
Determine all pairs $ (m,n)$ of natural numbers for which $ m^2\equal{}nk\plus{}2$ where $ k\equal{}\overline{n1}$.
EDIT. [color=#FF0000]It has been discovered the correct statement is with $ k\equal{}\overline{1n}$.[/color]
PEN O Problems, 45
Find all positive integers $n$ with the property that the set \[\{n,n+1,n+2,n+3,n+4,n+5\}\] can be partitioned into two sets such that the product of the numbers in one set equals the product of the numbers in the other set.
2003 Brazil National Olympiad, 1
Find the smallest positive prime that divides $n^2 + 5n + 23$ for some integer $n$.
2002 Germany Team Selection Test, 1
Determine the number of all numbers which are represented as $x^2+y^2$ with $x, y \in \{1, 2, 3, \ldots, 1000\}$ and which are divisible by 121.
1995 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 8
Let $ f(x) \equal{} x \minus{} \frac {1}{x}.$ How many different solutions are there to the equation $ f(f(f(x))) \equal{} 1$?
A. 1
B. 2
C. 3
D. 6
E. 8
2001 IMO Shortlist, 2
Consider the system \begin{align*}x + y &= z + u,\\2xy & = zu.\end{align*} Find the greatest value of the real constant $m$ such that $m \leq x/y$ for any positive integer solution $(x,y,z,u)$ of the system, with $x \geq y$.
2014 Moldova Team Selection Test, 1
Find all pairs of non-negative integers $(x,y)$ such that
\[\sqrt{x+y}-\sqrt{x}-\sqrt{y}+2=0.\]
2002 AMC 10, 18
For how many positive integers $n$ is $n^3-8n^2+20n-13$ a prime number?
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }\text{more than 4}$
2007 All-Russian Olympiad Regional Round, 9.1
Pete chooses $ 1004$ monic quadratic polynomial $ f_{1},\cdots,f_{1004}$, such that each integer from $ 0$ to $ 2007$ is a root of at least one of them. Vasya considers all equations of the form $ f_{i}\equal{}f_{j}(i\not \equal{}j)$ and computes their roots; for each such root , Pete has to pay to Vasya $ 1$ ruble . Find the least possible value of Vasya's income.
2004 Nicolae Coculescu, 2
Solve in the real numbers the equation:
$$ \cos^2 \frac{(x-2)\pi }{4} +\cos\frac{(x-2)\pi }{3} =\log_3 (x^2-4x+6) $$
[i]Gheorghe Mihai[/i]
2009 Math Prize For Girls Problems, 7
Compute the value of the expression
\[ 2009^4 \minus{} 4 \times 2007^4 \plus{} 6 \times 2005^4 \minus{} 4 \times 2003^4 \plus{} 2001^4 \, .\]
2014 Singapore Senior Math Olympiad, 21
Let $n$ be an integer, and let $\triangle ABC$ be a right-angles triangle with right angle at $C$. It is given that $\sin A$ and $\sin B$ are the roots of the quadratic equation \[(5n+8)x^2-(7n-20)x+120=0.\] Find the value of $n$
1995 India Regional Mathematical Olympiad, 5
Show that for any triangle $ABC$, the following inequality is true:
\[ a^2 + b^2 +c^2 > \sqrt{3} max \{ |a^2 - b^2|, |b^2 -c^2|, |c^2 -a^2| \} . \]
2004 Korea - Final Round, 2
Prove that the equation $3y^2 = x^4 + x$ has no positive integer solutions.
1999 India Regional Mathematical Olympiad, 7
Find the number of quadratic polynomials $ax^2 + bx +c$ which satisfy the following:
(a) $a,b,c$ are distinct;
(b) $a,b,c \in \{ 1,2,3,\cdots 1999 \}$;
(c) $x+1$ divides $ax^2 + bx+c$.
2007 National Olympiad First Round, 27
What is the sum of real roots of the equation
\[
\left ( x + 1\right )\left ( x + \dfrac 14\right )\left ( x + \dfrac 12\right )\left ( x + \dfrac 34\right )= \dfrac {45}{32}?
\]
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ -1
\qquad\textbf{(C)}\ -\dfrac {3}{2}
\qquad\textbf{(D)}\ -\dfrac {5}{4}
\qquad\textbf{(E)}\ -\dfrac {7}{12}
$
2020 Brazil National Olympiad, 2
The following sentece is written on a board:
[center]The equation $x^2-824x+\blacksquare 143=0$ has two integer solutions.[/center]
Where $\blacksquare$ represents algarisms of a blurred number on the board. What are the possible equations originally on the board?
2009 International Zhautykov Olympiad, 1
Find all pairs of integers $ (x,y)$, such that
\[ x^2 \minus{} 2009y \plus{} 2y^2 \equal{} 0
\]
1995 AIME Problems, 9
Triangle $ABC$ is isosceles, with $AB=AC$ and altitude $AM=11.$ Suppose that there is a point $D$ on $\overline{AM}$ with $AD=10$ and $\angle BDC=3\angle BAC.$ Then the perimeter of $\triangle ABC$ may be written in the form $a+\sqrt{b},$ where $a$ and $b$ are integers. Find $a+b.$
[asy] import graph; size(7cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-1.55,xmax=7.95,ymin=-4.41,ymax=5.3; draw((1,3)--(0,0)); draw((0,0)--(2,0)); draw((2,0)--(1,3)); draw((1,3)--(1,0)); draw((1,0.7)--(0,0)); draw((1,0.7)--(2,0)); label("$11$",(0.75,1.63),SE*lsf); dot((1,3),ds); label("$A$",(0.96,3.14),NE*lsf); dot((0,0),ds); label("$B$",(-0.15,-0.18),NE*lsf); dot((2,0),ds); label("$C$",(2.06,-0.18),NE*lsf); dot((1,0),ds); label("$M$",(0.97,-0.27),NE*lsf); dot((1,0.7),ds); label("$D$",(1.05,0.77),NE*lsf); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]
2012 District Olympiad, 2
Let $(A,+,\cdot)$ a 9 elements ring. Prove that the following assertions are equivalent:
(a) For any $x\in A\backslash\{0\}$ there are two numbers $a\in \{-1,0,1\}$ and $b\in \{-1,1\}$ such that $x^2+ax+b=0$.
(b) $(A,+,\cdot)$ is a field.
2000 National Olympiad First Round, 33
Let $K$ be a point on the side $[AB]$, and $L$ be a point on the side $[BC]$ of the square $ABCD$. If $|AK|=3$, $|KB|=2$, and the distance of $K$ to the line $DL$ is $3$, what is $|BL|:|LC|$?
$ \textbf{(A)}\ \frac78
\qquad\textbf{(B)}\ \frac{\sqrt 3}2
\qquad\textbf{(C)}\ \frac 87
\qquad\textbf{(D)}\ \frac 38
\qquad\textbf{(E)}\ \frac{\sqrt 2}2
$
2013 Benelux, 4
a) Find all positive integers $g$ with the following property: for each odd prime number $p$ there exists a positive integer $n$ such that $p$ divides the two integers
\[g^n - n\quad\text{ and }\quad g^{n+1} - (n + 1).\]
b) Find all positive integers $g$ with the following property: for each odd prime number $p$ there exists a positive integer $n$ such that $p$ divides the two integers
\[g^n - n^2\quad\text{ and }g^{n+1} - (n + 1)^2.\]