A paint brush is swept along both diagonals of a square to produce the symmetric painted area, as shown. Half the area of the square is painted. What is the ratio of the side length of the square to the brush width? [asy]unitsize(15mm); defaultpen(linewidth(.8pt)); path P=(-sqrt(2)/2,1)--(0,1-sqrt(2)/2)--(sqrt(2)/2,1); path Pc=(-sqrt(2)/2,1)--(0,1-sqrt(2)/2)--(sqrt(2)/2,1)--cycle; path S=(-1,-1)--(-1,1)--(1,1)--(1,-1)--cycle; fill(S,gray); for(int i=0;i<4;++i) { fill(rotate(90*i)*Pc,white); draw(rotate(90*i)*P); } draw(S);[/asy]$ \textbf{(A)}\ 2\sqrt {2} \plus{} 1 \qquad \textbf{(B)}\ 3\sqrt {2}\qquad \textbf{(C)}\ 2\sqrt {2} \plus{} 2 \qquad \textbf{(D)}\ 3\sqrt {2} \plus{} 1 \qquad \textbf{(E)}\ 3\sqrt {2} \plus{} 2$

In the triangle $ABC, \angle BAC$ is acute, the angle bisector of $\angle BAC$ meets $BC$ at $D, K$ is the foot of the perpendicular from $B$ to $AC$, and $\angle ADB = 45^o$. Point $P$ lies between $K$ and $C$ such that $\angle KDP = 30^o$. Point $Q$ lies on the ray $DP$ such that $DQ = DK$. The perpendicular at $P$ to $AC$ meets $KD$ at $L$. Prove that $PL^2 = DQ \cdot PQ$.

Let $O$ be the circumcenter of the acute triangle $ABC$ ($AB < AC$). Let $A_1$ and $P$ be the feet of the perpendicular lines drawn from $A$ and $O$ to $BC$, respectively. The lines $BO$ and $CO$ intersect $AA_1$ in $D$ and $E$, respectively. Let $F$ be the second intersection point of $\odot ABD$ and $\odot ACE$. Prove that the angle bisector od $\angle FAP$ passes through the incenter of $\triangle ABC$.

From a point $P$ outside of a circle with centre $O$, tangent segments $\overline{PA}$ and $\overline{PB}$ are drawn. If $\frac1{\left|\overline{OA}\right|^2}+\frac1{\left|\overline{PA}\right|^2}=\frac1{16}$, then $\left|\overline{AB}\right|=$? $\textbf{(A)}~4$ $\textbf{(B)}~6$ $\textbf{(C)}~8$ $\textbf{(D)}~10$

Given a triangle $ ABC$. Let $ G$, $ I$, $ H$ be the centroid, the incenter and the orthocenter of triangle $ ABC$, respectively. Prove that $ \angle GIH > 90^{\circ}$.

Let $A_{1}$, $B_{1}$, $C_{1}$ be the feet of the altitudes of an acute-angled triangle $ABC$ issuing from the vertices $A$, $B$, $C$, respectively. Let $K$ and $M$ be points on the segments $A_{1}C_{1}$ and $B_{1}C_{1}$, respectively, such that $\measuredangle KAM = \measuredangle A_{1}AC$. Prove that the line $AK$ is the angle bisector of the angle $C_{1}KM$.

[b]p1.[/b] What is $7\%$ of one half of $11\%$ of $20000$ ? [b]p2.[/b] Three circles centered at $A, B$, and $C$ are tangent to each other. Given that $AB = 8$, $AC = 10$, and $BC = 12$, find the radius of circle $ A$. [b]p3. [/b]How many positive integer values of $x$ less than $2012$ are there such that there exists an integer $y$ for which $\frac{1}{x} +\frac{2}{2y+1} =\frac{1}{y}$ ? [b]p4. [/b]The positive difference between $ 8$ and twice $x$ is equal to $11$ more than $x$. What are all possible values of $x$? [b]p5.[/b] A region in the coordinate plane is bounded by the equations $x = 0$, $x = 6$, $y = 0$, and $y = 8$. A line through $(3, 4)$ with slope $4$ cuts the region in half. Another line going through the same point cuts the region into fourths, each with the same area. What is the slope of this line? [b]p6.[/b] A polygon is composed of only angles of degrees $138$ and $150$, with at least one angle of each degree. How many sides does the polygon have? [b]p7.[/b] $M, A, T, H$, and $L$ are all not necessarily distinct digits, with $M \ne 0$ and $L \ne 0$. Given that the sum $MATH +LMT$, where each letter represents a digit, equals $2012$, what is the average of all possible values of the three-digit integer $LMT$? [b]p8. [/b]A square with side length $\sqrt{10}$ and two squares with side length $\sqrt{7}$ share the same center. The smaller squares are rotated so that all of their vertices are touching the sides of the larger square at distinct points. What is the distance between two such points that are on the same side of the larger square? [b]p9.[/b] Consider the sequence $2012, 12012, 20120, 20121, ...$. This sequence is the increasing sequence of all integers that contain “$2012$”. What is the $30$th term in this sequence? [b]p10.[/b] What is the coefficient of the $x^5$ term in the simplified expansion of $(x +\sqrt{x} +\sqrt[3]{x})^{10}$ ? PS. You had better use hide for answers.

