[b]p1.[/b] What is the largest distance between any two points on a regular hexagon with a side length of one? [b]p2.[/b] For how many integers $n \ge 1$ is $\frac{10^n - 1}{9}$ the square of an integer? [b]p3.[/b] A vector in $3D$ space that in standard position in the first octant makes an angle of $\frac{\pi}{3}$ with the $x$ axis and $\frac{\pi}{4}$ with the $y$ axis. What angle does it make with the $z$ axis? [b]p4.[/b] Compute $\sqrt{2012^2 + 2012^2 \cdot 2013^2 + 2013^2} - 2012^2$. [b]p5.[/b] Round $\log_2 \left(\sum^{32}_{k=0} {{32} \choose k} \cdot 3^k \cdot 5^k\right)$ to the nearest integer. [b]p6.[/b] Let $P$ be a point inside a ball. Consider three mutually perpendicular planes through $P$. These planes intersect the ball along three disks. If the radius of the ball is $2$ and $1/2$ is the distance between the center of the ball and $P$, compute the sum of the areas of the three disks of intersection. [b]p7.[/b] Find the sum of the absolute values of the real roots of the equation $x^4 - 4x - 1 = 0$. [b]p8.[/b] The numbers $1, 2, 3, ..., 2013$ are written on a board. A student erases three numbers $a, b, c$ and instead writes the number $$\frac12 (a + b + c)\left((a - b)^2 + (b - c)^2 + (c - a)^2\right).$$ She repeats this process until there is only one number left on the board. List all possible values of the remainder when the last number is divided by 3. [b]p9.[/b] How many ordered triples of integers $(a, b, c)$, where $1 \le a, b, c \le 10$, are such that for every natural number $n$, the equation $(a + n)x^2 + (b + 2n)x + c + n = 0$ has at least one real root? Problems' source (as mentioned on official site) is Gator Mathematics Competition. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The real numbers $a_1,a_2,\ldots,a_{100}$ satisfy the relationship \[ a_1^2+ a_2^2 + \cdots +a_{100}^2 + ( a_1+a_2 + \cdots + a_{100})^2 = 101. \] Prove that $|a_k|\leq 10$, for all $k=1,2,\ldots,100$.

Let $ABC$ be a triangle, $P$ the midpoint of $BC$, and $Q$ a point on segment $CA$ such that $|CQ| = 2|QA|$. Let $S$ be the intersection of $BQ$ and $AP$. Prove that $|AS| = |SP|$.

A graph $G$ with $n$ vertices is called [i]cool[/i] if we can label each vertex with a different positive integer not greater than $\frac{n^2}{4}$ and find a set of non-negative integers $D$ so that there is an edge between two vertices iff the difference between their labels is in $D$. Show that if $n$ is sufficiently large we can always find a graph with $n$ vertices which is not cool.

$16$ people sit around a circular table. After some time, they all stand up and sit down in either the chair they were previously sitting on or on a chair next to it. Determine the number of ways that this can be done. Note: two or more people cannot sit on the same chair.

Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes $1$ token or $4$ tokens from the stack. The player who removes the last token wins. Find the number of positive integers $n$ less than or equal to $2024$ such that there is a strategy that guarantees that Bob wins, regardless of Alice’s moves.

Solve equation $\sqrt{1+2x-xy}+\sqrt{1+2y-xy}=2.$

Some time ago there was a war across the world. In the plane $n$ lines are moving, with the regions contained by the lines being the territories of the countries at war. Each line moves parallel to itself with constant speed (each with its own speed), and no line can reverse its direction. Some of the original countries disappeared (a country disappears iff its area is converted to zero) and within the course of the time, other countries appeared. After some time, the presidents of the existing countries made a treaty to end the war, created the United Nations, and all borders ceased movement. The UN then counted the total numbers of sovereign states that were destroyed and the existing ones, obtaining a total of $k$. Prove that $k\leq \frac{n^3+5n}{6}+1$. Is is possible to have equality?

Prove that for every integer power of 2, there exists a multiple of it with all digits (in decimal expression) not zero.

There is a $m \times (m-1)$ board. (i.e. there are $m+1$ horizontal lines and $m$ vertical lines) A stone is put on an intersection of the lowest horizontal line. Now two players move this stone with the following rules. (i) Each players move the stone to a neighboring intersection along a segment, by turns. (ii) A segment, which is already passed by the stone, cannot be used more. (iii) One who cannot move the stone anymore loses. Prove that there is a winning strategy for the former player.

