Let $n$ be a positive integer, and $a_1,...,a_n, b_1,..., b_n$ be $2n$ positive real numbers such that $a_1 + ... + a_n = b_1 + ... + b_n = 1$. Find the minimal value of $ \frac {a_1^2} {a_1 + b_1} + \frac {a_2^2} {a_2 + b_2} + ...+ \frac {a_n^2} {a_n + b_n}$.

[b]p1.[/b] How many lines of symmetry does a square have? [b]p2.[/b] Compute$ 1/2 + 1/6 + 1/12 + 1/4$. [b]p3.[/b] What is the maximum possible area of a rectangle with integer side lengths and perimeter $8$? [b]p4.[/b] Given that $1$ printer weighs $400000$ pennies, and $80$ pennies weighs $2$ books, what is the weight of a printer expressed in books? [b]p5.[/b] Given that two sides of a triangle are $28$ and $3$ and all three sides are integers, what is the sum of the possible lengths of the remaining side? [b]p6.[/b] What is half the sum of all positive integers between $1$ and $15$, inclusive, that have an even number of positive divisors? [b]p7.[/b] Austin the Snowman has a very big brain. His head has radius $3$, and the volume of his torso is one third of his head, and the volume of his legs combined is one third of his torso. If Austin's total volume is $a\pi$ where $a$ is an integer, what is $a$? [b]p8.[/b] Neethine the Kiwi says that she is the eye of the tiger, a fighter, and that everyone is gonna hear her roar. She is standing at point $(3, 3)$. Neeton the Cat is standing at $(11,18)$, the farthest he can stand from Neethine such that he can still hear her roar. Let the total area of the region that Neeton can stand in where he can hear Neethine's roar be $a\pi$ where $a$ is an integer. What is $a$? [b]p9.[/b] Consider $2018$ identical kiwis. These are to be divided between $5$ people, such that the first person gets $a_1$ kiwis, the second gets $a_2$ kiwis, and so forth, with $a_1 \le a_2 \le a_3 \le a_4 \le a_5$. How many tuples $(a_1, a_2, a_3, a_4, a_5)$ can be chosen such that they form an arithmetic sequence? [b]p10.[/b] On the standard $12$ hour clock, each number from $1$ to $12$ is replaced by the sum of its divisors. On this new clock, what is the number of degrees in the measure of the non-reflex angle between the hands of the clock at the time when the hour hand is between $7$ and $6$ while the minute hand is pointing at $15$? [b]p11.[/b] In equiangular hexagon $ABCDEF$, $AB = 7$, $BC = 3$, $CD = 8$, and $DE = 5$. The area of the hexagon is in the form $\frac{a\sqrt{b}}{c}$ with $b$ square free and $a$ and $c$ relatively prime. Find $a+b+c$ where $a, b,$ and $c$ are integers. [b]p12.[/b] Let $\frac{p}{q} = \frac15 + \frac{2}{5^2} + \frac{3}{5^3} + ...$ . Find $p + q$, where $p$ and $q$ are relatively prime positive integers. [b]p13.[/b] Two circles $F$ and $G$ with radius $10$ and $4$ respectively are externally tangent. A square $ABMC$ is inscribed in circle $F$ and equilateral triangle $MOP$ is inscribed in circle $G$ (they share vertex $M$). If the area of pentagon $ABOPC$ is equal to $a + b\sqrt{c}$, where $a$, $b$, $c$ are integers $c$ is square free, then find $a + b + c$. [b]p14.[/b] Consider the polynomial $P(x) = x^3 + 3x^2 + ax + 8$. Find the sum of all integer $a$ such that the sum of the squares of the roots of $P(x)$ divides the sum of the coecients of $P(x)$. [b]p15.[/b] Nithin and Antonio play a number game. At the beginning of the game, Nithin picks a prime $p$ that is less than $100$. Antonio then tries to find an integer $n$ such that $n^6 + 2n^5 + 2n^4 + n^3 + (n^2 + n + 1)^2$ is a multiple of $p$. If Antonio can find such a number n, then he wins, otherwise, he loses. Nithin doesn't know what he is doing, and he always picks his prime randomly while Antonio always plays optimally. The probability of Antonio winning is $a/b$ where $a$ and $b$ are relatively prime positive integers. Find$a + b$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Wendy eats sushi for lunch. She wants to eat six pieces of sushi arranged in a $23$ rectangular grid, but sushi is sticky, and Wendy can only eat a piece if it is adjacent to (not counting diagonally) at most two other pieces. In how many orders can Wendy eat the six pieces of sushi, assuming that the pieces of sushi are distinguishable?

As shown in the figure, it is known that $BC = AC$ in $ABC$, $M$ is the midpoint of $AB$, points $D$ and $E$ lie on $AB$ satisfying $\angle DCE = \angle MCB$, the circumscribed circle of $\vartriangle BDC$ and the circumscribed circle of $\vartriangle AEC$ intersect at point $F$ (different from point $C$), point $H$ lies on $AB$ such that the straight line $CM$ bisects the line segment $HF$. Let the circumcenters of $\vartriangle HFE$ and $\vartriangle BFM$ be $O_1$ and $O_2$ respectively. Prove that $O_1O_2\perp CF$. [img]https://cdn.artofproblemsolving.com/attachments/e/4/e8fc62735b8cfbd382e490617f26d335c46823.png[/img]

If a worker receives a $ 20$ percent cut in wages, he may regain his original pay exactly by obtaining a raise of: $ \textbf{(A)}\ \text{20 percent} \qquad \textbf{(B)}\ \text{25 percent} \qquad \textbf{(C)}\ 22\frac{1}{2} \text{ percent} \qquad \textbf{(D)}\ \$20 \qquad \textbf{(E)}\ \$25$

Find all functions $f: \mathbb{Z} \to \mathbb{Z}$ such that \[ f(x) + f(y - 1) + f(f(y - f(x))) = 1 \] for all integers $x$ and $y$.