Points $X$ and $Y$ inside the rhombus $ABCD$ are such that $Y$ is inside the convex quadrilateral $BXDC$ and $2\angle XBY = 2\angle XDY = \angle ABC$. Prove that the lines $AX$ and $CY$ are parallel. [i]S. Berlov[/i]

Let $ABC$ be a triangle. Prove that there exists a point $D$ on the side $AB$ of the triangle $ABC$, such that $CD$ is the geometric mean of $AD$ and $DB$, iff the triangle $ABC$ satisfies the inequality $\sin A\sin B\le\sin^2\frac{C}{2}$. [hide="Comment"][i]Alternative formulation, from IMO ShortList 1974, Finland 2:[/i] We consider a triangle $ABC$. Prove that: $\sin(A) \sin(B) \leq \sin^2 \left( \frac{C}{2} \right)$ is a necessary and sufficient condition for the existence of a point $D$ on the segment $AB$ so that $CD$ is the geometrical mean of $AD$ and $BD$.[/hide]

[b]p1. [/b]Ed, Fred and George are playing on a see-saw that is slightly off center. When Ed sits on the left side and George, who weighs $100$ pounds, on the right side, they are perfectly balanced. Similarly, if Fred, who weighs $400$ pounds, sits on the left and Ed sits on the right, they are also perfectly balanced. Assuming the see-saw has negligible weight, what is the weight of Ed, in pounds? [b]p2.[/b] How many digits does the product $2^{42}\cdot 5^{38}$ have? [b]p3.[/b] Square $ABCD$ has equilateral triangles drawn external to each side, as pictured. If each triangle is folded upwards to meet at a point $E$, then a square pyramid can be made. If the center of square $ABCD$ is $O$, what is the measure of $\angle OEA$? [img]https://cdn.artofproblemsolving.com/attachments/9/a/39c0096ace5b942a9d3be1eafe7aa7481fbb9f.png[/img] [b]p4.[/b] How many solutions $(x, y)$ in the positive integers are there to $3x + 7y = 1337$ ? [b]p5.[/b] A trapezoid with height $12$ has legs of length $20$ and $15$ and a larger base of length $42$. What are the possible lengths of the other base? [b]p6.[/b] Let $f(x) = 6x + 7$ and $g(x) = 7x + 6$. Find the value of a such that $g^{-1}(f^{-1}(g(f(a)))) = 1$. [b]p7.[/b] Billy and Cindy want to meet at their favorite restaurant, and they have made plans to do so sometime between $1:00$ and $2:00$ this Sunday. Unfortunately, they didn’t decide on an exact time, so they both decide to arrive at a random time between $1:00$ and $2:00$. Silly Billy is impatient, though, and if he has to wait for Cindy, he will leave after $15$ minutes. Cindy, on the other hand, will happily wait for Billy from whenever she arrives until $2:00$. What is the probability that Billy and Cindy will be able to dine together? [b]p8.[/b] As pictured, lines are drawn from the vertices of a unit square to an opposite trisection point. If each triangle has legs with ratio $3 : 1$, what is the area of the shaded region? [img]https://cdn.artofproblemsolving.com/attachments/e/9/35a6340018edcddfcd7e085f8f6e56686a8e07.png[/img] [b]p9.[/b] For any positive integer $n$, let $f_1(n)$ denote the sum of the squares of the digits of $n$. For $k \ge 2$, let $f_k(n) = f_{k-1}(f_1(n))$. Then, $f_1(5) = 25$ and $f_3(5) = f_2(25) = 85$. Find $f_{2012}(15)$. [b]p10.[/b] Given that $2012022012$ has $ 8$ distinct prime factors, find its largest prime factor. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

When all vertex angles of a convex polygon are equal, call it equiangular. Prove that $ p > 2$ is a prime number, if and only if the lengths of all sides of equiangular $ p$ polygon are rational numbers, it is a regular $ p$ polygon.

