Let $ABC$ be an isosceles triangle with $\angle{ABC} = \angle{ACB} = 80^\circ$. Let $D$ be a point on $AB$ such that $\angle{DCB} = 60^\circ$ and $E$ be a point on $AC$ such that $\angle{ABE} = 30^\circ$. Find $\angle{CDE}$ in degrees.

One dimension of a cube is increased by $ 1$, another is decreased by $ 1$, and the third is left unchanged. The volume of the new rectangular solid is $ 5$ less than that of the cube. What was the volume of the cube? $ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 27 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ 125 \qquad \textbf{(E)}\ 216$

How many different prime numbers are factors of $ N$ if \[ \log_2 (\log_3 (\log_5 (\log_7 N))) \equal{} 11? \]$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 7$

Prove that the diagonals of a convex quadrilateral are perpendicular if and only if the sum of the squares of one pair of opposite sides is equal to the sum of the squares of the other pair.

Find the area of the figure bounded by two curves $y=x^4,\ y=x^2+2$.

Find the greatest positive integer $n$ for which there exist $n$ nonnegative integers $x_1, x_2,\ldots , x_n$, not all zero, such that for any $\varepsilon_1, \varepsilon_2, \ldots, \varepsilon_n$ from the set $\{-1, 0, 1\}$, not all zero, $\varepsilon_1 x_1 + \varepsilon_2 x_2 + \cdots + \varepsilon_n x_n$ is not divisible by $n^3$.

A perpendicular bisector of side $BC$ of triangle $ABC$ meets lines $AB$ and $AC$ at points $A_B$ and $A_C$ respectively. Let $O_a$ be the circumcenter of triangle $AA_BA_C$. Points $O_b$ and $O_c$ are defined similarly. Prove that the circumcircle of triangle $O_aO_bO_c$ touches the circumcircle of the original triangle.

Find the number of ways there are to permute the elements of the set $\{1,2,3,4,5,6,7,8,9\}$ such that no two adjacent numbers are both even or both odd. [i]Proposed by Ephram Chun[/i]

Show that if for non-negative integers $m$, $n$, $N$, $k$ the equation \[(n^2+1)^{2^k}\cdot(44n^3+11n^2+10n+2)=N^m\] holds, then $m = 1$.

