[b]p1. [/b]The longest diagonal of a regular hexagon is 12 inches long. What is the area of the hexagon, in square inches? [b]p2.[/b] When Al and Bob play a game, either Al wins, Bob wins, or they tie. The probability that Al does not win is $\frac23$ , and the probability that Bob does not win is $\frac34$ . What is the probability that they tie? [b]p3.[/b] Find the sum of the $a + b$ values over all pairs of integers $(a, b)$ such that $1 \le a < b \le 10$. That is, compute the sum $$(1 + 2) + (1 + 3) + (1 + 4) + ...+ (2 + 3) + (2 + 4) + ...+ (9 + 10).$$ [b]p4.[/b] A $3 \times 11$ cm rectangular box has one vertex at the origin, and the other vertices are above the $x$-axis. Its sides lie on the lines $y = x$ and $y = -x$. What is the $y$-coordinate of the highest point on the box, in centimeters? [b]p5.[/b] Six blocks are stacked on top of each other to create a pyramid, as shown below. Carl removes blocks one at a time from the pyramid, until all the blocks have been removed. He never removes a block until all the blocks that rest on top of it have been removed. In how many different orders can Carl remove the blocks? [img]https://cdn.artofproblemsolving.com/attachments/b/e/9694d92eeb70b4066b1717fedfbfc601631ced.png[/img] [b]p6.[/b] Suppose that a right triangle has sides of lengths $\sqrt{a + b\sqrt{3}}$,$\sqrt{3 + 2\sqrt{3}}$, and $\sqrt{4 + 5\sqrt{3}}$, where $a, b$ are positive integers. Find all possible ordered pairs $(a, b)$. [b]p7.[/b] Farmer Chong Gu glues together $4$ equilateral triangles of side length $ 1$ such that their edges coincide. He then drives in a stake at each vertex of the original triangles and puts a rubber band around all the stakes. Find the minimum possible length of the rubber band. [b]p8.[/b] Compute the number of ordered pairs $(a, b)$ of positive integers less than or equal to $100$, such that a $b -1$ is a multiple of $4$. [b]p9.[/b] In triangle $ABC$, $\angle C = 90^o$. Point $P$ lies on segment $BC$ and is not $B$ or $C$. Point $I$ lies on segment $AP$. If $\angle BIP = \angle PBI = \angle CAB = m^o$ for some positive integer $m$, find the sum of all possible values of $m$. [b]p10.[/b] Bob has $2$ identical red coins and $2$ identical blue coins, as well as $4$ distinguishable buckets. He places some, but not necessarily all, of the coins into the buckets such that no bucket contains two coins of the same color, and at least one bucket is not empty. In how many ways can he do this? [b]p11.[/b] Albert takes a $4 \times 4$ checkerboard and paints all the squares white. Afterward, he wants to paint some of the square black, such that each square shares an edge with an odd number of black squares. Help him out by drawing one possible configuration in which this holds. (Note: the answer sheet contains a $4 \times 4$ grid.) [b]p12.[/b] Let $S$ be the set of points $(x, y)$ with $0 \le x \le 5$, $0 \le y \le 5$ where $x$ and $y$ are integers. Let $T$ be the set of all points in the plane that are the midpoints of two distinct points in $S$. Let $U$ be the set of all points in the plane that are the midpoints of two distinct points in $T$. How many distinct points are in $U$? (Note: The points in $T$ and $U$ do not necessarily have integer coordinates.) [b]p13.[/b] In how many ways can one express $6036$ as the sum of at least two (not necessarily positive) consecutive integers? [b]p14.[/b] Let $a, b, c, d, e, f$ be integers (not necessarily distinct) between $-100$ and $100$, inclusive, such that $a + b + c + d + e + f = 100$. Let $M$ and $m$ be the maximum and minimum possible values, respectively, of $$abc + bcd + cde + def + ef a + f ab + ace + bdf.$$ Find $\frac{M}{m}$. [b]p15.[/b] In quadrilateral $ABCD$, diagonal $AC$ bisects diagonal $BD$. Given that $AB = 20$, $BC = 15$, $CD = 13$, $AC = 25$, find $DA$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$9$ points are given in the interior of the unit square. Prove there exists a triangle of area $\le \frac18$ whose vertices are three of the points.

Inside the acute-angled triangle $ ABC $ we consider the point $ P $ and its projections $ L, M, N $ to the sides $ BC, CA, AB $, respectively. Determine the point $ P $ for which the sum $ |BL|^2 + |CM|^2 + |AN|^2 $ is the smallest.

