The figure may be folded along the lines shown to form a number cube. Three number faces come together at each corner of the cube. What is the largest sum of three numbers whose faces come together at a corner? [asy] draw((0,0)--(0,1)--(1,1)--(1,2)--(2,2)--(2,1)--(4,1)--(4,0)--(2,0)--(2,-1)--(1,-1)--(1,0)--cycle); draw((1,0)--(1,1)--(2,1)--(2,0)--cycle); draw((3,1)--(3,0)); label("$1$",(1.5,1.25),N); label("$2$",(1.5,.25),N); label("$3$",(1.5,-.75),N); label("$4$",(2.5,.25),N); label("$5$",(3.5,.25),N); label("$6$",(.5,.25),N); [/asy] $\text{(A)}\ 11 \qquad \text{(B)}\ 12 \qquad \text{(C)}\ 13 \qquad \text{(D)}\ 14 \qquad \text{(E)}\ 15$

Find the maximum value of real number $A$ such that $$3x^2 + y^2 + 1 \geq A(x^2 + xy + x)$$ for all positive integers $x, y.$

[b]p1.[/b] Elsie M. is fixing a watch with three gears. Gear $A$ makes a full rotation every $5$ minutes, gear $B$ makes a full rotation every $8$ minutes, and gear $C$ makes a full rotation every $12$ minutes. The gears continue spinning until all three gears are in their original positions at the same time. How many minutes will it take for the gears to stop spinning? [b]p2.[/b] Optimus has to pick $10$ distinct numbers from the set of positive integers $\{2, 3, 4,..., 29, 30\}$. Denote the numbers he picks by $\{a_1, a_2, ...,a_{10}\}$. What is the least possible value of $$d(a_1 ) + d(a_2) + ... + d(a_{10}),$$ where $d(n)$ denotes the number of positive integer divisors of $n$? For example, $d(33) = 4$ since $1$, $3$, $11$, and $33$ divide $33$. [b]p3.[/b] Michael is given a large supply of both $1\times 3$ and $1\times 5$ dominoes and is asked to arrange some of them to form a $6\times 13$ rectangle with one corner square removed. What is the minimum number of $1\times 3$ dominoes that Michael can use? [img]https://cdn.artofproblemsolving.com/attachments/6/6/c6a3ef7325ecee417e37ec9edb5374aceab9fd.png[/img] [b]p4.[/b] Andy, Ben, and Chime are playing a game. The probabilities that each player wins the game are, respectively, the roots $a$, $b$, and $c$ of the polynomial $x^3 - x^2 + \frac{111}{400}x - \frac{9}{400} = 0$ with $a \le b \le c$. If they play the game twice, what is the probability of the same player winning twice? [b]p5.[/b] TongTong is doodling in class and draws a $3 \times 3$ grid. She then decides to color some (that is, at least one) of the squares blue, such that no two $1 \times 1$ squares that share an edge or a corner are both colored blue. In how many ways may TongTong color some of the squares blue? TongTong cannot rotate or reflect the board. [img]https://cdn.artofproblemsolving.com/attachments/6/0/4b4b95a67d51fda0f155657d8295b0791b3034.png[/img] [b]p6.[/b] Given a positive integer $n$, we define $f(n)$ to be the smallest possible value of the expression $$| \square 1 \square 2 ... \square n|,$$ where we may place a $+$ or a $-$ sign in each box. So, for example, $f(3) = 0$, since $| + 1 + 2 - 3| = 0$. What is $f(1) + f(2) + ... + f(2011)$? [b]p7.[/b] The Duke Men's Basketball team plays $11$ home games this season. For each game, the team has a $\frac34$ probability of winning, except for the UNC game, which Duke has a $\frac{9}{10}$ probability of winning. What is the probability that Duke wins an odd number of home games this season? [b]p8.[/b] What is the sum of all integers $n$ such that $n^2 + 2n + 2$ divides $n^3 + 4n^2 + 4n - 14$? [b]p9.[/b] Let $\{a_n\}^N_{n=1}$ be a finite sequence of increasing positive real numbers with $a_1 < 1$ such that $$a_{n+1} = a_n \sqrt{1 - a^2_1}+ a_1\sqrt{1 - a^2_n}$$ and $a_{10} = 1/2$. What is $a_{20}$? [b]p10.[/b] Three congruent circles are placed inside a unit square such that they do not overlap. What is the largest possible radius of one of these circles? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the locus of points in the plane, the sum of whose distances from the sides of a regular polygon is five times the inradius of the pentagon.

