[b]p1.[/b] Let $a$ and $b$ be complex numbers such that $a^3 + b^3 = -17$ and $a + b = 1$. What is the value of $ab$? [b]p2.[/b] Let $AEFB$ be a right trapezoid, with $\angle AEF = \angle EAB = 90^o$. The two diagonals $EB$ and $AF$ intersect at point $D$, and $C$ is a point on $AE$ such that $AE \perp DC$. If $AB = 8$ and $EF = 17$, what is the lenght of $CD$? [b]p3.[/b] How many three-digit numbers $abc$ (where each of $a$, $b$, and $c$ represents a single digit, $a \ne 0$) are there such that the six-digit number $abcabc$ is divisible by $2$, $3$, $5$, $7$, $11$, or $13$? [b]p4.[/b] Let $S$ be the sum of all numbers of the form $\frac{1}{n}$ where $n$ is a postive integer and $\frac{1}{n}$ terminales in base $b$, a positive integer. If $S$ is $\frac{15}{8}$, what is the smallest such $b$? [b]p5.[/b] Sysyphus is having an birthday party and he has a square cake that is to be cut into $25$ square pieces. Zeus gets to make the first straight cut and messes up badly. What is the largest number of pieces Zeus can ruin (cut across)? Diagram? [b]p6.[/b] Given $(9x^2 - y^2)(9x^2 + 6xy + y^2) = 16$ and $3x + y = 2$. Find $x^y$. [b]p7.[/b] What is the prime factorization of the smallest integer $N$ such that $\frac{N}{2}$ is a perfect square, $\frac{N}{3}$ is a perfect cube, $\frac{N}{5}$ is a perfect fifth power? [b]p8.[/b] What is the maximum number of pieces that an spherical watermelon can be divided into with four straight planar cuts? [b]p9.[/b] How many ordered triples of integers $(x,y,z)$ are there such that $0 \le x, y, z \le 100$ and $$(x - y)^2 + (y - z)^2 + (z - x)^2 \ge (x + y - 2z) + (y + z - 2x)^2 + (z + x - 2y)^2.$$ [b]p10.[/b] Find all real solutions to $(2x - 4)^2 + (4x - 2)^3 = (4x + 2x - 6)^3$. [b]p11.[/b] Let $f$ be a function that takes integers to integers that also has $$f(x)=\begin{cases} x - 5\,\, if \,\, x \ge 50 \\ f (f (x + 12)) \,\, if \,\, x < 50 \end{cases}$$ Evaluate $f (2) + f (39) + f (58).$ [b]p12.[/b] If two real numbers are chosen at random (i.e. uniform distribution) from the interval $[0,1]$, what is the probability that theit difference will be less than $\frac35$? [b]p13.[/b] Let $a$, $b$, and $c$ be positive integers, not all even, such that $a < b$, $b = c - 2$, and $a^2 + b^2 = c^2$. What is the smallest possible value for $c$? [b]p14.[/b] Let $ABCD$ be a quadrilateral whose diagonals intersect at $O$. If $BO = 8$, $OD = 8$, $AO = 16$, $OC = 4$, and $AB = 16$, then find $AD$. [b]p15.[/b] Let $P_0$ be a regular icosahedron with an edge length of $17$ units. For each nonnegative integer $n$, recursively construct $P_{n+1}$ from Pn by performing the following procedure on each face of $P_n$: glue a regular tetrahedron to that face such that three of the vertices of the tetrahedron are the midpoints of the three adjacent edges of the face, and the last vertex extends outside of $P_n$. Express the number of square units in the surface area of $P_{17}$ in the form $$\frac{u^v\cdot w \sqrt{x}}{y^z}$$ , where $u, v, w, x, y$, and $z$ are integers, all greater than or equal to $2$, that satisfy the following conditions: the only perfect square that evenly divides $x$ is $1$, the GCD of $u$ and y is $1$, and neither $u$ nor $y$ divides $w$. Answers written in any other form will not be considered correct! PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider a convex quadrilateral $ABCD$. Denote $M, \ N$ the points of tangency of the circle inscribed in $\triangle ABD$ with $AB, \ AD$, respectively and $P, \ Q$ the points of tangency of the circle inscribed in $\triangle CBD$ with the sides $CD, \ CB$, respectively. Assume that the circles inscribed in $\triangle ABD, \ \triangle CBD$ are tangent. Prove that: a) $ABCD$ is circumscriptible. b) $MNPQ$ is cyclic. c) The circles inscribed in $\triangle ABC, \ \triangle ADC$ are tangent.

