Found problems: 329
2009 AMC 10, 17
Rectangle $ ABCD$ has $ AB \equal{} 4$ and $ BC \equal{} 3$. Segment $ EF$ is constructed through $ B$ so that $ EF$ is perpendicular to $ DB$, and $ A$ and $ C$ lie on $ DE$ and $ DF$, respectively. What is $ EF$?
$ \textbf{(A)}\ 9\qquad \textbf{(B)}\ 10\qquad \textbf{(C)}\ \frac {125}{12}\qquad \textbf{(D)}\ \frac {103}{9}\qquad \textbf{(E)}\ 12$
2011 USAMTS Problems, 3
Find all integers $b$ such that there exists a positive real number $x$ with \[ \dfrac {1}{b} = \dfrac {1}{\lfloor 2x \rfloor} + \dfrac {1}{\lfloor 5x \rfloor} \] Here, $\lfloor y \rfloor$ denotes the greatest integer that is less than or equal to $y$.
2002 Manhattan Mathematical Olympiad, 2
Let us consider the sequence $1,2, 3, \ldots , 2002$. Somebody choses $1002$ numbers from the sequence. Prove that there are two of the chosen numbers which are relatively prime (i.e. do not have any common divisors except $1$).
2013 Online Math Open Problems, 21
Dirock has a very neat rectangular backyard that can be represented as a $32\times 32$ grid of unit squares. The rows and columns are each numbered $1,2,\ldots, 32$. Dirock is very fond of rocks, and places a rock in every grid square whose row and column number are both divisible by $3$. Dirock would like to build a rectangular fence with vertices at the centers of grid squares and sides parallel to the sides of the yard such that
[list] [*] The fence does not pass through any grid squares containing rocks; [*] The interior of the fence contains exactly 5 rocks. [/list]
In how many ways can this be done?
[i]Ray Li[/i]
2012 Iran Team Selection Test, 3
We call a subset $B$ of natural numbers [i]loyal[/i] if there exists natural numbers $i\le j$ such that $B=\{i,i+1,\ldots,j\}$. Let $Q$ be the set of all [i]loyal[/i] sets. For every subset $A=\{a_1<a_2<\ldots<a_k\}$ of $\{1,2,\ldots,n\}$ we set
\[f(A)=\max_{1\le i \le k-1}{a_{i+1}-a_i}\qquad\text{and}\qquad g(A)=\max_{B\subseteq A, B\in Q} |B|.\] Furthermore, we define \[F(n)=\sum_{A\subseteq \{1,2,\ldots,n\}} f(A)\qquad\text{and}\qquad G(n)=\sum_{A\subseteq \{1,2,\ldots,n\}} g(A).\] Prove that there exists $m\in \mathbb N$ such that for each natural number $n>m$ we have $F(n)>G(n)$. (By $|A|$ we mean the number of elements of $A$, and if $|A|\le 1$, we define $f(A)$ to be zero).
[i]Proposed by Javad Abedi[/i]
2003 Hong kong National Olympiad, 1
Find the greatest real number $K$ such that for all positive real number $u,v,w$ with $u^{2}>4vw$ we have $(u^{2}-4vw)^{2}>K(2v^{2}-uw)(2w^{2}-uv)$
2013 NIMO Problems, 1
At ARML, Santa is asked to give rubber duckies to $2013$ students, one for each student. The students are conveniently numbered $1,2,\cdots,2013$, and for any integers $1 \le m < n \le 2013$, students $m$ and $n$ are friends if and only if $0 \le n-2m \le 1$.
Santa has only four different colors of duckies, but because he wants each student to feel special, he decides to give duckies of different colors to any two students who are either friends or who share a common friend. Let $N$ denote the number of ways in which he can select a color for each student. Find the remainder when $N$ is divided by $1000$.
[i]Proposed by Lewis Chen[/i]
2006 Poland - Second Round, 1
Let $c$ be fixed natural number. Sequence $(a_n)$ is defined by:
$a_1=1$, $a_{n+1}=d(a_n)+c$ for $n=1,2,...$.
where $d(m)$ is number of divisors of $m$. Prove that there exist $k$ natural such that sequence $a_k,a_{k+1},...$ is periodic.