The points $M$ and $N$ are chosen on the angle bisector $AL$ of a triangle $ABC$ such that $\angle ABM=\angle ACN=23^{\circ}$. $X$ is a point inside the triangle such that $BX=CX$ and $\angle BXC=2\angle BML$. Find $\angle MXN$.

Consider a triangle $ABC$, inscribed in $O$ and $\angle A < \angle B$. Some point $P$ outside the circle satisfies $$\angle A=\angle PBA =180^{\circ}- \angle PCB$$ Let $D$ be the intersection of line $PB$ and $O$(different from $B$), and $Q$ the intersection of the tangent line of $O$ passing through $A$ and line $CD$. Show that $CQ : AB=AQ^2:AD^2$.

Let $ ABC$ be an isoceles triangle with $ AC\equal{}BC$. A point $ P$ lies inside $ ABC$ such that \[ \angle PAB \equal{} \angle PBC, \angle PAC \equal{} \angle PCB.\] Let $ M$ be the midpoint of $ AB$ and $ K$ be the intersection of $ BP$ and $ AC$. Prove that $ AP$ and $ PK$ trisect $ \angle MPC$.

Let $ABC$ be a triangle. The incircle of triangle $ABC$ touches the side $BC$ at $A^{\prime}$, and the line $AA^{\prime}$ meets the incircle again at a point $P$. Let the lines $CP$ and $BP$ meet the incircle of triangle $ABC$ again at $N$ and $M$, respectively. Prove that the lines $AA^{\prime}$, $BN$ and $CM$ are concurrent.