[i]25 problems for 30 minutes[/i] [b]p1.[/b] Alisha has $6$ cupcakes and Tyrone has $10$ brownies. Tyrone gives some of his brownies to Alisha so that she has three times as many desserts as Tyrone. How many desserts did Tyrone give to Alisha? [b]p2.[/b] Bisky adds one to her favorite number. She then divides the result by $2$, and gets $56$. What is her favorite number? [b]p3.[/b] What is the maximum number of points at which a circle and a square can intersect? [b]p4.[/b] An integer $N$ leaves a remainder of 66 when divided by $120$. Find the remainder when $N$ is divided by $24$. [b]p5.[/b] $7$ people are chosen to run for student council. How many ways are there to pick $1$ president, $1$ vice president, and $1$ secretary? [b]p6.[/b] Anya, Beth, Chloe, and Dmitri are all close friends, and like to make group chats to talk. How many group chats can be made if Dmitri, the gossip, must always be in the group chat and Anya is never included in them? Group chats must have more than one person. [b]p7.[/b] There exists a telephone pole of height $24$ feet. From the top of this pole, there are two wires reaching the ground in opposite directions, with one wire $25$ feet, and the other wire 40 feet. What is the distance (in feet) between the places where the wires hit the ground? [b]p8.[/b] Tarik is dressing up for a job-interview. He can wear a chill, business, or casual outfit. If he wears a chill oufit, he must wear a t-shirt, shorts, and flip-flops. He has eight of the first, seven of the second, and three of the third. If he wears a business outfit, he must wear a blazer, a tie, and khakis; he has two of the first, six of the second, and five of the third; finally, he can also choose the casual style, for which he has three hoodies, nine jeans, and two pairs of sneakers. How many different combinations are there for his interview? [b]p9.[/b] If a non-degenerate triangle has sides $11$ and $13$, what is the sum of all possibilities for the third side length, given that the third side has integral length? [b]p10.[/b] An unknown disease is spreading fast. For every person who has the this illness, it is spread on to $3$ new people each day. If Mary is the only person with this illness at the start of Monday, how many people will have contracted the illness at the end of Thursday? [b]p11.[/b] Gob the giant takes a walk around the equator on Mars, completing one lap around Mars. If Gob’s head is $\frac{13}{\pi}$ meters above his feet, how much farther (in meters) did his head travel than his feet? [b]p12.[/b] $2022$ leaves a remainder of $2$, $6$, $9$, and $7$ when divided by $4$, $7$, $11$, and $13$ respectively. What is the next positive integer which has the same remainders to these divisors? [b]p13.[/b] In triangle $ABC$, $AB = 20$, $BC = 21$, and $AC = 29$. Let D be a point on $AC$ such that $\angle ABD = 45^o$. If the length of $AD$ can be represented as $\frac{a}{b}$ , what is $a + b$? [b]p14.[/b] Find the number of primes less than $100$ such that when $1$ is added to the prime, the resulting number has $3$ divisors. [b]p15.[/b] What is the coefficient of the term $a^4z^3$ in the expanded form of $(z - 2a)^7$? [b]p16.[/b] Let $\ell$ and $m$ be lines with slopes $-2$, $1$ respectively. Compute $|s_1 \cdot s_2|$ if $s_1$, $s_2$ represent the slopes of the two distinct angle bisectors of $\ell$ and $m$. [b]p17.[/b] R1D2, Lord Byron, and Ryon are creatures from various planets. They are collecting monkeys for King Avanish, who only understands octal (base $8$). R1D2 only understands binary (base $2$), Lord Byron only understands quarternary (base $4$), and Ryon only understands decimal (base $10$). R1D2 says he has $101010101$ monkeys and adds his monkey to the pile. Lord Byron says he has $3231$ monkeys and adds them to the pile. Ryon says he has $576$ monkeys and adds them to the pile. If King Avanish says he has $x$ monkeys, what is the value of $x$? [b]p18.[/b] A quadrilateral is defined by the origin, $(3, 0)$, $(0, 10)$, and the vertex of the graph of $y = x^2 -8x+22$. What is the area of this quadrilateral? [b]p19.[/b] There is a sphere-container, filled to the brim with fruit punch, of diameter $6$. The contents of this container are poured into a rectangular prism container, again filled to the brim, of dimensions $2\pi$ by $4$ by $3$. However, there is an excess amount in the original container. If all the excess drink is poured into conical containers with diameter $4$ and height $3$, how many containers will be used? [b]p20.[/b] Brian is shooting arrows at a target, made of concurrent circles of radius $1$, $2$, $3$, and $4$. He gets $10$ points for hitting the innermost circle, $8$ for hitting between the smallest and second smallest circles, $5$ for between the second and third smallest circles, $2$ points for between the third smallest and outermost circle, and no points for missing the target. Assume for each shot he takes, there is a $20\%$ chance Brian will miss the target, but otherwise the chances of hitting each target are proportional to the area of the region. The chance that after three shots, Brian will have scored $15$ points can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$. Find $m + n$. [b]p21.[/b] What is the largest possible integer value of $n$ such that $\frac{2n^3+n^2+7n-15}{2n+1}$ is an integer? [b]p22.[/b] Let $f(x, y) = x^3 + x^2y + xy^2 + y^3$. Compute $f(0, 2) + f(1, 3) +... f(9, 11).$ [b]p23.[/b] Let $\vartriangle ABC$ be a triangle. Let $AM$ be a median from $A$. Let the perpendicular bisector of segment $\overline{AM}$ meet $AB$ and $AC$ at $D$, $E$ respectively. Given that $AE = 7$, $ME = MC$, and $BDEC$ is cyclic, then compute $AM^2$. [b]p24.[/b] Compute the number of ordered triples of positive integers $(a, b, c)$ such that $a \le 10$, $b \le 11$, $c \le 12$ and $a > b - 1$ and $b > c - 1$. [b]p25.[/b] For a positive integer $n$, denote by $\sigma (n)$ the the sum of the positive integer divisors of $n$. Given that $n + \sigma (n)$ is odd, how many possible values of $n$ are there from $1$ to $2022$, inclusive? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\gamma$ and $\Gamma$ be two circles such that $\gamma$ is internally tangent to $\Gamma$ at a point $X$. Let $P$ be a point on the common tangent of $\gamma$ and $\Gamma$ and $Y$ be the point on $\gamma$ other than $X$ such that $PY$ is tangent to $\gamma$ at $Y$. Let $PY$ intersect $\Gamma$ at $A$ and $B$, such that $A$ is in between $P$ and $B$ and let the tangents to $\Gamma$ at $A$ and $B$ intersect at $C$. $CX$ intersects $\Gamma$ again at $Z$ and $ZY$ intersects $\Gamma$ again at $Q$. If $AQ = 6, AB = 10$ and $\tfrac{AX}{XB} = \tfrac{1}{4}$. The length of $QZ = \tfrac{p}{q}\sqrt{r}$ where $p$ and $q$ are coprime positive integers, and $r$ is square free positive integer. Find $p + q + r$.

