How many line segments have both their endpoints located at the vertices of a given cube? $\text{(A)}\ 12 \qquad \text{(B)}\ 15 \qquad \text{(C)}\ 24 \qquad \text{(D)}\ 28\qquad \text{(E)}\ 56$

Let $D$ be a point on the small arc $AC$ of the circumcircle of an equilateral triangle $ABC$, different from $A$ and $C$. Let $E$ and $F$ be the projections of $D$ onto $BC$ and $AC$ respectively. Find the locus of the intersection point of $EF$ and $OD$, where $O$ is the center of $ABC$.

Let $A,B,C,D$ denote four points in space such that at most one of the distances $AB,AC,AD,BC,BD,CD$ is greater than $1$. Determine the maximum value of the sum of the six distances.

Let $ABCD$ be a rectangle. Let $K,L$ be the midpoints of $BC, AD$ respectively. From point $B$ the perpendicular line on $AK$, intersects $AK$ at point $E$ and $CL$ at point $Z$. a) Prove that the quadrilateral $AKZL$ is an isosceles trapezoid b) Prove that $2S_{ABKZ}=S_{ABCD}$ c) If quadrilateral $ABCD$ is a square of side $a$, calculate the area of the isosceles trapezoid $AKZL$ in terms of side $BC=a$

[b]p10.[/b] Three rectangles of dimension $X \times 2$ and four rectangles of dimension $Y \times 1$ are the pieces that form a rectangle of area $3XY$ where $X$ and $Y$ are positive, integer values. What is the sum of all possible values of $X$? [b]p11.[/b] Suppose we have a polynomial $p(x) = x^2 + ax + b$ with real coefficients $a + b = 1000$ and $b > 0$. Find the smallest possible value of $b$ such that $p(x)$ has two integer roots. [b]p12.[/b] Ten square slips of paper of the same size, numbered $0, 1, 2, ..., 9$, are placed into a bag. Four of these squares are then randomly chosen and placed into a two-by-two grid of squares. What is the probability that the numbers in every pair of blocks sharing a side have an absolute difference no greater than two? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\times$ represent the cross product in $\mathbb{R}^3.$ For what positive integers $n$ does there exist a set $S \subset \mathbb{R}^3$ with exactly $n$ elements such that $$S=\{v \times w: v, w \in S\}?$$

Determine all integers $n > 1$ such that \[\frac{2^{n}+1}{n^{2}}\] is an integer.

Let $\omega_1$ and $\omega_2$ be two circles with centers $O_1$ and $O_2$. The two circles intersect at $A$ and $B$. $\ell$ is the circles' common external tangent that is closer to $B$, and it meets $\omega_1$ at $T_1$ and $\omega_2$ at $T_2$. Let $C$ be the point on line $AB$ not equal to $A$ that is the same distance from $\ell$ as $A$ is. Given that $O_1O_2=15$, $AT_1=5$ and $AT_2=12$, find $AC^2+{T_1T_2}^2$. [i]Proposed by Zachary Perry[/i]

Let $ f(0) \equal{} f(1) \equal{} 0$ and \[ f(n\plus{}2) \equal{} 4^{n\plus{}2} \cdot f(n\plus{}1) \minus{} 16^{n\plus{}1} \cdot f(n) \plus{} n \cdot 2^{n^2}, \quad n \equal{} 0, 1, 2, \ldots\] Show that the numbers $ f(1989), f(1990), f(1991)$ are divisible by $ 13.$

For positive integers $ n,$ the numbers $ f(n)$ are defined inductively as follows: $ f(1) \equal{} 1,$ and for every positive integer $ n,$ $ f(n\plus{}1)$ is the greatest integer $ m$ such that there is an arithmetic progression of positive integers $ a_1 < a_2 < \ldots < a_m \equal{} n$ for which \[ f(a_1) \equal{} f(a_2) \equal{} \ldots \equal{} f(a_m).\] Prove that there are positive integers $ a$ and $ b$ such that $ f(an\plus{}b) \equal{} n\plus{}2$ for every positive integer $ n.$

Let $\mathcal L$ be the set of all lines in the plane and let $f$ be a function that assigns to each line $\ell\in\mathcal L$ a point $f(\ell)$ on $\ell$. Suppose that for any point $X$, and for any three lines $\ell_1,\ell_2,\ell_3$ passing through $X$, the points $f(\ell_1),f(\ell_2),f(\ell_3)$, and $X$ lie on a circle. Prove that there is a unique point $P$ such that $f(\ell)=P$ for any line $\ell$ passing through $P$. [i]Australia[/i]

For a point $P$ in the interior of a triangle $ABC$ let $D$ be the intersection of $AP$ with $BC$, let $E$ be the intersection of $BP$ with $AC$ and let $F$ be the intersection of $CP$ with $AB$.Furthermore let $Q$ and $R$ be the intersections of the parallel to $AB$ through $P$ with the sides $AC$ and $BC$, respectively. Likewise, let $S$ and $T$ be the intersections of the parallel to $BC$ through $P$ with the sides $AB$ and $AC$, respectively.In a given triangle $ABC$, determine all points $P$ for which the triangles $PRD$, $PEQ$and $PTE$ have the same area.

