Let $ABCD$ be a parallelogram with an acute angle at $A$. Let $G$ be a point on the line $AB$, distinct from $B$, such that $|CG| = |CB|$. Let $H$ be a point on the line $BC$, distinct from $B$, such that $|AB| =|AH|$. Prove that triangle $DGH$ is isosceles. [asy] unitsize(1.5 cm); pair A, B, C, D, G, H; A = (0,0); B = (2,0); D = (0.5,1.5); C = B + D - A; G = reflect(A,B)*(C) + C - B; H = reflect(B,C)*(H) + A - B; draw(H--A--D--C--G); draw(interp(A,G,-0.1)--interp(A,G,1.1)); draw(interp(C,H,-0.1)--interp(C,H,1.1)); draw(D--G--H--cycle, dashed); dot("$A$", A, SW); dot("$B$", B, SE); dot("$C$", C, E); dot("$D$", D, NW); dot("$G$", G, NE); dot("$H$", H, SE); [/asy]

In triangle $\Delta ADC$ we got $AD=DC$ and $D=100^\circ$. In triangle $\Delta CAB$ we got $CA=AB$ and $A=20^\circ$. Prove that $AB=BC+CD$.

For each positive integer $n$, let $S(n)$ denote the sum of the digits of $n.$ For how many values of $n$ is $n + S(n) + S(S(n)) = 2007?$ $\mathrm{(A)}\ 1 \qquad \mathrm{(B)}\ 2 \qquad \mathrm{(C)}\ 3 \qquad \mathrm{(D)}\ 4 \qquad \mathrm{(E)}\ 5$

A function $D(n)$ of the positive integral variable $n$ is defined by the following properties: $D(1) = 0, D(p) = 1$ if $p$ is a prime, $D(uv) = u D(v) + v D(u)$ for any two positive integers $u$ and $v.$ Answer all three parts below. (i) Show that these properties are compatible and determine uniquely $D(n).$ (Derive a formula for $D(n) /n,$ assuming that $n = p_1^{\alpha_1} p_2^{\alpha_2} \cdots p_k^{\alpha_k}$ where $p_1, p_2, \ldots, p_k$ are different primes.) (ii) For what values of $n$ is $D(n) = n?$ (iii) Define $D^2 (n) = D[D(n)],$ etc., and find the limit of $D^m (63)$ as $m$ tends to $\infty.$

For natural numbers $a, b$, define $Z(a,b)=\frac{(3a)!\cdot (4b)!}{a!^4 \cdot b!^3}$. [b](a)[/b] Prove that $Z(a, b)$ is an integer for $a \leq b$. [b](b)[/b] Prove that for each natural number $b$ there are infinitely many natural numbers a such that $Z(a, b)$ is not an integer.[/list]

Find all prime numbers $ p $ for which $ 1 + p\cdot 2^{p} $ is a perfect square.

[b]Q8.[/b] Given a triangle $ABC$ and $2$ point $K \in AB, \; N \in BC$ such that $BK=2AK, \; CN=2BN$ and $Q$ is the common point of $AN$ and $CK$. Compute $\dfrac{ S_{ \triangle ABC}}{S_{\triangle BCQ}}.$

Find the product of all possible real values for $k$ such that the system of equations $$x^2+y^2= 80$$ $$x^2+y^2= k+2x-8y$$ has exactly one real solution $(x,y)$. [i]Proposed by Nathan Xiong[/i]

The graph shows the constant rate at which Suzanna rides her bike. If she rides a total of a half an hour at the same speed, how many miles would she have ridden? [asy] size(200); import graph; /* this is a label */ Label f; f.p=fontsize(0); xaxis(-0.9,20,Ticks(f, 5.0, 5.0)); yaxis(-0.9,20, Ticks(f, 22.0,5.0)); // real f(real x) { return x; } draw(graph(f,-1,22),black+linewidth(1)); label("1", (-1,5), black); label("2", (-1, 10), black); label("3", (-1, 15), black); label("4", (-1, 20), black); dot((5,5), black+linewidth(5)); dot((10,10), black+linewidth(5)); dot((15, 15), black+linewidth(5)); dot((20,20), black+linewidth(5)); label("MINUTES", (11,-5), S); label(rotate(90)*"MILES", (-5,11), W);[/asy] $ \textbf{(A) }5\qquad\textbf{(B) }5.5\qquad\textbf{(C) }6\qquad\textbf{(D) }6.5\qquad\textbf{(E) }7 $

Let $ABC$ be a scalene triangle, $\Gamma$ its circumscribed circle and $H$ the point where the altitudes of triangle $ABC$ meet. The circumference with center at $H$ passing through $A$ cuts $\Gamma$ at a second point $D$. In the same way, the circles with center at $H$ and passing through $B$ and $C$ cut $\Gamma$ again at points $E$ and $F$, respectively. Prove that $H$ is also the point in which the altitudes of the triangle $DEF$ meet.

$ABC$ is a triangle and $R,Q,P$ are midpoints of $AB,AC,BC$. Line $AP$ intersects $RQ$ in $E$ and circumcircle of $ABC$ in $F$. $T,S$ are on $RP,PQ$ such that $ES\perp PQ,ET\perp RP$. $F'$ is on circumcircle of $ABC$ that $FF'$ is diameter. The point of intersection of $AF'$ and $BC$ is $E'$. $S',T'$ are on $AB,AC$ that $E'S'\perp AB,E'T'\perp AC$. Prove that $TS$ and $T'S'$ are perpendicular.