[u]Round 1[/u] [b]1.1.[/b] Suppose a certain menu has $3$ sandwiches and $5$ drinks. How many ways are there to pick a meal so that you have exactly a drink and a sandwich? [b]1.2.[/b] If $a + b = 4$ and $a + 3b = 222222$, find $10a + b$. [b]1.3.[/b] Compute $$\left\lfloor \frac{2019 \cdot 2017}{2018} \right\rfloor $$ where $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$. [u]Round 2[/u] [b]2.1.[/b] Andrew has $10$ water bottles, each of which can hold at most $10$ cups of water. Three bottles are thirty percent filled, five are twenty-four percent filled, and the rest are empty. What is the average amount of water, in cups, contained in the ten water bottles? [b]2.2.[/b] How many positive integers divide $195$ evenly? [b]2.3.[/b] Square $A$ has side length $\ell$ and area $128$. Square $B$ has side length $\ell/2$. Find the length of the diagonal of Square $B$. [u]Round 3[/u] [b]3.1.[/b] A right triangle with area $96$ is inscribed in a circle. If all the side lengths are positive integers, what is the area of the circle? Express your answer in terms of $\pi$. [b]3.2.[/b] A circular spinner has four regions labeled $3, 5, 6, 10$. The region labeled $3$ is $1/3$ of the spinner, $5$ is $1/6$ of the spinner, $6$ is $1/10$ of the spinner, and the region labeled $10$ is $2/5$ of the spinner. If the spinner is spun once randomly, what is the expected value of the number on which it lands? [b]3.3.[/b] Find the integer k such that $k^3 = 8353070389$ [u]Round 4[/u] [b]4.1.[/b] How many ways are there to arrange the letters in the word [b]zugzwang [/b] such that the two z’s are not consecutive? [b]4.2.[/b] If $O$ is the circumcenter of $\vartriangle ABC$, $AD$ is the altitude from $A$ to $BC$, $\angle CAB = 66^o$ and $\angle ABC = 44^o$, then what is the measure of $\angle OAD$ ? [b]4.3.[/b] If $x > 0$ satisfies $x^3 +\frac{1}{x^3} = 18$, find $x^5 +\frac{1}{x^5}$ [u]Round 5[/u] [b]5.1.[/b] Let $C$ be the answer to Question $3$. Neethen decides to run for school president! To be entered onto the ballot, however, Neethen needs $C + 1$ signatures. Since no one else will support him, Neethen gets the remaining $C$ other signatures through bribery. The situation can be modeled by $k \cdot N = 495$, where $k$ is the number of dollars he gives each person, and $N$ is the number of signatures he will get. How many dollars does Neethen have to bribe each person with to get exactly C signatures? [b]5.2.[/b] Let $A$ be the answer to Question $1$. With $3A - 1$ total votes, Neethen still comes short in the election, losing to Serena by just $1$ vote. Darn! Neethen sneaks into the ballot room, knowing that if he destroys just two ballots that voted for Serena, he will win the election. How many ways can Neethen choose two ballots to destroy? [b]5.3.[/b] Let $B$ be the answer to Question $2$. Oh no! Neethen is caught rigging the election by the principal! For his punishment, Neethen needs to run the perimeter of his school three times. The school is modeled by a square of side length $k$ furlongs, where $k$ is an integer. If Neethen runs $B$ feet in total, what is $k + 1$? (Note: one furlong is $1/8$ of a mile). [u]Round 6[/u] [b]6.1.[/b] Find the unique real positive solution to the equation $x =\sqrt{6 + 2\sqrt6 + 2x}- \sqrt{6 - 2\sqrt6 - 2x} -\sqrt6$. [b]6.2.[/b] Consider triangle ABC with $AB = 13$ and $AC = 14$. Point $D$ lies on $BC$, and the lengths of the perpendiculars from $D$ to $AB$ and $AC$ are both $\frac{56}{9}$. Find the largest possible length of $BD$. [b]6.3.[/b] Let $f(x, y) = \frac{m}{n}$, where $m$ is the smallest positive integer such that $x$ and $y$ divide $m$, and $n$ is the largest positive integer such that $n$ divides both $x$ and $y$. If $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$, what is the median of the distinct values that $f(a, b)$ can take, where $a, b \in S$? [u]Round 7[/u] [b]7.1.[/b] The polynomial $y = x^4 - 22x^2 - 48x - 23$ can be written in the form $$y = (x - \sqrt{a} - \sqrt{b} - \sqrt{c})(x - \sqrt{a} +\sqrt{b} +\sqrt{c})(x +\sqrt{a} -\sqrt{b} +\sqrt{c})(x +\sqrt{a} +\sqrt{b} -\sqrt{c})$$ for positive integers $a, b, c$ with $a \le b \le c$. Find $(a + b)\cdot c$. [b]7.2.[/b] Varun is grounded for getting an $F$ in every class. However, because his parents don’t like him, rather than making him stay at home they toss him onto a number line at the number $3$. A wall is placed at $0$ and a door to freedom is placed at $10$. To escape the number line, Varun must reach 10, at which point he walks through the door to freedom. Every $5$ minutes a bell rings, and Varun may walk to a different number, and he may not walk to a different number except when the bell rings. Being an $F$ student, rather than walking straight to the door to freedom, whenever the bell rings Varun just randomly chooses an adjacent integer with equal chance and walks towards it. Whenever he is at $0$ he walks to $ 1$ with a $100$ percent chance. What is the expected number of times Varun will visit $0$ before he escapes through the door to freedom? [b]7.3.[/b] Let $\{a_1, a_2, a_3, a_4, a_5, a_6\}$ be a set of positive integers such that every element divides $36$ under the condition that $a_1 < a_2 <... < a_6$. Find the probability that one of these chosen sets also satisfies the condition that every $a_i| a_j$ if $i|j$. [u]Round 8[/u] [b]8.[/b] How many numbers between $1$ and $100, 000$ can be expressed as the product of at most $3$ distinct primes? Your answer will be scored according to the following formula, where $X$ is the correct answer and $I$ is your input. $$max \left\{ 0, \left\lceil min \left\{13 - \frac{|I-X|}{0.1 |I|}, 13 - \frac{|I-X|}{0.1 |I-2X|} \right\} \right\rceil \right\}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Inside triangle $ABC$, point $P$ is taken so that angles $\angle ARB= \angle BPC = \angle CPA= 120^o$. Lines $BP$ and $CP$ intersect lines $AC$ and $AB$ at points $M$ and $K$. It is known that the quadrilateral $AMPK$ has same areq with the triangle $BCP$. What is the angle $\angle BAC$?

In a forest there are $5$ trees $A, B, C, D, E$ that are in that order on a straight line. At the midpoint of $AB$ there is a daisy, at the midpoint of $BC$ there is a rose bush, at the midpoint of $CD$ there is a jasmine, and at the midpoint of $DE$ there is a carnation. The distance between $A$ and $E$ is $28$ m; the distance between the daisy and the carnation is $20$ m. Calculate the distance between the rose bush and the jasmine.