A large cube of size \(4 \times 4 \times 4\) is made up of 64 small unit cubes. Exactly 16 of these small cubes must be colored red, subject to the following condition: In each block of \(1 \times 1 \times 4\), \(1 \times 4 \times 1\), and \(4 \times 1 \times 1\) cubes, there must be exactly one red cube. Determine how many different ways it is possible to choose the 16 small cubes to be colored red. Note: Two colorings are considered different even if one can be obtained from the other by rotations or symmetries of the cube.

$\textbf{N4:} $ Let $a,b$ be two positive integers such that for all positive integer $n>2020^{2020}$, there exists a positive integer $m$ coprime to $n$ with \begin{align*} \text{ $a^n+b^n \mid a^m+b^m$} \end{align*} Show that $a=b$ [i]Proposed by ltf0501[/i]

=A subset $ S$ of $ \mathbb R^2$ is called an algebraic set if and only if there is a polynomial $ p(x,y)\in\mathbb R[x,y]$ such that \[ S \equal{} \{(x,y)\in\mathbb R^2|p(x,y) \equal{} 0\} \] Are the following subsets of plane an algebraic sets? 1. A square [img]http://i36.tinypic.com/28uiaep.png[/img] 2. A closed half-circle [img]http://i37.tinypic.com/155m155.png[/img]

On a chessboard a square is chosen . The sum of the squares of distances from its centre to the centre of all black squares is designated by $a$ and to the centre of all white squares by $b$. Prove that $a = b$. (A. Andj ans, Riga)

The sequence $x_1,x_2, ...$ is defined by the following equations: $$x_1=19, \ \ x_2=97, \ \ x_{n+2} =x_n - \frac{1}{x_{n+1}}$$ for $n \ge 1$. Prove that there exists a positive integer $k$ such that $x_k=0$ and find $k$. (A Berzinsh)

Function $f: R\to R$ is said periodic , if $f$ is not a constant function and there is a number real positive $p$ with the property of $f (x) = f (x + p)$ for every $x \in R$. The smallest positive real number p which satisfies the condition $f (x) = f (x + p)$ for each $x \in R$ is named period of $f$. Given $a$ and $b$ real positive numbers, show that there are periodic functions $f_1$ and $f_2$, with periods $a$ and $b$ respectively, so that $f_1 (x)\cdot f_2 (x)$ is also a periodic function.

The diagonals of the square $ABCD$ intersect at $P$ and the midpoint of the side $AB$ is $E$. Segment $ED$ intersects the diagonal $AC$ at point $F$ and segment $EC$ intersects the diagonal $BD$ at $G$. Inside the quadrilateral $EFPG$, draw a circle of radius $r$ tangent to all the sides of this quadrilateral. Prove that $r = | EF | - | FP |$.

Consider an acute triangle $ABC$ with circumcircle $\mathcal{C}$. Let $H$ be the orthocenter of $ABC$ and $M$ the midpoint of $BC$. Lines $AH$, $BH$ and $CH$ cut $\mathcal{C}$ again at points $D$, $E$, and $F$ respectively; line $MH$ cuts $\mathcal{C}$ at $J$ such that $H$ lies between $J$ and $M$. Let $K$ and $L$ be the incenters of triangles $DEJ$ and $DFJ$ respectively. Prove $KL$ is parallel to $BC$.

Let $(a_1,b_1),(a_2,b_2),\ldots ,(a_n,b_n)$ be the vertices of a convex polygon containing the origin in its interior. Prove that there are positive real numbers $x,y$ such that : \[ (a_1,b_1)x^{a_1}y^{b_1}+(a_2,b_2)x^{a_2}y^{b_2}+\ldots +(a_n,b_n)x^{a_n}y^{b_n}=(0,0) \]

Assume there are $ n\ge3$ points in the plane, Prove that there exist three points $ A,B,C$ satisfying $ 1\le\frac{AB}{AC}\le\frac{n\plus{}1}{n\minus{}1}.$

There are two letters in a language. Every word consists of seven letters, and two different words always have different letters on at least three places. a. Show that such a language cannot have more than $16$ words. b. Can there be $16$ words in the language?

Given $ABC$ triangle with incircle $L_1$ and circumcircle $L_2$. If points $X, Y, Z$ lie on $L_2$, such that $XY, XZ$ are tangent to $L_1$, then prove that $YZ$ is also tangent to $L_1$.

