Found problems: 329
1951 AMC 12/AHSME, 26
In the equation $ \frac {x(x \minus{} 1) \minus{} (m \plus{} 1)}{(x \minus{} 1)(m \minus{} 1)} \equal{} \frac {x}{m}$ the roots are equal when
$ \textbf{(A)}\ m \equal{} 1 \qquad\textbf{(B)}\ m \equal{} \frac {1}{2} \qquad\textbf{(C)}\ m \equal{} 0 \qquad\textbf{(D)}\ m \equal{} \minus{} 1 \qquad\textbf{(E)}\ m \equal{} \minus{} \frac {1}{2}$
2010 Princeton University Math Competition, 7
Square $ABCD$ is divided into four rectangles by $EF$ and $GH$. $EF$ is parallel to $AB$ and $GH$ parallel to $BC$. $\angle BAF = 18^\circ$. $EF$ and $GH$ meet at point $P$. The area of rectangle $PFCH$ is twice that of rectangle $AGPE$. Given that the value of $\angle FAH$ in degrees is $x$, find the nearest integer to $x$.
[asy]
size(100); defaultpen(linewidth(0.7)+fontsize(10));
pair D2(pair P) {
dot(P,linewidth(3)); return P;
}
// NOTE: I've tampered with the angles to make the diagram not-to-scale. The correct numbers should be 72 instead of 76, and 45 instead of 55.
pair A=(0,1), B=(0,0), C=(1,0), D=(1,1), F=intersectionpoints(A--A+2*dir(-76),B--C)[0], H=intersectionpoints(A--A+2*dir(-76+55),D--C)[0], E=F+(0,1), G=H-(1,0), P=intersectionpoints(E--F,G--H)[0];
draw(A--B--C--D--cycle);
draw(F--A--H); draw(E--F); draw(G--H);
label("$A$",D2(A),NW);
label("$B$",D2(B),SW);
label("$C$",D2(C),SE);
label("$D$",D2(D),NE);
label("$E$",D2(E),plain.N);
label("$F$",D2(F),S);
label("$G$",D2(G),W);
label("$H$",D2(H),plain.E);
label("$P$",D2(P),SE);
[/asy]
2004 Junior Balkan Team Selection Tests - Romania, 1
We consider the following triangular array
\[ \begin{array}{cccccccc}
0 & 1 & 1 & 2 & 3 & 5 & 8 & \ldots \\
\ & 0 & 1 & 1 & 2 & 3 & 5 & \ldots \\
\ & \ & 2 & 3 & 5 & 8 & 13 & \ldots \\
\ & \ & \ & 4 & 7 & 11 & 18 & \ldots \\
\ & \ & \ & \ & 12 & 19 & 31 & \ldots \\
\end{array} \]
which is defined by the conditions
i) on the first two lines, each element, starting with the third one, is the sum of the preceding two elements;
ii) on the other lines each element is the sum of the two numbers found on the same column above it.
a) Prove that all the lines satisfy the first condition i);
b) Let $a,b,c,d$ be the first elements of 4 consecutive lines in the array. Find $d$ as a function of $a,b,c$.
2008 AIME Problems, 6
The sequence $ \{a_n\}$ is defined by
\[ a_0 \equal{} 1,a_1 \equal{} 1, \text{ and } a_n \equal{} a_{n \minus{} 1} \plus{} \frac {a_{n \minus{} 1}^2}{a_{n \minus{} 2}}\text{ for }n\ge2.
\]The sequence $ \{b_n\}$ is defined by
\[ b_0 \equal{} 1,b_1 \equal{} 3, \text{ and } b_n \equal{} b_{n \minus{} 1} \plus{} \frac {b_{n \minus{} 1}^2}{b_{n \minus{} 2}}\text{ for }n\ge2.
\]Find $ \frac {b_{32}}{a_{32}}$.
2002 Manhattan Mathematical Olympiad, 4
A triangle has sides with lengths $a,b,c$ such that
\[ a^2 + b^2 = 5c^2 \]
Prove that medians to the sides of lengths $a$ and $b$ are perpendicular.
2005 Morocco TST, 2
Let $a,b,c$ be positive real numbers. Prove the inequality
\[\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\geq a+b+c+\frac{4(a-b)^2}{a+b+c}.\]
When does equality occur?
