I'd really appreciate help on this. (a) Given a set $X$ of points in the plane, let $f_{X}(n)$ be the largest possible area of a polygon with at most $n$ vertices, all of which are points of $X$. Prove that if $m, n$ are integers with $m \geq n > 2$ then $f_{X}(m) + f_{X}(n) \geq f_{X}(m + 1) + f_{X}(n - 1)$. (b) Let $P_0$ be a $1 \times 2$ rectangle (including its interior) and inductively define the polygon $P_i$ to be the result of folding $P_{i-1}$ over some line that cuts $P_{i-1}$ into two connected parts. The diameter of a polygon $P_i$ is the maximum distance between two points of $P_i$. Determine the smallest possible diameter of $P_{2013}$.

Define $\{p_n\}_{n=0}^\infty\subset\mathbb N$ and $\{q_n\}_{n=0}^\infty\subset\mathbb N$ to be sequences of natural numbers as follows: [list] [*]$p_0=q_0=1$; [*]For all $n\in\mathbb N$, $q_n$ is the smallest natural number such that there exists a natural number $p_n$ with $\gcd(p_n,q_n)=1$ satisfying \[\dfrac{p_{n-1}}{q_{n-1}} < \dfrac{p_n}{q_n} < \sqrt 2.\] [/list] Find $q_3$.

Prove that the edges of a finite simple planar graph (with no loops, multiple edges) may be oriented in such a way that at most three fourths of the total number of dges of any cycle share the same orientation. Moreover, show that this is the best global bound possible. Comment: The actual problem in the TST asked to prove that the edges can be $2$-colored so that the same conclusion holds. Under this circumstances, the problem is wrong and a counterexample was found in the contest by Marius Tiba.

Let $n \geq 5$ be a positive integer. There are $n$ stars with values $1$ to $n$, respectively. Anya and Becky play a game. Before the game starts, Anya places the $n$ stars in a row in whatever order she wishes. Then, starting from Becky, each player takes the left-most or right-most star in the row. After all the stars have been taken, the player with the highest total value of stars wins; if their total values are the same, then the game ends in a draw. Find all $n$ such that Becky has a winning strategy. [i] Proposed by Ho-Chien Chen[/i]

p1. A telephone number with $7$ digits is called a [i]Beautiful Number [/i]if the digits are which appears in the first three numbers (the three must be different) repeats on the next three digits or the last three digits. For example some beautiful numbers: $7133719$, $7131735$, $7130713$, $1739317$, $5433354$. If the numbers are taken from $0, 1, 2, 3, 4, 5, 6, 7, 8$ or $9$, but the number the first cannot be $0$, how many Beautiful Numbers can there be obtained? p2. Find the number of natural numbers $n$ such that $n^3 + 100$ is divisible by $n +10$ p3. A function $f$ is defined as in the following table. [img]https://cdn.artofproblemsolving.com/attachments/5/5/620d18d312c1709b00be74543b390bfb5a8edc.png[/img] Based on the definition of the function $f$ above, then a sequence is defined on the general formula for the terms is as follows: $U_1=2$ and $U_{n+1}=f(U_n)$ , for $n = 1, 2, 3, ...$ p4. In a triangle $ABC$, point $D$ lies on side $AB$ and point $E$ lies on side $AC$. Prove for the ratio of areas: $\frac{ADE }{ABC}=\frac{AD\times AE}{AB\times AC}$ p5. In a chess tournament, a player only plays once with another player. A player scores $1$ if he wins, $0$ if he loses, and $\frac12$ if it's a draw. After the competition ended, it was discovered that $\frac12$ of the total value that earned by each player is obtained from playing with 10 different players who got the lowest total points. Especially for those in rank bottom ten, $\frac12$ of the total score one gets is obtained from playing with $9$ other players. How many players are there in the competition?

What is the minimal value of $\frac{b}{c + d} + \frac{c}{a + b}$ for positive real numbers $b$ and $c$ and non-negative real numbers $a$ and $d$ such that $b + c\ge a + d$?

Find the volume of the solid of a circle $x^2+(y-1)^2=4$ generated by a rotation about the $x$-axis.

Let us consider all rectangles with sides of length $a,b$ both of which are whole numbers. Do more of these rectangles have perimeter $2000$ or perimeter $2002$?

If $\log_MN=\log_NM$, $M\ne N$, $MN>0$, $M\ne 1$, $N\ne 1$, then $MN$ equals: $\text{(A)} \ \frac12 \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ 10 \qquad \text{(E)} \ \text{a number greater than 2 and less than 10}$

(a) Find all pairs $(m, n)$ of integers satisfying $m^3-4mn^2=8n^3-2m^2n$ (b) Among such pairs find those for which $m+n^2=3$

Let $ H $ be the point of intersection of the altitudes $ AP $ and $ CQ $ of the acute-angled triangle $ABC$. The points $ E $ and $ F $ are marked on the median $ BM $ such that $ \angle APE = \angle BAC $, $ \angle CQF = \angle BCA $, with point $ E $ lying inside the triangle $APB$ and point $ F $ is inside the triangle $CQB$. Prove that the lines $AE, CF$, and $BH$ intersect at one point.