Let $n$  be a positive integer. A row of $n+ 1$ squares is written from left to right, numbered $0, 1, 2, \cdots, n$ Two frogs, named Alphonse and Beryl, begin a race starting at square 0. For each second that passes, Alphonse and Beryl make a jump to the right according to the following rules: if there are at least eight squares to the right of Alphonse, then Alphonse jumps eight squares to the right. Otherwise, Alphonse jumps one square to the right. If there are at least seven squares to the right of Beryl, then Beryl jumps seven squares to the right. Otherwise, Beryl jumps one square to the right. Let A(n) and B(n) respectively denote the number of seconds for Alphonse and Beryl to reach square n. For example, A(40) = 5 and B(40) = 10. (a) Determine an integer n>200 for which $B(n) <A(n)$. (b) Determine the largest integer n for which$ B(n) \le A(n)$.

An HMMT party has $m$ MIT students and $h$ Harvard students for some positive integers $m$ and $h,$ For every pair of people at the party, they are either friends or enemies. If every MIT student has $16$ MIT friends and $8$ Harvard friends, and every Harvard student has $7$ MIT enemies and $10$ Harvard enemies, compute how many pairs of friends there are at the party.

A tetrahedron $ABCD$ is give. A line $\ell$ meets the planes $ABC,BCD,CDA,DAB$ at points $D_0,A_0,B_0,C_0$ respectively. Let $P$ be an arbitrary point not lying on $\ell$ and the planes of the faces, and $A_1,B_1,C_1,D_1$ be the second common points of lines $PA_0,PB_0,PC_0,PD_0$ with the spheres $PBCD,PCDA,PDAB,PABC$ respectively. Prove $P,A_1,B_1,C_1,D_1$ lie on a circle.

Do there exist a infinite sequence of positive integers $(a_{n})$ ,such that for any $n\ge 1$ the relation $ a_{n+2}=\sqrt{a_{n+1}}+a_{n} $?

Find $a^2+b^2$ given that $a, b$ are real and satisfy \[a=b+\frac{1}{a+\frac{1}{b+\frac{1}{a+\cdots}}}; b=a-\frac{1}{b+\frac{1}{a-\frac{1}{b+\cdots}}}\]

The expression $\circ \ 1\ \circ \ 2 \ \circ 3 \ \circ \dots \circ \ 2012$ is written on a blackboard. Catherine places a $+$ sign or a $-$ sign into each blank. She then evaluates the expression, and finds the remainder when it is divided by 2012. How many possible values are there for this remainder? [i]Proposed by Aaron Lin[/i]

The sides of $\triangle ABC$ form an arithmetic sequence of integers. Incircle $I$ is tangent to $AB$, $BC$, and $CA$ at $D$, $E$, and $F$, respectively. Given that $DB = \tfrac32$, $FA = \tfrac12$, find the radius of $I$. $\textbf{(A) } \dfrac12\qquad\textbf{(B) } \dfrac{\sqrt{15}}7\qquad\textbf{(C) } \dfrac{\sqrt{15}}6\qquad\textbf{(D) } \dfrac{2\sqrt{15}}{9}\qquad\textbf{(E) } \dfrac{\sqrt{15}}{4}$

The fraction $\tfrac1{2015}$ has a unique "(restricted) partial fraction decomposition'' of the form \[\dfrac1{2015}=\dfrac a5+\dfrac b{13}+\dfrac c{31},\] where $a$, $b$, and $c$ are integers with $0\leq a<5$ and $0\leq b<13$. Find $a+b$.

