Let $m,n\in\mathbb Z_{\ge 0},$ $a_0,a_1,\ldots ,a_m,b_0,b_1,\ldots ,b_n\in\mathbb R_{\ge 0}$ For any integer $0\le k\le m+n,$ define $c_k:=\max_{i+j=k}a_ib_j.$ Proof $$\frac 1{m+n+1}\sum_{k=0}^{m+n}c_k\ge\frac 1{(m+1)(n+1)}\sum_{i=0}^{m}a_i\sum_{j=0}^{n}b_j.$$ [i]Created by Yinghua Ai[/i]

Let $D, E$, and $F$ respectively be the feet of the altitudes from $A, B$, and $C$ of acute triangle $\vartriangle ABC$ such that $AF = 28, FB = 35$ and $BD = 45$. Let $P$ be the point on segment $BE$ such that $AP = 42$. Find the length of $CP$.

Let $A$ be a set with $1000$ members and $\mathcal F =${$A_1,A_2,\ldots,A_n$} a family of subsets of A such that (a) Each element in $\mathcal F$ consists of 3 members (b) For every five elements in $\mathcal F$, the union of them all will have at least $12$ members Find the largest value of $n$

Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $$(y + 1)f(x) + f(xf(y) + f(x + y))= y$$ for all $x, y \in \mathbb{R}$.

Suppose that the function $h:\mathbb{R}^2\to\mathbb{R}$ has continuous partial derivatives and satisfies the equation \[h(x,y)=a\frac{\partial h}{\partial x}(x,y)+b\frac{\partial h}{\partial y}(x,y)\] for some constants $a,b.$ Prove that if there is a constant $M$ such that $|h(x,y)|\le M$ for all $(x,y)$ in $\mathbb{R}^2,$ then $h$ is identically zero.

Problem 19 The diagram below shows a 24×24 square ABCD. Points E and F lie on sides AD and CD, respectively, so that DE = DF = 8. Set X consists of the shaded triangle ABC with its interior, while set Y consists of the shaded triangle DEF with its interior. Set Z consists of all the points that are midpoints of segments connecting a point in set X with a point in set Y . That is, Z = {z | z is the midpoint of xy for x ∈ X and y ∈ Y}. Find the area of the set Z. For diagram to http://www.purplecomet.org/welcome/practice

$ ABCD$ is a quadrilateral with $ BC$ parallel to $ AD$. $ M$ is the midpoint of $ CD$, $ P$ is the midpoint of $ MA$ and $ Q$ is the midpoint of $ MB$. The lines $ DP$ and $ CQ$ meet at $ N$. Prove that $ N$ is inside the quadrilateral $ ABCD$.

Let $ABC$ be an equilateral triangle of side $1$, $D$ be a point on $BC$, and $r_1, r_2$ be the inradii of triangles $ABD$ and $ADC$. Express $r_1r_2$ in terms of $p = BD$ and find the maximum of $r_1r_2$.

Let $a$ be a real number. 1) Prove that there is a triangle with side lengths $\sqrt{a^2-a + 1}$, $\sqrt{a^2+a + 1}$, and $\sqrt{4a^2 + 3}$. 2) Prove that the area of this triangle does not depend on $a$.

Let $\{a_k\}_{k\geq 1}$ be a sequence of non-negative integers, such that $a_k \geq a_{2k} + a_{2k+1}$, for all $k\geq 1$. a) Prove that for all positive integers $n\geq 1$ there exist $n$ consecutive terms equal with 0 in the sequence $\{a_k\}_k$; b) State an example of sequence with the property in the hypothesis which contains an infinite number of non-zero terms.

Find all pairs of positive integers $(a, b)$, such that $S(a^{b+1})=a^b$, where $S(m)$ denotes the digit sum of $m$.

If $N$ is the number of ways to place $16$ [i]jumping [/i]rooks on an $8 \times 8$ chessboard such that each rook attacks exactly two other rooks, find the remainder when $N$ is divided by $1000$. (A jumping rook is said to [i]attack [/i]a square if the square is in the same row or in the same column as the rook.)

Consider a coordinate system in the plane, with the origin $O$. We call a lattice point $A{}$ [i]hidden[/i] if the open segment $OA$ contains at least one lattice point. Prove that for any positive integer $n$ there exists a square of side-length $n$ such that any lattice point lying in its interior or on its boundary is hidden.

Let $ABC$ be a scalene triangle with $\angle A = 60^{\circ}$. Let $E$ and $F$ be the feet of the angle bisectors of $\angle ABC$ and $\angle ACB$, respectively, and let $I$ be the incenter of $\triangle ABC$. Let $P,Q$ be distinct points such that $\triangle PEF$ and $\triangle QEF$ are equilateral. If $O$ is the circumcenter of of $\triangle APQ$, show that $\overline{OI}\perp \overline{BC}$. [i]Proposed by Vincent Huang

Prove that there exists a convex 1990-gon with the following two properties : [b]a.)[/b] All angles are equal. [b]b.)[/b] The lengths of the 1990 sides are the numbers $ 1^2$, $ 2^2$, $ 3^2$, $ \cdots$, $ 1990^2$ in some order.

The sum of $3$ real numbers is known to be zero. If the sum of their cubes is $\pi^e$, what is their product equal to?