Let $A_1B_1C_1D_1A_2B_2C_2D_2$ be a unit cube, with $A_1B_1C_1D_1$ and $A_2B_2C_2D_2$ opposite square faces, and let $M$ be the center of face $A_2 B_2 C_2 D_2$. Rectangular pyramid $MA_1B_1C_1D_1$ is cut out of the cube. If the surface area of the remaining solid can be expressed in the form $a + \sqrt{b}$, where $a$ and $b$ are positive integers and $b$ is not divisible by the square of any prime, find $a+b$. [i]Author: Alex Zhu[/i]

For a complex number $ z\equal{}x\plus{}iy$ we denote by $ P(z)$ the corresponding point $ (x,y)$ in the plane. Suppose $ z_1,z_2,z_3,z_4,z_5,\alpha$ are nonzero complex numbers such that: $ (i)$ $ P(z_1),...,P(z_5)$ are vertices of a complex pentagon $ Q$ containing the origin $ O$ in its interior, and $ (ii)$ $ P(\alpha z_1),...,P(\alpha z_5)$ are all inside $ Q$. If $ \alpha\equal{}p\plus{}iq$ $ (p,q \in \mathbb{R})$, prove that $ p^2\plus{}q^2 \le 1$ and $ p\plus{}q \tan \frac{\pi}{5} \le 1$.

In $\Delta ABC$, $AC=kAB$, with $k>1$. The internal angle bisector of $\angle BAC$ meets $BC$ at $D$. The circle with $AC$ as diameter cuts the extension of $AD$ at $E$. Express $\dfrac{AD}{AE}$ in terms of $k$.

Let $D_1$, $D_2$, ..., $D_n$ be closed discs in the plane. (A closed disc is the region limited by a circle, taken jointly with this circle.) Suppose that every point in the plane is contained in at most $2003$ discs $D_i$. Prove that there exists a disc $D_k$ which intersects at most $7\cdot 2003 - 1 = 14020$ other discs $D_i$.

The internal bisector of angle $A$ in a triangle $ABC$ with $AC > AB$ meets the circumcircle $\Gamma$ of the triangle in $D$. Join$D$ to the center $O$ of the circle $\Gamma$ and suppose that $DO$ meets $AC$ in $E$, possibly when extended. Given that $BE$ is perpendicular to $AD$, show that $AO$ is parallel to $BD$.

In a multiplication table, the entry in the $i$-th row and the $j$-th column is the product $ij$ From an $m\times n$ subtable with both $m$ and $n$ odd, the interior $(m-2) (n-2)$ rectangle is removed, leaving behind a frame of width $1$. The squares of the frame are painted alternately black and white. Prove that the sum of the numbers in the black squares is equal to the sum of the numbers in the white squares.

[asy] size(200); dotfactor=3; pair A=(0,0),B=(1,0),C=(2,0),D=(3,0),X=(1.2,0.7); draw(A--D); dot(A);dot(B);dot(C);dot(D); draw(arc((0.4,0.4),0.4,180,110),arrow = Arrow(TeXHead)); draw(arc((2.6,0.4),0.4,0,70),arrow = Arrow(TeXHead)); draw(B--X,dotted); draw(C--X,dotted); label("$A$",A,SW); label("$B$",B,S); label("$C$",C,S); label("$D$",D,S); label("x",X,fontsize(5pt)); //Credit to TheMaskedMagician for the diagram [/asy] Points $A , B, C$, and $D$ are distinct and lie, in the given order, on a straight line. Line segments $AB, AC$, and $AD$ have lengths $x, y$, and $z$ , respectively. If line segments $AB$ and $CD$ may be rotated about points $B$ and $C$, respectively, so that points $A$ and $D$ coincide, to form a triangle with positive area, then which of the following three inequalities must be satisfied? $\textbf{I. }x<\frac{z}{2}\qquad\textbf{II. }y<x+\frac{z}{2}\qquad\textbf{III. }y<\frac{z}{2}$ $\textbf{(A) }\textbf{I. }\text{only}\qquad\textbf{(B) }\textbf{II. }\text{only}\qquad$ $\textbf{(C) }\textbf{I. }\text{and }\textbf{II. }\text{only}\qquad\textbf{(D) }\textbf{II. }\text{and }\textbf{III. }\text{only}\qquad\textbf{(E) }\textbf{I. },\textbf{II. },\text{and }\textbf{III. }$