Three different one-digit positive integers are placed in the bottom row of cells. Numbers in adjacent cells are added and the sum is placed in the cell above them. In the second row, continue the same process to obtain a number in the top cell. What is the difference between the largest and smallest numbers possible in the top cell? [asy]path cell=((0,0)--(1,0)--(1,1)--(0,1)--cycle); path sw=((0,0)--(1,sqrt(3))); path se=((5,0)--(4,sqrt(3))); draw(cell, linewidth(1)); draw(shift(2,0)*cell, linewidth(1)); draw(shift(4,0)*cell, linewidth(1)); draw(shift(1,3)*cell, linewidth(1)); draw(shift(3,3)*cell, linewidth(1)); draw(shift(2,6)*cell, linewidth(1)); draw(shift(0.45,1.125)*sw, EndArrow); draw(shift(2.45,1.125)*sw, EndArrow); draw(shift(1.45,4.125)*sw, EndArrow); draw(shift(-0.45,1.125)*se, EndArrow); draw(shift(-2.45,1.125)*se, EndArrow); draw(shift(-1.45,4.125)*se, EndArrow); label("$+$", (1.5,1.5)); label("$+$", (3.5,1.5)); label("$+$", (2.5,4.5));[/asy] $ \textbf{(A)}\ 16 \qquad \textbf{(B)}\ 24 \qquad \textbf{(C)}\ 25 \qquad \textbf{(D)}\ 26 \qquad \textbf{(E)}\ 35$

Let $n=2^{31}3^{19}.$ How many positive integer divisors of $n^2$ are less than $n$ but do not divide $n$?

A test contains $5$ multiple choice questions which have $4$ options in each. Suppose each examinee chose one option for each question. There exists a number $n$, such that for any $n$ sheets among $2000$ sheets of answer papers, there are $4$ sheets of answer papers such that any two of them have at most $3$ questions with the same answers. Find the minimum value of $n$.

Find the largest integer $n$ for which $2^n$ divides \[ \binom 21 \binom 42 \binom 63 \dots \binom {128}{64}. \][i]Proposed by Evan Chen[/i]

Let $0<a<b$.Prove the following inequaliy. \[\frac{1}{b-a}\int_a^b \left(\ln \frac{b}{x}\right)^2 dx<2\]

Given a finite tree $ T$ and isomorphism $ f: T\rightarrow T$. Prove that either there exist a vertex $ a$ such that $ f(a)\equal{}a$ or there exist two neighbor vertices $ a$, $ b$ such that $ f(a)\equal{}b$, $ f(b)\equal{}a$.