2013 NIMO Problems, 8
A pair of positive integers $(m,n)$ is called [i]compatible[/i] if $m \ge \tfrac{1}{2} n + 7$ and $n \ge \tfrac{1}{2} m + 7$. A positive integer $k \ge 1$ is called [i]lonely[/i] if $(k,\ell)$ is not compatible for any integer $\ell \ge 1$. Find the sum of all lonely integers.
[i]Proposed by Evan Chen[/i]
1985 IMO Longlists, 15
[i]Superchess[/i] is played on on a $12 \times 12$ board, and it uses [i]superknights[/i], which move between opposite corner cells of any $3\times4$ subboard. Is it possible for a [i]superknight[/i] to visit every other cell of a superchessboard exactly once and return to its starting cell ?
2013 Online Math Open Problems, 37
Let $M$ be a positive integer. At a party with 120 people, 30 wear red hats, 40 wear blue hats, and 50 wear green hats. Before the party begins, $M$ pairs of people are friends. (Friendship is mutual.) Suppose also that no two friends wear the same colored hat to the party.
During the party, $X$ and $Y$ can become friends if and only if the following two conditions hold:
[list] [*] There exists a person $Z$ such that $X$ and $Y$ are both friends with $Z$. (The friendship(s) between $Z,X$ and $Z,Y$ could have been formed during the party.) [*] $X$ and $Y$ are not wearing the same colored hat. [/list]
Suppose the party lasts long enough so that all possible friendships are formed. Let $M_1$ be the largest value of $M$ such that regardless of which $M$ pairs of people are friends before the party, there will always be at least one pair of people $X$ and $Y$ with different colored hats who are not friends after the party. Let $M_2$ be the smallest value of $M$ such that regardless of which $M$ pairs of people are friends before the party, every pair of people $X$ and $Y$ with different colored hats are friends after the party. Find $M_1+M_2$.
[hide="Clarifications"]
[list]
[*] The definition of $M_2$ should read, ``Let $M_2$ be the [i]smallest[/i] value of $M$ such that...''. An earlier version of the test read ``largest value of $M$''.[/list][/hide]
[i]Victor Wang[/i]
2009 AIME Problems, 5
Equilateral triangle $ T$ is inscribed in circle $ A$, which has radius $ 10$. Circle $ B$ with radius $ 3$ is internally tangent to circle $ A$ at one vertex of $ T$. Circles $ C$ and $ D$, both with radius $ 2$, are internally tangent to circle $ A$ at the other two vertices of $ T$. Circles $ B$, $ C$, and $ D$ are all externally tangent to circle $ E$, which has radius $ \frac {m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$.
[asy]unitsize(2.2mm);
defaultpen(linewidth(.8pt)+fontsize(8pt));
dotfactor=4;
pair A=(0,0), D=8*dir(330), C=8*dir(210), B=7*dir(90);
pair Ep=(0,4-27/5);
pair[] dotted={A,B,C,D,Ep};
draw(Circle(A,10));
draw(Circle(B,3));
draw(Circle(C,2));
draw(Circle(D,2));
draw(Circle(Ep,27/5));
dot(dotted);
label("$E$",Ep,E);
label("$A$",A,W);
label("$B$",B,W);
label("$C$",C,W);
label("$D$",D,E);[/asy]
2013 Purple Comet Problems, 24
Find the remainder when $333^{333}$ is divided by $33$.
1995 Flanders Math Olympiad, 2
How many values of $x\in\left[ 1,3 \right]$ are there, for which $x^2$ has the same decimal part as $x$?
1997 AMC 12/AHSME, 4
If $ a$ is $ 50\%$ larger than $ c$, and $ b$ is $ 25\%$ larger than $ c$,then $ a$ is what percent larger than $ b$?
$ \textbf{(A)}\ 20\%\qquad \textbf{(B)}\ 25\%\qquad \textbf{(C)}\ 50\%\qquad \textbf{(D)}\ 100\%\qquad \textbf{(E)}\ 200\%$
1999 India Regional Mathematical Olympiad, 6
Find all solutions in integers $m,n$ of the equation \[ (m-n)^2 = \frac{4mn}{ m+n-1}. \]
2011 Romania Team Selection Test, 1
Determine all real-valued functions $f$ on the set of real numbers satisfying
\[2f(x)=f(x+y)+f(x+2y)\]
for all real numbers $x$ and all non-negative real numbers $y$.