Let $a$ and $b$ be distinct real numbers for which \[\dfrac ab+\dfrac{a+10b}{b+10a}=2.\] Find $\dfrac ab$. $\textbf{(A) }0.6\qquad\textbf{(B) }0.7\qquad\textbf{(C) }0.8\qquad\textbf{(D) }0.9\qquad\textbf{(E) }1$

Find all functions $f$ from the set of reals to itself so that for all reals $x,y,$ $$f(x)f(f(x)+y) = f(x^2) + f(xy).$$ [i]Proposed by Culver Kwan[/i]

Let $ABC$ be a triangle such that $\angle A=90$ and $\angle B < \angle C$. The tangent at $A$ to its circumcircle $\Gamma$ meets the line $BC$ at $D$. Let $E$ be the reflection of $A$ across $BC$, $X$ the foot of the perpendicular from $A$ to $BE$, and $Y$ be the midpoint of $AX$. Let the line $BY$ meet $\Gamma$ again at $Z$. Prove that the line $BD$ is tangent to circumcircle of triangle $ADZ$ .

A $2022 \times 2022$ squareboard was divided into $L$ and $Z$ tetrominoes. Each tetromino consists of four squares, which can be rotated or flipped. Determine the least number of $Z$-tetrominoes necessary to cover the $2022 \times 2022$ squareboard.

Suppose that $|ax^2+bx+c| \geq |x^2-1|$ for all real numbers x. Prove that $|b^2-4ac|\geq 4$.

What is the value of $\frac{2a^{-1}+\frac{a^{-1}}{2}}{a}$ when $a= \frac{1}{2}$? $\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \frac{5}{2}\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 20$

The digits $1$, $2$, $3$, $4$, and $5$ are each used once to write a five-digit number $PQRST$. The three-digit number $PQR$ is divisible by $4$, the three-digit number $QRS$ is divisible by $5$, and the three-digit number $RST$ is divisible by $3$. What is $P$? $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad \textbf{(E) }5$

Solve for $x$ if $\sqrt{x + 1}+ \sqrt{x} = 5.$ [i]Proposed by Eric Oh[/i]

For any integer $n\geq1$, we consider a set $P_{2n}$ of $2n$ points placed equidistantly on a circle. A [i]perfect matching[/i] on this point set is comprised of $n$ (straight-line) segments whose endpoints constitute $P_{2n}$. Let $\mathcal{M}_{n}$ denote the set of all non-crossing perfect matchings on $P_{2n}$. A perfect matching $M\in \mathcal{M}_{n}$ is said to be [i]centrally symmetric[/i], if it is invariant under point reflection at the circle center. Determine, as a function of $n$, the number of centrally symmetric perfect matchings within $\mathcal{M}_{n}$. [i]Proposed by Mirko Petrusevski[/i]

For a set $S$ of integers, define $\max (S)$ to be the maximal element of $S$. How many non-empty subsets $S \subseteq \{1, 2, 3, ... , 10\}$ satisfy $\max (S) \le |S| + 2$?

Let $A_1A_2 . . .A_{100}$ be the vertices of a regular $100$-gon. Let $\pi$ be a randomly chosen permutation of the numbers from $1$ through $100$. The segments $A_{\pi (1)}A_{\pi (2)}$, $A_{\pi (2)}A_{\pi (3)}$, $...$ ,$A_{\pi (99)}A_{\pi (100)}, A_{\pi (100)}A_{\pi (1)}$ are drawn. Find the expected number of pairs of line segments that intersect at a point in the interior of the $100$-gon.

Abby, Bridget, and four of their classmates will be seated in two rows of three for a group picture, as shown. \begin{eqnarray*} \text{X}&\quad\text{X}\quad&\text{X} \\ \text{X}&\quad\text{X}\quad&\text{X} \end{eqnarray*} If the seating positions are assigned randomly, what is the probability that Abby and Bridget are adjacent to each other in the same row or the same column? $\textbf{(A) } \frac{1}{3} \qquad \textbf{(B) } \frac{2}{5} \qquad \textbf{(C) } \frac{7}{15} \qquad \textbf{(D) } \frac{1}{2} \qquad \textbf{(E) } \frac{2}{3}$

Given positive integer $ n > 1$ and define \[ S \equal{} \{1,2,\dots,n\}. \] Suppose \[ T \equal{} \{t \in S: \gcd(t,n) \equal{} 1\}. \] Let $ A$ be arbitrary non-empty subset of $ A$ such thar for all $ x,y \in A$, we have $ (xy\mod n) \in A$. Prove that the number of elements of $ A$ divides $ \phi(n)$. ($ \phi(n)$ is Euler-Phi function)

Let $O$ be a point interior to triangle $ABC$ such that $\angle BAO = 30^o$, $\angle CBO = 20^o$ and $\angle ABO = \angle ACO = 40^o$ . Knowing that triangle $ABC$ is not equilateral, find the measures of its interior angles.