For polynomials $P(x) = a_nx^n + \cdots + a_0$, let $f(P) = a_n\cdots a_0$ be the product of the coefficients of $P$. The polynomials $P_1,P_2,P_3,Q$ satisfy $P_1(x) = (x-a)(x-b)$, $P_2(x) = (x-a)(x-c)$, $P_3(x) = (x-b)(x-c)$, $Q(x) = (x-a)(x-b)(x-c)$ for some complex numbers $a,b,c$. Given $f(Q) = 8$, $f(P_1) + f(P_2) + f(P_3) = 10$, and $abc>0$, find the value of $f(P_1)f(P_2)f(P_3)$. [i]Proposed by Justin Hsieh[/i]

A triangle has sides of length $\sqrt{13}$, $\sqrt{17}$, and $2 \sqrt{5}$. Compute the area of the triangle.

(a) On $[0, 1]$, let $f(x)$ have a continuous derivative satisfying $0 <f'(x) \leq1$. Also suppose that $f(0) = 0.$ Prove that $$ \left( \int_{0}^{1} f(x)\; dx \right)^{2} \geq \int_{0}^{1} f(x)^{3}\; dx.$$ (b) Show an example in which equality occurs.

Joe and Penny play a game. Initially there are $5000$ stones in a pile, and the two players remove stones from the pile by making a sequence of moves. On the $k$-th move, any number of stones between $1$ and $k$ inclusive may be removed. Joe makes the odd-numbered moves and Penny makes the even-numbered moves. The player who removes the very last stone is the winner. Who wins if both players play perfectly?

[u]Round 1 [/u] [b]p1.[/b] In the Hundred Acre Wood, all the animals are either knights or liars. Knights always tell the truth and liars always lie. One day in the Wood, Winnie-the-Pooh, a knight, decides to visit his friend Rabbit, also a noble knight. Upon arrival, Pooh finds his friend sitting at a round table with $5$ other guests. One-by-one, Pooh asks each person at the table how many of his two neighbors are knights. Surprisingly, he gets the same answer from everybody! "Oh bother!" proclaims Pooh. "I still don't have enough information to figure out how many knights are at this table." "But it's my birthday," adds one of the guests. "Yes, it's his birthday!" agrees his neighbor. Now Pooh can tell how many knights are at the table. Can you? [b]p2.[/b] Harry has an $8 \times 8$ board filled with the numbers $1$ and $-1$, and the sum of all $64$ numbers is $0$. A magical cut of this board is a way of cutting it into two pieces so that the sum of the numbers in each piece is also $0$. The pieces should not have any holes. Prove that Harry will always be able to find a magical cut of his board. (The picture shows an example of a proper cut.) [img]https://cdn.artofproblemsolving.com/attachments/4/b/98dec239cfc757e6f2996eef7876cbfd79d202.png[/img] [b]p3.[/b] Several girls participate in a tennis tournament in which each player plays each other player exactly once. At the end of the tournament, it turns out that each player has lost at least one of her games. Prove that it is possible to find three players $A$, $B$, and $C$ such that $A$ defeated $B$, $B$ defeated $C$, and $C$ defeated $A$. [b]p4.[/b] $120$ bands are participating in this year's Northwest Grunge Rock Festival, and they have $119$ fans in total. Each fan belongs to exactly one fan club. A fan club is called crowded if it has at least $15$ members. Every morning, all the members of one of the crowded fan clubs start arguing over who loves their favorite band the most. As a result of the fighting, each of them leaves the club to join another club, but no two of them join the same one. Is it true that, no matter how the clubs are originally arranged, all these arguments will eventually stop? [b]p5.[/b] In Infinite City, the streets form a grid of squares extending infinitely in all directions. Bonnie and Clyde have just robbed the Infinite City Bank, located at the busiest intersection downtown. Bonnie sets off heading north on her bike, and, $30$ seconds later, Clyde bikes after her in the same direction. They each bike at a constant speed of $1$ block per minute. In order to throw off any authorities, each of them must turn either left or right at every intersection. If they continue biking in this manner, will they ever be able to meet? [u]Round 2 [/u] [b]p6.[/b] In a certain herd of $33$ cows, each cow weighs a whole number of pounds. Farmer Dan notices that if he removes any one of the cows from the herd, it is possible to split the remaining $32$ cows into two groups of equal total weight, $16$ cows in each group. Show that all $33$ cows must have the same weight. [b]p7.[/b] Katniss is thinking of a positive integer less than $100$: call it $x$. Peeta is allowed to pick any two positive integers $N$ and $M$, both less than $100$, and Katniss will give him the greatest common divisor of $x+M$ and $N$ . Peeta can do this up to seven times, after which he must name Katniss' number $x$, or he will die. Can Peeta ensure his survival? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Evaluate the sum $$\sum_{k=0}^{r} {r \choose k}{{12-r} \choose {6-k}} $$

find all polynomials with integer coefficients that $P(\mathbb{Z})= ${$p(a):a\in \mathbb{Z}$} has a Geometric progression.