Problem-2: Given a circle with center $O$, the two tangent lines from a point $S$ outside the circle touch the circle at points $P$ and $Q$. Line $SO$ intersects the circle at $A$ and $B$, with $B$ closer to $S$. Let $X$ be an interior point of minor arc $PB$, and let line $OS$ intersect lines $QX$ and $PX$ at $C$ and $D$, respectively. Prove that $\frac{1}{\left| AC \right|}+\frac{1}{\left| AD \right|}=\frac{2}{\left| AB \right|}$.

Several people visited a library yesterday. Each one visited the library just once (in the course of yesterday). Amongst any three of them, there were two who met in the library. Prove that there were two moments $T$ and $T$' yesterday such that everyone who visited the library yesterday was in the library at $T$ or $T'$ (or both).

p1. Find an integer that has the following properties: a) Every two adjacent digits in the number are prime. b) All prime numbers referred to in item (a) above are different. p2. Determine all integers up to $\sqrt{50+\sqrt{n}}+\sqrt{50-\sqrt{n}}$ p3. The following figure shows the path to form a series of letters and numbers “OSN2015”. Determine as many different paths as possible to form the series of letters and numbers by following the arrows. [img]https://cdn.artofproblemsolving.com/attachments/6/b/490a751457871184a506c2966f8355f20cebbd.png[/img] p4. Given an acute triangle $ABC$ with $L$ as the circumcircle. From point $A$, a perpendicular line is drawn on the line segment $BC$ so that it intersects the circle $L$ at point $X$. In a similar way, a perpendicular line is made from point $B$ and point $C$ so that it intersects the circle $L$, at point $Y$ and point $Z$, respectively. Is arc length $AY$ = arc length $AZ$ ? p5. The students of class VII.3 were divided into five groups: $A, B, C, D$ and $E$. Each group conducted five science experiments for five weeks. Each week each group performs an experiment that is different from the experiments conducted by other groups. Determine at least two possible trial schedules in week five, based on the following information: $\bullet$ In the first week, group$ D$ did experiment $4$. $\bullet$ In the second week, group $C$ did the experiment $5$. $\bullet$ In the third week, group $E$ did the experiment $5$. $\bullet$ In the fourth week, group $A$ did experiment $4$ and group $D$ did experiment $2$.

Given a right triangle $ABC$ with the hypotenuse $AB$. Let $K$ be any interior point of triangle $ABC$ and points $L, M$ are symmetric of point $K$ wrt lines $BC, AC$ respectively. Specify all possible values for $S_{ABLM} / S_{ABC}$, where $S_{XY ... Z}$ indicates the area of the polygon $XY...Z$ .

Let $\left( A,+, \cdot \right)$ be a ring that verifies the following properties: (i) it has a unit, $1$, and its order is $p$, a prime number; (ii) there is $B \subset A, \, |B| = p$, such that: for all $x,y \in A$, there is $b \in B$ such that $xy = byx$. Prove that $A$ is commutative. [i]Ion Savu[/i]

For any positive integer $n$, let $r_n$ denote the greatest odd divisor of $n$. Compute $T =r_{100}+ r_{101} + r_{102}+...+r_{200}$

A cyclic quadrilateral $ABCD$ is given. The lines $AB$ and $DC$ meet at point $E$, and the lines $BC$ and $AD$ meet at point $F$. Let $I$ be the incenter of triangle $AED$, and a ray with origin $F$ be perpendicular to the bisector of angle AID. In which ratio this ray dissects the angle $AFB$?

Let $ABC$ be a triangle. Find all the triangles $XYZ$ with vertices inside triangle $ABC$ such that $XY,YZ,ZX$ and six non-intersecting segments from the following $AX, AY, AZ, BX, BY, BZ, CX, CY, CZ$ divide the triangle $ABC$ into seven regions with equal areas.