2003 Iran MO (3rd Round), 15
Assume $m\times n$ matrix which is filled with just 0, 1 and any two row differ in at least $n/2$ members, show that $m \leq 2n$.
( for example the diffrence of this two row is only in one index
110
100)
[i]Edited by Myth[/i]
2004 Putnam, B2
Let $m$ and $n$ be positive integers. Show that
$\frac{(m+n)!}{(m+n)^{m+n}} < \frac{m!}{m^m}\cdot\frac{n!}{n^n}$
1979 AMC 12/AHSME, 30
[asy]
/*Using regular asymptote, this diagram would take 30 min to make. Using cse5, this takes 5 minutes. Conclusion? CSE5 IS THE BEST PACKAGE EVER CREATED!!!!*/
size(100);
import cse5;
pathpen=black;
anglefontpen=black;
pointpen=black;
anglepen=black;
dotfactor=3;
pair A=(0,0),B=(0.5,0.5*sqrt(3)),C=(3,0),D=(1.7,0),EE;
EE=(B+C)/2;
D(MP("$A$",A,W)--MP("$B$",B,N)--MP("$C$",C,E)--cycle);
D(MP("$E$",EE,N)--MP("$D$",D,S));
D(D);D(EE);
MA("80^\circ",8,D,EE,C,0.1);
MA("20^\circ",8,EE,C,D,0.3,2,shift(1,3)*C);
draw(arc(shift(-0.1,0.05)*C,0.25,100,180),arrow =ArcArrow());
MA("100^\circ",8,A,B,C,0.1,0);
MA("60^\circ",8,C,A,B,0.1,0);
//Credit to TheMaskedMagician for the diagram
[/asy]
In $\triangle ABC$, $E$ is the midpoint of side $BC$ and $D$ is on side $AC$. If the length of $AC$ is $1$ and $\measuredangle BAC = 60^\circ$, $\measuredangle ABC = 100^\circ$, $\measuredangle ACB = 20^\circ$ and $\measuredangle DEC = 80^\circ$, then the area of $\triangle ABC$ plus twice the area of $\triangle CDE$ equals
$\textbf{(A) }\frac{1}{4}\cos 10^\circ\qquad\textbf{(B) }\frac{\sqrt{3}}{8}\qquad\textbf{(C) }\frac{1}{4}\cos 40^\circ\qquad\textbf{(D) }\frac{1}{4}\cos 50^\circ\qquad\textbf{(E) }\frac{1}{8}$
1985 IMO Longlists, 2
We are given a triangle $ABC$ and three rectangles $R_1,R_2,R_3$ with sides parallel to two fixed perpendicular directions and such that their union covers the sides $AB,BC$, and $CA$; i.e., each point on the perimeter of $ABC$ is contained in or on at least one of the rectangles. Prove that all points inside the triangle are also covered by the union of $R_1,R_2,R_3.$
1989 China Team Selection Test, 2
Let $v_0 = 0, v_1 = 1$ and $v_{n+1} = 8 \cdot v_n - v_{n-1},$ $n = 1,2, ...$. Prove that in the sequence $\{v_n\}$ there aren't terms of the form $3^{\alpha} \cdot 5^{\beta}$ with $\alpha, \beta \in \mathbb{N}.$
2008 AMC 10, 14
Triangle $ OAB$ has $ O \equal{} (0,0)$, $ B \equal{} (5,0)$, and $ A$ in the first quadrant. In addition, $ \angle{ABO} \equal{} 90^\circ$ and $ \angle{AOB} \equal{} 30^\circ$. Suppose that $ \overline{OA}$ is rotated $ 90^\circ$ counterclockwise about $ O$. What are the coordinates of the image of $ A$?