Fuzzy likes isosceles trapezoids. He can choose lengths from $1, 2, \dots, 8$, where he may choose any amount of each length. He takes a multiset of three integers from $1, \dots, 8$. From this multiset, one length will become a base length, one will become a diagonal length, and one will become a leg length. He uses each element as either a diagonal, leg, or base length exactly once. Fuzzy is happy if he can use these lengths to make an isosceles trapezoid such that the undecided base has nonzero rational length. How many multiset choices can he make? (Multisets are unordered) [i]Proposed by Timothy Qian[/i]

The cells of a $1\times 2n$ board are labelled $1,2,...,, n, -n,..., -2, -1$ from left to right. A marker is placed on an arbitrary cell. If the label of the cell is positive, the marker moves to the right a number of cells equal to the value of the label. If the label is negative, the marker moves to the left a number of cells equal to the absolute value of the label. Prove that if the marker can always visit all cells of the board, then $2n + 1$ is prime.

Let $I$ be the incenter of $\triangle ABC$ with $AB>AC$. Let $\Gamma$ be the circle with diameter $AI$. The circumcircle of $\triangle ABC$ intersects $\Gamma$ at points $A,D$, with point $D$ lying on $\overarc{AC}$ (not containing $B$). Let the line passing through $A$ and parallel to $BC$ intersect $\Gamma$ at points $A,E$. If $DI$ is the angle bisector of $\angle CDE$, and $\angle ABC = 33^{\circ}$, find the value of $\angle BAC$.

let $a_1,a_2,...a_n$ a sequence of real numbers such that $a_1+....+a_n=0$. define $b_i=a_1+a_2+....a_i$ for all $1 \leq i \leq n$ .suppose $b_i(a_{j+1}-a_{i+1}) \geq 0$ for all $1 \leq i \leq j \leq n-1$. Show that $$\max_{1 \leq l \leq n} |a_l| \geq \max_{1 \leq m \leq n} |b_m|$$

Let $ABCDEF$ be a regular hexagon with side length two. Extend $FE$ and $BD$ to meet at $G$. Compute the area of $ABGF$.

Sierpinski's triangle is formed by taking a triangle, and drawing an upside down triangle inside each upright triangle that appears. A snake sees the fractal, but decides that the triangles need circles inside them. Therefore, she draws a circle inscribed in every upside down triangle she sees (assume that the snake can do an infinite amount of work). If the original triangle had side length $1$, what is the total area of all the individual circles? [i]2015 CCA Math Bonanza Lightning Round #4.4[/i]

Convex quadrilateral $ABCD$ has sides $AB=BC=7$, $CD=5$, and $AD=3$. Given additionally that $m\angle ABC=60^\circ$, find $BD$.

Find all two-variable polynomials $p(x,y)$ such that for each $a,b,c\in\mathbb R$: \[p(ab,c^2+1)+p(bc,a^2+1)+p(ca,b^2+1)=0\]

At the start of this problem, six frogs are sitting at each of the six vertices of a regular hexagon, one frog per vertex. Every minute, we choose a frog to jump over another frog using one of the two rules illustrated below. If a frog at point $F$ jumps over a frog at point $P$ the frog will land at point $F'$ such that $F, P,$ and $F'$ are collinear and: - using Rule 1, $F'P = 2FP$. - using Rule 2, $F'P = FP/2$. [img]https://cdn.artofproblemsolving.com/attachments/7/0/2936bda61eb60c7b89bcd579386041022ba81f.png[/img] It is up to us to choose which frog to take the leap and which frog to jump over. (a) If we only use Rule 1, is it possible for some frog to land at the center of the original hexagon after a finite amount of time? (b) If both Rule 1 and Rule 2 are allowed (freely choosing which rule to use, which frog to jump, and which frog it jumps over), is it possible for some frog to land at the center of the original hexagon after a finite amount of time?

Let $P(x)=8x^3+ax+b+1$ for $a,b\in\mathbb{Z}$. It is known that $P$ has a root $x_0=p+\sqrt{q}+\sqrt[3]{r}$, where $p,q,r\in\mathbb{Q}$, $q\ge0$; however, $P$ has no $\textit{rational}$ roots. Find the smallest possible value of $a+b$.

Given an $ n\times n$ chessboard. a) Find the number of rectangles on the chessboard. b) Assume there exists an $ r\times r$ square (label $ B$) with $ r<n$ which is located on the upper left corner of the board. Define "inner border" of $ A$ as the border of $ A$ which is not the border of the chessboard. How many rectangles in $ B$ that touch exactly one inner border of $ B$?

Consider a $5 \times 5$ grid of squares. Vladimir colors some of these squares red, such that the centers of any four red squares do $\textbf{not}$ form an axis-parallel rectangle (i.e. a rectangle whose sides are parallel to those of the squares). What is the maximum number of squares he could have colored red?

Let $ a > b > 1, b$ is an odd number, let $ n$ be a positive integer. If $ b^n|a^n\minus{}1,$ then $ a^b > \frac {3^n}{n}.$