Suppose that $S$ is a finite set of real numbers with the property that any two distinct elements of $S$ form an arithmetic progression with another element in $S$. Give an example of such a set with 5 elements and show that no such set exists with more than $5$ elements.

Let $n\geq 2$ be a natural number. A unit square is removed from a square $n$ x $n$ and the remaining figure is cut into squares 2 x 2 and 3 x 3. Determine all possible values of $n$.

Let $C$ denote the curve $y^2 =\frac{x(x+1)(2x+1)}{6}$. The points $(1/2, a)$,$(b, c)$, and $(24, d)$ lie on $C$ and are collinear, and $ad < 0$. Given that $b, c$ are rational numbers, find $100b^2 + c^2$.

[u]Round 1[/u] [b]p1.[/b] Alice has a pizza with eight slices. On each slice, she either adds only salt, only pepper, or leaves it plain. Determine how many ways there are for Alice to season her entire pizza. [b]p2.[/b] Call a number almost prime if it has exactly three divisors. Find the number of almost prime numbers less than $100$. [b]p3.[/b] Determine the maximum number of points of intersection between a circle and a regular pentagon. [u]Round 2[/u] [b]p4.[/b] Let $d(n)$ denote the number of positive integer divisors of $n$. Find $d(d(20^{18}))$. [b]p5.[/b] $20$ chubbles are equal to $19$ flubbles. $20$ flubbles are equal to $18$ bubbles. How many bubbles are $1000$ chubbles worth? [b]p6.[/b] Square $ABCD$ and equilateral triangle $EFG$ have equal area. Compute $\frac{AB}{EF}$ . [u]Round 3[/u] [b]p7.[/b] Find the minimumvalue of $y$ such that $y = x^2 -6x -9$ where x is a real number. [b]p8.[/b] I have $2$ pairs of red socks, $5$ pairs of white socks, and $7$ pairs of blue socks. If I randomly pull out one sock at a time without replacement, how many socks do I need to draw to guarantee that I have drawn $3$ pairs of socks of the same color? [b]p9. [/b]There are $23$ paths from my house to the school, $29$ paths from the school to the library, and $3$ paths fromthe library to town center. Additionally, there are $6$ paths directly from my house to the library. If I have to pass through the library to get to town center, howmany ways are there to travel from my house all the way to the town center? [u]Round 4[/u] [b]p10.[/b] A circle of radius $25$ and a circle of radius $4$ are externally tangent. A line is tangent to the circle of radius $25$ at $A$ and the circle of radius $4$ at $B$, where $A \ne B$. Compute the length of $AB$. [b]p11.[/b] A gambler spins two wheels, one numbered $1$ to $4$ and another numbered $1$ to $5$, and the amount of money he wins is the sum of the two numbers he spins in dollars. Determine the expected amount of money he will win. [b]p12.[/b] Find the remainder when $20^{19}$ is divided by $18$. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3166012p28809547]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166099p28810427]here[/url].Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Alex the Kat has written $61$ problems for a math contest, and there are a total of $187$ problems submitted. How many more problems does he need to write (and submit) before he has written half of the total problems?

Let $ A_{1}A_{2}A_{3}A_{4}A_{5}$ be a convex pentagon, such that \[ [A_{1}A_{2}A_{3}] \equal{} [A_{2}A_{3}A_{4}] \equal{} [A_{3}A_{4}A_{5}] \equal{} [A_{4}A_{5}A_{1}] \equal{} [A_{5}A_{1}A_{2}].\] Prove that there exists a point $ M$ in the plane of the pentagon such that \[ [A_{1}MA_{2}] \equal{} [A_{2}MA_{3}] \equal{} [A_{3}MA_{4}] \equal{} [A_{4}MA_{5}] \equal{} [A_{5}MA_{1}].\] Here $ [XYZ]$ stands for the area of the triangle $ \Delta XYZ$.

Let $\alpha_1,\alpha_2,\alpha_3$ be three congruent circles that are tangent to each other. A third circle $\beta$ is tangent to them at points $A_1,A_2,A_3$ respectively. Let $P$ be a point on $\beta$ which is different from $A_1,A_2,A_3$. For $i=1,2,3$, let $B_i$ be the second intersection point of the line $PA_i$ with circle $\alpha_i$. Prove that $\Delta B_1B_2B_3$ is equilateral.

Two graphs $G_1$ and $G_2$ of quadratic polynomials intersect at points $A$ and $B$. Let $O$ be the vertex of $G_1$. Lines $OA$ and $OB$ intersect $G_2$ again at points $C$ and $D$. Prove that $CD$ is parallel to the $x$-axis.

In the diagram below, $AX$ is parallel to $BY$, $AB$ is perpendicular to $BY$, and $AZB$ is an isosceles right triangle. If $AB = 7$ and $XY =25$, what is the length of $AX$? [asy] size(7cm); draw((0,0)--(0,7)--(17/2,7)--(-31/2,0)--(0,0)); draw((0,0)--(-7/2,7/2)--(0,7)); label("$B$",(0,0),S); label("$Y$",(-31/2,0),S); label("$A$",(0,7),N); label("$X$",(17/2,7),N); label("$Z$",(-7/2,7/2),N); [/asy] $\textbf{(A) }\frac{17}{3}\qquad\textbf{(B) }\frac{17}{2}\qquad\textbf{(C) }9\qquad\textbf{(D) }\frac{51}{4}\qquad\textbf{(E) }12$