$ \textbf{(A)}\ \left( \minus{} \frac {10}{3}\sqrt {3},5\right) \qquad \textbf{(B)}\ \left( \minus{} \frac {5}{3}\sqrt {3},5\right) \qquad \textbf{(C)}\ \left(\sqrt {3},5\right) \qquad \textbf{(D)}\ \left(\frac {5}{3}\sqrt {3},5\right) \\ \textbf{(E)}\ \left(\frac {10}{3}\sqrt {3},5\right)$
2006 Team Selection Test For CSMO, 4
All the squares of a board of $(n+1)\times(n-1)$ squares
are painted with [b]three colors[/b] such that, for any two different
columns and any two different rows, the 4 squares in their
intersections they don't have all the same color. Find the
greatest possible value of $n$.
2012 Pre-Preparation Course Examination, 5
Suppose that for the linear transformation $T:V \longrightarrow V$ where $V$ is a vector space, there is no trivial subspace $W\subset V$ such that $T(W)\subseteq W$. Prove that for every polynomial $p(x)$, the transformation $p(T)$ is invertible or zero.
2008 Baltic Way, 9
Suppose that the positive integers $ a$ and $ b$ satisfy the equation $ a^b\minus{}b^a\equal{}1008$ Prove that $ a$ and $ b$ are congruent modulo 1008.
2005 Romania Team Selection Test, 1
Solve the equation $3^x=2^xy+1$ in positive integers.
2006 China Northern MO, 5
$a,b,c$ are positive numbers such that $a+b+c=3$, show that: \[\frac{a^{2}+9}{2a^{2}+(b+c)^{2}}+\frac{b^{2}+9}{2b^{2}+(a+c)^{2}}+\frac{c^{2}+9}{2c^{2}+(a+b)^{2}}\leq 5\]
PEN H Problems, 21
Prove that the equation \[6(6a^{2}+3b^{2}+c^{2}) = 5n^{2}\] has no solutions in integers except $a=b=c=n=0$.
1957 AMC 12/AHSME, 14
If $ y \equal{} \sqrt{x^2 \minus{} 2x \plus{} 1} \plus{} \sqrt{x^2 \plus{} 2x \plus{} 1}$, then $ y$ is:
$ \textbf{(A)}\ 2x\qquad
\textbf{(B)}\ 2(x \plus{} 1)\qquad
\textbf{(C)}\ 0\qquad
\textbf{(D)}\ |x \minus{} 1| \plus{} |x \plus{} 1|\qquad
\textbf{(E)}\ \text{none of these}$
2012 Online Math Open Problems, 36
Let $s_n$ be the number of solutions to $a_1 + a_2 + a_3 +a _4 + b_1 + b_2 = n$, where $a_1,a_2,a_3$ and $a_4$ are elements of the set $\{2, 3, 5, 7\}$ and $b_1$ and $b_2$ are elements of the set $\{ 1, 2, 3, 4\}$. Find the number of $n$ for which $s_n$ is odd.
[i]Author: Alex Zhu[/i]
[hide="Clarification"]$s_n$ is the number of [i]ordered[/i] solutions $(a_1, a_2, a_3, a_4, b_1, b_2)$ to the equation, where each $a_i$ lies in $\{2, 3, 5, 7\}$ and each $b_i$ lies in $\{1, 2, 3, 4\}$. [/hide]
2009 AMC 10, 3
Which of the following is equal to $ 1\plus{}\frac{1}{1\plus{}\frac{1}{1\plus{}1}}$?
$ \textbf{(A)}\ \frac{5}{4} \qquad
\textbf{(B)}\ \frac{3}{2} \qquad
\textbf{(C)}\ \frac{5}{3} \qquad
\textbf{(D)}\ 2 \qquad
\textbf{(E)}\ 3$
1985 IMO Longlists, 8
Let $K $ be a convex set in the $xy$-plane, symmetric with respect to the origin and having area greater than $4 $. Prove that there exists a point $(m, n) \neq (0, 0)$ in $K$ such that $m$ and $n$ are integers.
2004 CHKMO, 1
Find the greatest real number $K$ such that for all positive real number $u,v,w$ with $u^{2}>4vw$ we have $(u^{2}-4vw)^{2}>K(2v^{2}-uw)(2w^{2}-uv)$