Let $z_1, z_2, z_3$ be pairwise distinct complex numbers satisfying $|z_1| = |z_2| = |z_3| = 1$ and \[\frac{1}{2 + |z_1 + z_2|}+\frac{1}{2 + |z_2 + z_3|}+\frac{1}{2 + |z_3 + z_1|} =1.\] If the points $A(z_1),B(z_2),C(z_3)$ are vertices of an acute-angled triangle, prove that this triangle is equilateral.

In regular hexagon $ABCDEF$, $AC$, $CE$ are two diagonals. Points $M$, $N$ are on $AC$, $CE$ respectively and satisfy $AC: AM = CE: CN = r$. Suppose $B, M, N$ are collinear, find $100r^2$. [asy] size(120); defaultpen(linewidth(0.7)+fontsize(10)); pair D2(pair P) { dot(P,linewidth(3)); return P; } pair A=dir(0), B=dir(60), C=dir(120), D=dir(180), E=dir(240), F=dir(300), N=(4*E+C)/5,M=intersectionpoints(A--C,B--N)[0]; draw(A--B--C--D--E--F--cycle); draw(A--C--E); draw(B--N); label("$A$",D2(A),plain.E); label("$B$",D2(B),NE); label("$C$",D2(C),NW); label("$D$",D2(D),W); label("$E$",D2(E),SW); label("$F$",D2(F),SE); label("$M$",D2(M),(0,-1.5)); label("$N$",D2(N),SE); [/asy]

Let $ABCDEF$ be a convex hexagon satisfying $\overline{AB} \parallel \overline{DE}$, $\overline{BC} \parallel \overline{EF}$, $\overline{CD} \parallel \overline{FA}$, and \[ AB \cdot DE = BC \cdot EF = CD \cdot FA. \] Let $X$, $Y$, and $Z$ be the midpoints of $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$. Prove that the circumcenter of $\triangle ACE$, the circumcenter of $\triangle BDF$, and the orthocenter of $\triangle XYZ$ are collinear.

A convicted person will be released when he reaches the top of a $100$-step staircase. But he cannot advance as he pleases, since he is obliged to go up one step each day of the odd-numbered months and go down one step each day of the even-numbered months. If it begins on January $ 1$, $2001$, what day will it be released?

Find all prime number $p$ such that the number of positive integer pair $(x,y)$ satisfy the following is not $29$. [list] [*]$1\le x,y\le 29$ [*]$29\mid y^2-x^p-26$ [/list]

The diagram below shows an isosceles triangle with base $21$ and height $28$. Inscribed in the triangle is a square. Find the area of the shaded region inside the triangle and outside of the square. [asy] size(170); defaultpen(linewidth(0.8)); draw((0,0)--(1,1)); pair A=(5,0),B=(-5,0),C=(0,14), invis[]={(1,2),(-1,2)}; pair intsquare[]={extension(origin,invis[0],A,C),extension(origin,invis[1],B,C)}; path triangle=A--B--C--cycle,square=(intsquare[0]--intsquare[1]--(intsquare[1].x,0)--(intsquare[0].x,0)--cycle); fill(triangle,gray); unfill(square); draw(triangle^^square); [/asy]

Define a function $f_n(x)\ (n=0,\ 1,\ 2,\ \cdots)$ by \[f_0(x)=\sin x,\ f_{n+1}(x)=\int_0^{\frac{\pi}{2}} f_n\prime (t)\sin (x+t)dt.\] (1) Let $f_n(x)=a_n\sin x+b_n\cos x.$ Express $a_{n+1},\ b_{n+1}$ in terms of $a_n,\ b_n.$ (2) Find $\sum_{n=0}^{\infty} f_n\left(\frac{\pi}{4}\right).$

Two triangles $ABC$ and $BDE$ have vertexes $C$ and $E$ on the same side of the line $AB{}$ and $AB=a<BD$. Denote $\{P\}=CE\cap AB$ and $\gamma=m(\angle CPA)$. Let $r_1$ be the radius of the inscribed cricle of triangle $PAC$ and $r_2$ the radius of the excircle of triangle $PDE$, tangent to the side $DE$. Find $r_1+r_2$.

$x$, $y$ and $z$ are positive reals such that $x+y+z=xyz$. Find the minimum value of: \[ x^7(yz-1)+y^7(zx-1)+z^7(xy-1) \]

Problem 4. Angel has a warehouse, which initially contains $100$ piles of $100$ pieces of rubbish each. Each morning, Angel performs exactly one of the following moves: (a) He clears every piece of rubbish from a single pile. (b) He clears one piece of rubbish from each pile. However, every evening, a demon sneaks into the warehouse and performs exactly one of the following moves: (a) He adds one piece of rubbish to each non-empty pile. (b) He creates a new pile with one piece of rubbish. What is the first morning when Angel can guarantee to have cleared all the rubbish from the warehouse?