On side $ BC $ of the triangle $ ABC, $ consider points $ M,N, $ such that $ MN=\frac{1}{n}BC. $ Let $ L $ on side $ AC $ such that $ ML $ is parallel with $ AB, Q $ on side $ AB $ such that $ NQ $ is parallel with $ AC, $ and $ O $ is the intersection of $ NQ $ with $ ML. $ The parallel of $ BC $ through $ O $ intersects $ AB,BC $ in $ P, $ respectively, $ K. $ Determine the location of $ M $ so that the sum of the areas of the triangles $ OMN, OKL $ and $ OPQ $ is minimum, and calculate this minimum in function of $ n $ and the area of $ ABC. $

a) (version for grades 10-11) Let $P$ be a point lying in the interior of a triangle. Show that the product of the distances from $P$ to the sides of the triangle is at least $8$ times less than the product of the distances from $P$ to the tangents to the circumcircle at the vertices of the triangle. b) (version for grades 8-9) Is it true that for any triangle there exists a point $P$ for which equality in the inequality from a) holds?

Let $a, b, c, d$ be fixed integers with $d$ not divisible by $5$. Assume that $m$ is an integer for which $$am3 +bm2 +cm+d$$ is divisible by $5$. Prove that there exists an integer $n$ for which $$dn3 +cn2 +bn+a$$ is also divisible by $5$.

Let $a_1 < a_2< ...< a_n$ are real numbers, $$f(x) = \sum_{i=1}^n|x-a_i|,$$ for $n$ even. Find the minimum of this function.

Numbers from $1$ to $49$ are randomly placed in a $35 \times 35$ table such that number $i$ is used exactly $i$ times. Some random cells of the table are removed so that table falls apart into several connected (by sides) polygons. Among them, the one with the largest area is chosen (if there are several of the same largest areas, a random one of them is chosen). What is the largest number of cells that can be removed that guarantees that in the chosen polygon there is a number which occurs at least $15$ times?

A connected graph has $1998$ points and each point has degree $3$. If $200$ points, no two of them joined by an edge, are deleted, show that the result is a connected graph.

A [i]binary palindrome[/i] is a positive integer whose standard base 2 (binary) representation is a palindrome (reads the same backward or forward). (Leading zeroes are not permitted in the standard representation.) For example, 2015 is a binary palindrome, because in base 2 it is 11111011111. How many positive integers less than 2015 are binary palindromes?

Find all integers $n$ such that there exists a polynomial $P(x)$ with integer coefficients satisfying $$P(\sqrt[3]{n^2} + \sqrt[3]{ n}) = 2016n + 20\sqrt[3]{n^2} + 16\sqrt[3]{n}$$

Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and the last two digits just happen to be my age." Which of the following is not the age of one of Mr. Jones's children? $ \textbf{(A) } 4 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 8$

The interior of a quadrilateral is bounded by the graphs of $(x+ay)^2 = 4a^2$ and $(ax-y)^2 = a^2$, where $a$ is a positive real number. What is the area of this region in terms of $a$, valid for all $a > 0$? $\textbf{(A)}\ \frac{8a^2}{(a+1)^2}\qquad\textbf{(B)}\ \frac{4a}{a+1}\qquad\textbf{(C)}\ \frac{8a}{a+1}\qquad\textbf{(D)}\ \frac{8a^2}{a^2+1}\qquad\textbf{(E)}\ \frac{8a}{a^2+1}$

We are given in the plane a line $\ell$ and three circles with centres $A,B,C$ such that they are all tangent to $\ell$ and pairwise externally tangent to each other. Prove that the triangle $ABC$ has an obtuse angle and find all possible values of this this angle. [i]Mircea Becheanu[/i]

A triangle with area $60\text{ units}^2$ has vertices with coordinates of $(-15,x)$, $(0,x)$, and $(25,0)$. Find the largest possible value of $x$. $\textbf{(A) } {-}8\qquad\textbf{(B) } {-}4\qquad\textbf{(C) } 4\qquad\textbf{(D) } 8\qquad\textbf{(E) } 16$

2. Let $\Gamma_{1}$ and $\Gamma_{2}$ be externally tangent circles with radii $\frac{1}{2}$ and $\frac{1}{8}$, respectively. The line $\ell$ is a common external tangent to $\Gamma_{1}$ and $\Gamma_{2}$. For $n \geq 3$, we define $\Gamma_{n}$ as the smallest circle tangent to $\Gamma_{n-1}, \Gamma_{n-2}$, and $\ell$. The radius of $\Gamma_{10}$ can be expressed as $\frac{a}{b}$ where $a, b$ are relatively prime positive integers. Find $a+b$.