In the adjoining figure, the triangle $ABC$ is a right triangle with $\angle BCA=90^\circ$. Median $CM$ is perpendicular to median $BN$, and side $BC=s$. The length of $BN$ is [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10));real r=54.72; pair B=origin, C=dir(r), A=intersectionpoint(B--(9,0), C--C+4*dir(r-90)), M=midpoint(B--A), N=midpoint(A--C), P=intersectionpoint(B--N, C--M); draw(M--C--A--B--C^^B--N); pair point=P; markscalefactor=0.005; draw(rightanglemark(C,P,B)); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$M$", M, S); label("$N$", N, dir(C--A)*dir(90)); label("$s$", B--C, NW);[/asy] $\textbf {(A) } s\sqrt 2 \qquad \textbf {(B) } \frac 32s\sqrt2 \qquad \textbf {(C) } 2s\sqrt2 \qquad \textbf {(D) } \frac 12s\sqrt5 \qquad \textbf {(E) } \frac 12s\sqrt6$

Let $\alpha$, $\beta$ and $\gamma$ be three acute angles such that $\sin \alpha+\sin \beta+\sin \gamma = 1$. Show that \[\tan^{2}\alpha+\tan^{2}\beta+\tan^{2}\gamma \geq \frac{3}{8}. \]

Let $A,B\in M_n(C)$ be two square matrices satisfying $A^2+B^2 = 2AB$. 1.Prove that $\det(AB-BA)=0$. 2.If $rank(A-B)=1$, then prove that $AB=BA$.

Find the number of ordered triplets $(a,b,c)$ of positive integers such that $abc=2008$ (the product of $a$, $b$, and $c$ is $2008$).

Fix positive integers $k,n$. A candy vending machine has many different colours of candy, where there are $2n$ candies of each colour. A couple of kids each buys from the vending machine $2$ candies of different colours. Given that for any $k+1$ kids there are two kids who have at least one colour of candy in common, find the maximum number of kids.

From a paper quadrilateral like the one in the figure, you have to cut out a new quadrilateral whose area is equal to half the area of the original quadrilateral.You can only bend one or more times and cut by some of the lines of the folds. Describe the folds and cuts and justify that the area is half. [img]https://2.bp.blogspot.com/-btvafZuTvlk/XNY8nba0BmI/AAAAAAAAKLo/nm4c21A1hAIK3PKleEwt6F9cd6zv4XffwCK4BGAYYCw/s400/may%2B2012%2Bl1.png[/img]

Determine if there exists a triangle that can be cut into $101$ congruent triangles.

For each positive integer $ m > 1$, let $ P(m)$ denote the greatest prime factor of $ m$. For how many positive integers $ n$ is it true that both $ P(n) \equal{} \sqrt{n}$ and $ P(n \plus{} 48) \equal{} \sqrt{n \plus{} 48}$? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 3\qquad \textbf{(D)}\ 4\qquad \textbf{(E)}\ 5$

How many numbers between $1$ and $1,000,000$ are perfect squares but not perfect cubes?

Let $\mathcal S$ be a set of $16$ points in the plane, no three collinear. Let $\chi(S)$ denote the number of ways to draw $8$ lines with endpoints in $\mathcal S$, such that no two drawn segments intersect, even at endpoints. Find the smallest possible value of $\chi(\mathcal S)$ across all such $\mathcal S$. [i]Ankan Bhattacharya[/i]

Let $C_1$ and $C_2$ be the outer circumferences of centers $O_1$ and $O_2$, respectively. The two tangents to the circumference $C_2$ are drawn by $O_1$, intersecting $C_1$ at $P$ and $P'$. The two tangents to the circumference $C_1$ are drawn by $O_2$, intersecting $C_2$ at $Q$ and $Q'$. Prove that the segment $PP'$ is equal to the segment $QQ'$.

Let $ S$ be a sequence that: \[ \left\{ \begin{array}{cc} S_0\equal{}0\hfill\\ S_1\equal{}1\hfill\\ S_n\equal{}S_{n\minus{}1}\plus{}S_{n\minus{}2}\plus{}F_n& (n>1) \end{array} \right.\] such that $ F_n$ is Fibonacci sequence such that $ F_1\equal{}F_2\equal{}1$. Find $ S_n$ in terms of Fibonacci numbers.

Find all four-digit numbers $\overline{abcd}$ (in decimal system) such that $\overline{abcd}= (\overline{ac}+1).(\overline{bd} +1)$

Show that if $a,b$ are positive real numbers such that $a^{2000}+b^{2000}=a^{1998}+b^{1998}$ then $a^2+b^2 \le 2$.

Let $p$ be a prime number. Prove that there exists $n \in Z$ such that $p | n^2 -n+3$ if and only if there exists $m \in Z$ such that $p | m^2 -m+25$.