How many $f:\mathbb{R} \rightarrow \mathbb{R}$ are there satisfying $f(x)f(y)f(z)=12f(xyz)-16xyz$ for every real $x,y,z$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 0 \qquad \textbf{(E)}\ \text{None}$

The number obtained from the last two nonzero digits of $ 90!$ is equal to $ n$. What is $ n$? $ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 32 \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ 52 \qquad \textbf{(E)}\ 68$

For positive integers $ n$ let $ D(n)$ denote the set of positive integers that divide $ n$ and let $ S(n)\equal{}\Sigma_{k \in D(n)} \frac{1}{k}$. What is $ S(2006)$? Answer with a fraction reduced to lowest terms.

Upper content of a subset $E$ of the plane $\mathbb{R}^{2}$ is defined as $$\mathcal{C}(E)=\inf\{\sum_{i=1}^{n} \text{diam}(E_{i})\}$$ where $\inf$ is taken over all finite families of sets $E_{1},\dots,E_{n}$ $n\in \mathbb{N}$, in $\mathbb{R}^{2}$ such that $E\subset \bigcup_{i=1}^{n}E_{i}$. Lower content of $E$ is defined as $$\mathcal{K}(E)=\sup\{\text{length}(L) |\, L \text{ is a closed line segment onto which $E$ can be contracted}\}$$. Prove that i) $\mathcal{C}(L)=\text{length}(L)$ if $L$ is a closed line segment; ii) $\mathcal{C}(E) \geq \mathcal{K}(E)$; iii) the equality in ii) is not always true even if $E$ is compact.

A finite set of line segments, of total length $18$, belongs to a square of unit side length (we assume that the square includes its boundary and that a line segment includes its end points). The line segments are parallel to the sides of the square and may intersect one another. Prove that among the regions into which the square is divided by the line segments, at least one of these must have area not less than $0.01$. (A Berzinsh, Riga)

On the sides $AB, BC$ and $CA$ of triangle $ABC$, points $L, M$ and $N$ are chosen, respectively, such that the lines $CL, AM$ and $BN$ intersect at a common point O inside the triangle and the quadrilaterals $ALON, BMOL$ and $CNOM$ have incircles. Prove that $$\frac{1}{AL\cdot BM} +\frac{1}{BM\cdot CN} +\frac{1}{CN \cdot AL} =\frac{1}{AN\cdot BL} +\frac{1}{BL\cdot CM} +\frac{1}{CM\cdot AN} $$

Points $E,F$ are chosen on the sides $CA,AB$ of triangle $ABC$. Let $(K)$ be the circumcircle of triangle $AEF$. The tangents at $E, F$ of $(K)$ intersect at $T$ . Prove that (a) $T$ is on $BC$ if and only if $BE$ meets $CF$ at a point on the circle $(K)$, (b) $EF, PQ,BC$ are concurrent given that $BE$ meets $FT$ at $M, CF$ meets $ET$ at $N, AM$ and $AN$ intersects $(K)$ at $P,Q$ distinct from $A$. Trần Quang Hùng

[b]p7.[/b] An ant starts at the point $(0, 0)$. It travels along the integer lattice, at each lattice point choosing the positive $x$ or $y$ direction with equal probability. If the ant reaches $(20, 23)$, what is the probability it did not pass through $(20, 20)$? [b]p8.[/b] Let $a_0 = 2023$ and $a_n$ be the sum of all divisors of $a_{n-1}$ for all $n \ge 1$. Compute the sum of the prime numbers that divide $a_3$. [b]p9.[/b] Five circles of radius one are stored in a box of base length five as in the following diagram. How far above the base of the box are the upper circles touching the sides of the box? [img]https://cdn.artofproblemsolving.com/attachments/7/c/c20b5fa21fbd8ce791358fd888ed78fcdb7646.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Positive integer numbers a and b satisfy $(a^2- 9b^2)^2 - 33b = 1$. a) Prove $|a -3b|\ge 1$. b) Find all pairs of positive integers $(a, b)$ satisfying the equality.

Let $ ABC$ be a triangle, and $ D$ the midpoint of the side $ BC$. On the sides $ AB$ and $ AC$ we consider the points $ M$ and $ N$, respectively, both different from the midpoints of the sides, such that \[ AM^2\plus{}AN^2 \equal{}BM^2 \plus{} CN^2 \textrm{ and } \angle MDN \equal{} \angle BAC.\] Prove that $ \angle BAC \equal{} 90^\circ$.