Found problems: 1132
2012 Cono Sur Olympiad, 6
6. Consider a triangle $ABC$ with $1 < \frac{AB}{AC} < \frac{3}{2}$. Let $M$ and $N$, respectively, be variable points of the sides $AB$ and $AC$, different from $A$, such that $\frac{MB}{AC} - \frac{NC}{AB} = 1$. Show that circumcircle of triangle $AMN$ pass through a fixed point different from $A$.
2014 Contests, 3
Let $ x,y,z $ be three non-negative real numbers such that \[x^2+y^2+z^2=2(xy+yz+zx). \] Prove that \[\dfrac{x+y+z}{3} \ge \sqrt[3]{2xyz}.\]
1969 IMO Shortlist, 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 $x^2+px+ q = 0.$
1961 AMC 12/AHSME, 29
Let the roots of $ax^2+bx+c=0$ be $r$ and $s$. The equation with roots $ar+b$ and $as+b$ is:
$ \textbf{(A)}\ x^2-bx-ac=0$
$\qquad\textbf{(B)}\ x^2-bx+ac=0$
$\qquad\textbf{(C)}\ x^2+3bx+ca+2b^2=0$
${\qquad\textbf{(D)}\ x^2+3bx-ca+2b^2=0 }$
${\qquad\textbf{(E)}\ x^2+bx(2-a)+a^2c+b^2(a+1)=0} $
2005 Germany Team Selection Test, 1
Let $k$ be a fixed integer greater than 1, and let ${m=4k^2-5}$. Show that there exist positive integers $a$ and $b$ such that the sequence $(x_n)$ defined by \[x_0=a,\quad x_1=b,\quad x_{n+2}=x_{n+1}+x_n\quad\text{for}\quad n=0,1,2,\dots,\] has all of its terms relatively prime to $m$.
[i]Proposed by Jaroslaw Wroblewski, Poland[/i]
2005 Romania National Olympiad, 1
Let $ABCD$ be a convex quadrilateral with $AD\not\parallel BC$. Define the points $E=AD \cap BC$ and $I = AC\cap BD$. Prove that the triangles $EDC$ and $IAB$ have the same centroid if and only if $AB \parallel CD$ and $IC^{2}= IA \cdot AC$.
[i]Virgil Nicula[/i]
2011 Canadian Open Math Challenge, 12
Let $f(x)=x^2-ax+b$, where $a$ and $b$ are positive integers.
(a) Suppose that $a=2$ and $b=2$. Determine the set of real roots of $f(x)-x$, and the set of real roots of $f(f(x))-x$.
(b) Determine the number of positive integers $(a,b)$ with $1\le a,b\le 2011$ for which every root of $f(f(x))-x$ is an integer.
2000 AMC 12/AHSME, 8
Figures $ 0$, $ 1$, $ 2$, and $ 3$ consist of $ 1$, $ 5$, $ 13$, and $ 25$ nonoverlapping squares, respectively. If the pattern were continued, how many nonoverlapping squares would there be in figure $ 100$?
[asy]
unitsize(8);
draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);
draw((9,0)--(10,0)--(10,3)--(9,3)--cycle);
draw((8,1)--(11,1)--(11,2)--(8,2)--cycle);
draw((19,0)--(20,0)--(20,5)--(19,5)--cycle);
draw((18,1)--(21,1)--(21,4)--(18,4)--cycle);
draw((17,2)--(22,2)--(22,3)--(17,3)--cycle);
draw((32,0)--(33,0)--(33,7)--(32,7)--cycle);
draw((29,3)--(36,3)--(36,4)--(29,4)--cycle);
draw((31,1)--(34,1)--(34,6)--(31,6)--cycle);
draw((30,2)--(35,2)--(35,5)--(30,5)--cycle);
label("Figure",(0.5,-1),S);
label("$0$",(0.5,-2.5),S);
label("Figure",(9.5,-1),S);
label("$1$",(9.5,-2.5),S);
label("Figure",(19.5,-1),S);
label("$2$",(19.5,-2.5),S);
label("Figure",(32.5,-1),S);
label("$3$",(32.5,-2.5),S);[/asy]$ \textbf{(A)}\ 10401 \qquad \textbf{(B)}\ 19801 \qquad \textbf{(C)}\ 20201 \qquad \textbf{(D)}\ 39801 \qquad \textbf{(E)}\ 40801$
2000 Korea - Final Round, 1
Prove that for any prime $p$, there exist integers $x,y,z,$ and $w$ such that $x^2+y^2+z^2-wp=0$ and $0<w<p$
2015 AMC 10, 14
Let $a$, $b$, and $c$ be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation $(x-a)(x-b)+(x-b)(x-c)=0$?
$\textbf{(A) } 15
\qquad\textbf{(B) } 15.5
\qquad\textbf{(C) } 16
\qquad\textbf{(D) } 16.5
\qquad\textbf{(E) } 17
$
1993 All-Russian Olympiad, 3
Quadratic trinomial $f(x)$ is allowed to be replaced by one of the trinomials $x^2f(1+\frac{1}{x})$ or $(x-1)^2f(\frac{1}{x-1})$. With the use of these operations, is it possible to go from $x^2+4x+3$ to $x^2+10x+9$?