2007 India National Olympiad, 6
If $ x$, $ y$, $ z$ are positive real numbers, prove that
\[ \left(x \plus{} y \plus{} z\right)^2 \left(yz \plus{} zx \plus{} xy\right)^2 \leq 3\left(y^2 \plus{} yz \plus{} z^2\right)\left(z^2 \plus{} zx \plus{} x^2\right)\left(x^2 \plus{} xy \plus{} y^2\right) .\]
1998 China Team Selection Test, 3
For any $h = 2^{r}$ ($r$ is a non-negative integer), find all $k \in \mathbb{N}$ which satisfy the following condition: There exists an odd natural number $m > 1$ and $n \in \mathbb{N}$, such that $k \mid m^{h} - 1, m \mid n^{\frac{m^{h}-1}{k}} + 1$.
2009 Croatia Team Selection Test, 1
Prove for all positive reals a,b,c,d:
$ \frac{a\minus{}b}{b\plus{}c}\plus{}\frac{b\minus{}c}{c\plus{}d}\plus{}\frac{c\minus{}d}{d\plus{}a}\plus{}\frac{d\minus{}a}{a\plus{}b} \geq 0$
PEN G Problems, 15
Prove that for any $ p, q\in\mathbb{N}$ with $ q > 1$ the following inequality holds:
\[ \left\vert\pi\minus{}\frac{p}{q}\right\vert\ge q^{\minus{}42}.\]
2008 Turkey MO (2nd round), 3
Let a.b.c be positive reals such that their sum is 1. Prove that
$ \frac{a^{2}b^{2}}{c^{3}(a^{2}\minus{}ab\plus{}b^{2})}\plus{}\frac{b^{2}c^{2}}{a^{3}(b^{2}\minus{}bc\plus{}c^{2})}\plus{}\frac{a^{2}c^{2}}{b^{3}(a^{2}\minus{}ac\plus{}c^{2})}\geq \frac{3}{ab\plus{}bc\plus{}ac}$
2014 AIME Problems, 8
Circle $C$ with radius $2$ has diameter $\overline{AB}$. Circle $D$ is internally tangent to circle $C$ at $A$. Circle $E$ is internally tangent to circle $C,$ externally tangent to circle $D,$ and tangent to $\overline{AB}$. The radius of circle $D$ is three times the radius of circle $E$ and can be written in the form $\sqrt{m} - n,$ where $m$ and $n$ are positive integers. Find $m+n$.
1971 AMC 12/AHSME, 31
[asy]
size(2.5inch);
pair A = (-2,0), B = 2dir(150), D = (2,0), C;
draw(A..(0,2)..D--cycle);
C = intersectionpoint(A..(0,2)..D,Circle(B,arclength(A--B)));
draw(A--B--C--D--cycle);
label("$A$",A,W);
label("$B$",B,NW);
label("$C$",C,N);
label("$D$",D,E);
label("$4$",A--D,S);
label("$1$",A--B,E);
label("$1$",B--C,SE);
//Credit to chezbgone2 for the diagram[/asy]
Quadrilateral $ABCD$ is inscribed in a circle with side $AD$, a diameter of length $4$. If sides $AB$ and $BC$ each have length $1$, then side $CD$ has length
$\textbf{(A) }\frac{7}{2}\qquad\textbf{(B) }\frac{5\sqrt{2}}{2}\qquad\textbf{(C) }\sqrt{11}\qquad\textbf{(D) }\sqrt{13}\qquad \textbf{(E) }2\sqrt{3}$
2000 All-Russian Olympiad, 8
All points in a $100 \times 100$ array are colored in one of four colors red, green, blue or yellow in such a way that there are $25$ points of each color in each row and in any column. Prove that there are two rows and two columns such that their four intersection points are all in different colors.
2007 AIME Problems, 7
Let \[N= \sum_{k=1}^{1000}k(\lceil \log_{\sqrt{2}}k\rceil-\lfloor \log_{\sqrt{2}}k \rfloor).\] Find the remainder when N is divided by 1000. (Here $\lfloor x \rfloor$ denotes the greatest integer that is less than or equal to x, and $\lceil x \rceil$ denotes the least integer that is greater than or equal to x.)
1994 USAMO, 1
Let $\, k_1 < k_2 < k_3 < \cdots \,$ be positive integers, no two consecutive, and let $\, s_m = k_1 + k_2 + \cdots + k_m \,$ for $\, m = 1,2,3, \ldots \; \;$. Prove that, for each positive integer $\, n, \,$ the interval $\, [s_n, s_{n+1}) \,$ contains at least one perfect square.