The number of scalene triangles having all sides of integral lengths, and perimeter less than $ 13$ is: $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 18$

Given the tetrahedron $ABCD$ whose edges $AB$ and $CD$ have lengths $a$ and $b$ respectively. The distance between the skew lines $AB$ and $CD$ is $d$, and the angle between them is $\omega$. Tetrahedron $ABCD$ is divided into two solids by plane $\epsilon$, parallel to lines $AB$ and $CD$. The ratio of the distances of $\epsilon$ from $AB$ and $CD$ is equal to $k$. Compute the ratio of the volumes of the two solids obtained.

Given an acute triangle $ABC$ such that $\angle C< \angle B< \angle A$. Let $I$ be the incenter of $ABC$. Let $M$ be the midpoint of the smaller arc $BC$, $N$ be the midpoint of the segment $BC$ and let $E$ be a point such that $NE=NI$. The line $ME$ intersects circumcircle of $ABC$ at $Q$ (different from $A, B$, and $C$). Prove that [b](i)[/b] The point $Q$ is on the smaller arc $AC$ of circumcircle of $ABC$. [b](ii)[/b] $BQ=AQ+CQ$

Into an angle $A$ of size $a$, a circle is inscribed tangent to its sides at points $B$ and $C$. A line tangent to this circle at a point M meets the segments $AB$ and $AC$ at points $P$ and $Q$ respectively. What is the minimum $a$ such that the inequality $S_{PAQ}<S_{BMC}$ is possible?

Let $ABC$ be a right-angled triangle where ∠$ACB$=90°.Let $CD$ be an altitude of that triangle and points $M$ and $N$ be the midpoints of $CD$ and $BC$, respectively.If $S$ is the circumcenter of the triangle $AMN$, prove that $AS$ and $BC$ are paralel.

A simplified model of a bicycle of mass $M$ has two tires that each comes into contact with the ground at a point. The wheelbase of this bicycle (the distance between the points of contact with the ground) is $w$, and the center of mass of the bicycle is located midway between the tires and a height h above the ground. The bicycle is moving to the right, but slowing down at a constant rate. The acceleration has a magnitude $a$. Air resistance may be ignored. [asy] size(175); pen dps = linewidth(0.7) + fontsize(4); defaultpen(dps); draw(circle((0,0),1),black+linewidth(2.5)); draw(circle((3,0),1),black+linewidth(2.5)); draw((1.5,0)--(0,0)--(1,1.5)--(2.5,1.5)--(1.5,0)--(1,1.5),black+linewidth(1)); draw((3,0)--(2.4,1.8),black+linewidth(1)); filldraw(circle((1.5,2/3),0.05),gray); draw((1.3,1.6)--(0.7,1.6)--(0.7,1.75)--cycle,black+linewidth(1)); label("center of mass of bicycle",(2.5,1.9)); draw((1.55,0.85)--(1.8,1.8),BeginArrow); draw((4.5,-1)--(4.5,2/3),BeginArrow,EndArrow); label("$h$",(4.5,-1/6),E); draw((1.5,2/3)--(4.5,2/3),dotted); draw((0,-1)--(4.5,-1),dotted); draw((0,-5/4)--(3,-5/4),BeginArrow,EndArrow); label("$w$",(3/2,-5/4),S); draw((0,-1)--(0,-6/4),dotted); draw((3,-1)--(3,-6/4),dotted); [/asy] Case 1 ([b][u]Questions 28 - 29[/u][/b]): Assume that the coefficient of sliding friction between each tire and the ground is $\mu$, and that both tires are skidding: sliding without rotating. Express your answers in terms of $w$, $h$, $M$, and $g$. What is the maximum value of $\mu$ so that both tires remain in contact with the ground? $ \textbf{(A)}\ \frac{w}{2h} $ $ \textbf{(B)}\ \frac{h}{2w} $ $ \textbf{(C)}\ \frac{2h}{w} $ $ \textbf{(D)}\ \frac{w}{h}$ $ \textbf{(E)}\ \text{none of the above} $

Astrophysicists have discovered a minor planet of radius $30$ kilometers whose surface is completely covered in water. A spherical meteor hits this planet and is submerged in the water. This incidence causes an increase of $1$ centimeters to the height of the water on this planet. What is the radius of the meteor in meters?

Point $A,B,C,D,E$ lie on a line in this order, such that $BC=CD=\sqrt{AB\cdot DE},$ $P$ doesn't lie on the line, and satisfys that $PB=PD.$ Point $K,L$ lie on the segment $PB,PD,$ respectively, such that $KC$ bisects $\angle BKE,$ and $LC$ bisects $\angle ALD.$ Prove that $A,K,L,E$ are concyclic.

In how many ways can $n^2$ distinct real numbers be arranged into an $n\times n$ array $(a_{ij })$ such that max$_{j}$ min $_i \,\, a_{ij} $= min$_i$ max$_j \,\, a_{ij}$?