2009 Moldova Team Selection Test, 1
Let $ m,n\in \mathbb{N}^*$. Find the least $ n$ for which exists $ m$, such that rectangle $ (3m \plus{} 2)\times(4m \plus{} 3)$ can be covered with $ \dfrac{n(n \plus{} 1)}{2}$ squares, among which exist $ n$ squares of length $ 1$, $ n \minus{} 1$ of length $ 2$, $ ...$, $ 1$ square of length $ n$. For the found value of $ n$ give the example of covering.
1999 Flanders Math Olympiad, 3
Determine all $f: \mathbb{R}\rightarrow\mathbb{R}$ for which
\[ 2\cdot f(x)-g(x)=f(y)-y \textrm{ and } f(x)\cdot g(x) \geq x+1. \]
2006 China National Olympiad, 3
Positive integers $k, m, n$ satisfy $mn=k^2+k+3$, prove that at least one of the equations $x^2+11y^2=4m$ and $x^2+11y^2=4n$ has an odd solution.
2004 IberoAmerican, 3
Given a set $ \mathcal{H}$ of points in the plane, $ P$ is called an "intersection point of $ \mathcal{H}$" if distinct points $ A,B,C,D$ exist in $ \mathcal{H}$ such that lines $ AB$ and $ CD$ are distinct and intersect in $ P$.
Given a finite set $ \mathcal{A}_{0}$ of points in the plane, a sequence of sets is defined as follows: for any $ j\geq0$, $ \mathcal{A}_{j+1}$ is the union of $ \mathcal{A}_{j}$ and the intersection points of $ \mathcal{A}_{j}$.
Prove that, if the union of all the sets in the sequence is finite, then $ \mathcal{A}_{i}=\mathcal{A}_{1}$ for any $ i\geq1$.
2007 Estonia National Olympiad, 1
Find all real numbers a such that all solutions to the quadratic equation $ x^2 \minus{} ax \plus{} a \equal{} 0$ are integers.
2006 Polish MO Finals, 2
Find all positive integers $k$ for which number $3^k+5^k$ is a power of some integer with exponent greater than $1$.
2008 Tournament Of Towns, 2
Solve the system of equations $(n > 2)$
\[\begin{array}{c}\ \sqrt{x_1}+\sqrt{x_2+x_3+\cdots+x_n}=\sqrt{x_2}+\sqrt{x_3+x_4+\cdots+x_n+x_1}=\cdots=\sqrt{x_n}+\sqrt{x_1+x_2+\cdots+x_{n-1}} \end{array}, \] \[x_1-x_2=1.\]
2012 Baltic Way, 18
Find all triples $(a,b,c)$ of integers satisfying $a^2 + b^2 + c^2 = 20122012$.
2009 USAMO, 4
For $ n\geq2$ let $ a_1, a_2, \ldots a_n$ be positive real numbers such that
\[ (a_1 \plus{} a_2 \plus{} \cdots \plus{} a_n)\left(\frac {1}{a_1} \plus{} \frac {1}{a_2} \plus{} \cdots \plus{} \frac {1}{a_n}\right) \leq \left(n \plus{} \frac {1}{2}\right)^2.
\]
Prove that $ \max(a_1, a_2, \ldots, a_n)\leq 4\min(a_1, a_2, \ldots, a_n)$.
2004 AMC 12/AHSME, 15
Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run $ 100$ meters. They next meet after Sally has run $ 150$ meters past their first meeting point. Each girl runs at a constant speed. What is the length of the track in meters?
$ \textbf{(A)}\ 250 \qquad \textbf{(B)}\ 300 \qquad \textbf{(C)}\ 350 \qquad \textbf{(D)}\ 400\qquad \textbf{(E)}\ 500$
PEN B Problems, 1
Let $n$ be a positive integer. Show that there are infinitely many primes $p$ such that the smallest positive primitive root of $p$ is greater than $n$.
2006 All-Russian Olympiad, 8
Given a quadratic trinomial $f\left(x\right)=x^2+ax+b$. Assume that the equation $f\left(f\left(x\right)\right)=0$ has four different real solutions, and that the sum of two of these solutions is $-1$. Prove that $b\leq -\frac14$.
2014 Harvard-MIT Mathematics Tournament, 5
Find the sum of all real numbers $x$ such that $5x^4-10x^3+10x^2-5x-11=0$.
2010 AIME Problems, 15
In $ \triangle{ABC}$ with $ AB = 12$, $ BC = 13$, and $ AC = 15$, let $ M$ be a point on $ \overline{AC}$ such that the incircles of $ \triangle{ABM}$ and $ \triangle{BCM}$ have equal radii. Let $ p$ and $ q$ be positive relatively prime integers such that $ \tfrac{AM}{CM} = \tfrac{p}{q}$. Find $ p + q$.