In the $ 6\times4$ grid shown, $ 12$ of the $ 24$ squares are to be shaded so that there are two shaded squares in each row and three shaded squares in each column. Let $ N$ be the number of shadings with this property. Find the remainder when $ N$ is divided by $ 1000$. [asy]size(100); defaultpen(linewidth(0.7)); int i; for(i=0; i<5; ++i) { draw((i,0)--(i,6)); } for(i=0; i<7; ++i) { draw((0,i)--(4,i)); }[/asy]

Let $ABCD$ be a quadrilateral, and $P,Q,R,S$ are the midpoints of $AB, BC, CD, DA$ respectively. Let $O$ be the intersection between $PR$ and $QS$. Prove that $PO = OR$ and $QO = OS$.

Find all pairs of prime numbers $p, q$ for which there exist positive integers $(m, n)$ such that $$(p+q)^m=(p-q)^n$$.

Let $ABC$ be an acute-angled triangle with circumradius $R$ less than the sides of $ABC$, let $H$ and $O$ be the orthocenter and circuncenter of $ABC$, respectively. The angle bisectors of $\angle ABH$ and $\angle ACH$ intersects in the point $A_1$, analogously define $B_1$ and $C_1$. If $E$ is the midpoint of $HO$, prove that $EA_1+EB_1+EC_1=p-\frac{3R}{2}$ where $p$ is the semiperimeter of $ABC$

Given a triangular pyramid $ABCD$. Sphere $S_1$ passing through points $A$, $B$, $C$, intersects edges $AD$, $BD$, $CD$ at points $K$, $L$, $M$, respectively; sphere $S_2$ passing through points $A$, $B$, $D$ intersects the edges $AC$, $BC$, $DC$ at points $P$, $Q$, $M$ respectively. It turned out that $KL \parallel PQ$. Prove that the bisectors of plane angles $KMQ$ and $LMP$ are the same.

$AB$ is the chord of the circle $S$. Circles $S_1$ and $S_2$ touch the circle $S$ at points $P$ and $Q$, respectively, and the segment $AB$ at point $K$. It turned out that $\angle{PBA}=\angle{QBA}$. Prove that $AB$ is the diameter of the circle $S$.

Into triangle $ABC$ gives point $K$ lies on bisector of $ \angle BAC$. Line $CK$ intersect circumcircle $ \omega$ of triangle $ABC$ at $M \neq C$. Circle $ \Omega$ passes through $A$, touch $CM$ at $K$ and intersect segment $AB$ at $P \neq A$ and $\omega $ at $Q \neq A$. Prove, that $P$, $Q$, $M$ lies at one line.

Square $ABCD$ has side length $5$ and arc $BD$ with center $A$. $E$ is the midpoint of $AB$ and $CE$ intersects arc $BD$ at $F$. $G$ is placed onto $BC$ such that $FG$ is perpendicular to $BC$. What is the length of $FG$?

Two $5\times1$ rectangles have 2 vertices in common as on the picture. (a) Determine the area of overlap (b) Determine the length of the segment between the other 2 points of intersection, $A$ and $B$. [img]https://cdn.artofproblemsolving.com/attachments/9/0/4f1721c7ccdecdfe4d9cc05a17a553a0e9f670.png[/img]

The diagonals of a convex quadrilateral dissect it into four similar triangles. Prove that this quadrilateral can also be dissected into two congruent triangles.

For an $n\times n$ matrix with real entries let $||M||=\sup_{x\in \mathbb{R}^{n}\setminus\{0\}}\frac{||Mx||_{2}}{||x||_{2}}$, where $||\cdot||_{2}$ denotes the Euclidean norm on $\mathbb{R}^{n}$. Assume that an $n\times n$ matrxi $A$ with real entries satisfies $||A^{k}-A^{k-1}||\leq\frac{1}{2002k}$ for all positive integers $k$. Prove that $||A^{k}||\leq 2002$ for all positive integers $k$.

Juku invented an apparatus that can divide any segment into three equal segments. How can you find the midpoint of any segment, using only the Juku made, a ruler and pencil?

Call an $n$-gon to be [i]lattice[/i] if its vertices are lattice points. Prove that inside every lattice convex pentagon there exists a lattice point.

Given that in a triangle $ABC$, $AB=3$, $BC=4$ and the midpoints of the altitudes of the triangle are collinear, find all possible values of the length of $AC$.