Professor Chang has nine different language books lined up on a bookshelf: two Arabic, three German, and four Spanish. How many ways are there to arrange the nine books on the shelf keeping the Arabic books together and keeping the Spanish books together? $\textbf{(A) }1440\qquad\textbf{(B) }2880\qquad\textbf{(C) }5760\qquad\textbf{(D) }182,440\qquad \textbf{(E) }362,880$

Isosceles triangles with a fixed angle $\alpha$ at the vertex opposite to the base are being inscribed into a rectangle $ABCD$ so that this vertex lies on the side $BC$ and the vertices of the base lie on the sides $AB$ and $CD$. Prove that the midpoints of the bases of all such triangles coincide. (Igor Zhizhilkin)

Let $ ABC$ be a triangle and $ PQRS$ a square with $ P$ on $ AB$, $ Q$ on $ AC$, and $ R$ and $ S$ on $ BC$. Let $ H$ on $ BC$ such that $ AH$ is the altitude of the triangle from $ A$ to base $ BC$. Prove that: (a) $ \frac{1}{AH} \plus{}\frac{1}{BC}\equal{}\frac{1}{PQ}$ (b) the area of $ ABC$ is twice the area of $ PQRS$ iff $ AH\equal{}BC$

Consider these two operations: \begin{align*} a \, \blacklozenge \, b &= a^2 - b^2\\ a \, \bigstar \, b &= (a - b)^2 \end{align*} What is the value of $(5 \, \blacklozenge \, 3) \, \bigstar \, 6?$ $\textbf{(A) } {-}20\qquad\textbf{(B) } 4\qquad\textbf{(C) } 16\qquad\textbf{(D) } 100\qquad\textbf{(E) } 220$

Let $(a_n)^\infty_{n=1}$ be a non-decreasing sequence of natural numbers with $a_{20}=100$. A sequence $(b_n)$ is defined by $b_m=\min\{n|an\ge m\}$. Find the maximum value of $a_1+a_2+\ldots+a_{20}+b_1+b_2+\ldots+b_{100}$ over all such sequences $(a_n)$.

If $\arccos\frac{4}{5}-\arccos\left(-\frac{4}{5}\right)=\arcsin x$, then $\text{(A)}x=\frac{24}{25}\qquad\text{(B)}x=-\frac{24}{25}\qquad\text{(C)}x=0\qquad\text{(D)}$ No such $x$

Prove that for any prime $p\ge5$, the number $$\sum_{0<k<\frac{2p}3}\binom pk$$is divisible by $p^2$.

Find, with proof, all polynomials $f$ such that $f$ has nonnegative integer coefficients, $f$($1$) = $8$ and $f$($2$) = $2012$.

Can two perfect cubes fit between two consecutive perfect squares? In other words, do there exist positive integers $a$, $b$, $n$ such that $n^2 < a^3 < b^3 < (n + 1)^2$? [i](3 points)[/i]

A group of $5$ rappers wants to make a song together. They each make their own parts for the song and then arrange the $5$ parts. J Cole wants to be friends with both Drake and Kendrick, so he wants his part to be adjacent to both of theirs. Find the number of possible songs (distinct orders) that can be made.

Let $a,b,c$ be pairwise distinct real numbers. Show that \[ (\frac{2a-b}{a-b})^2+(\frac{2b-c}{b-c})^2+(\frac{2c-a}{c-a})^2 \ge 5. \]

Consider the following sequence of positive real numbers $\dots<a_{-2}<a_{-1}<a_0<a_1<a_2<\dots$ infinite in both directions. For each positive integer $k$ let $b_k$ be the least integer such that the ratio between the sum of $k$ consecutive terms and the greatest of these $k$ terms is less than or equal to $b_k$(This fact occurs for any sequence of $k$ consecutive numbers). Prove that the sequence $b_1,b_2,b_3,...$ coincides with the sequence $1,2,3,...$ or is eventually constant.

In the diagram below, square $ABCD$ with side length 23 is cut into nine rectangles by two lines parallel to $\overline{AB}$ and two lines parallel to $\overline{BC}$. The areas of four of these rectangles are indicated in the diagram. Compute the largest possible value for the area of the central rectangle. [asy] size(250); defaultpen (linewidth (0.7) + fontsize (10)); draw ((0,0)--(23,0)--(23,23)--(0,23)--cycle); label("$A$", (0,23), NW); label("$B$", (23, 23), NE); label("$C$", (23,0), SE); label("$D$", (0,0), SW); draw((0,6)--(23,6)); draw((0,19)--(23,19)); draw((5,0)--(5,23)); draw((12,0)--(12,23)); label("13", (17/2, 21)); label("111",(35/2,25/2)); label("37",(17/2,3)); label("123",(2.5,12.5));[/asy] [i]Proposed by Lewis Chen[/i]

Let \[P(m)=\frac{m}{2} + \frac{m^2}{4}+ \frac{m^4}{8} + \frac{m^8}{8}.\] How many of the values of $P(2022)$, $P(2023)$, $P(2024)$, and $P(2025)$ are integers? $ \textbf{(A) }0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qquad \textbf{(E) }4 \qquad $