How many ways are there to divide $10$ candies between $3$ Berkeley students and $4$ Stanford students, if each Berkeley student must get at least one candy? All students are distinguishable from each other; all candies are indistinguishable.

Consider a regular octahedron $ABCDEF$ with lower vertex $E$, upper vertex $F$, middle cross-section $ABCD$, midpoint $M$ and circumscribed sphere $k$. Further, let $X$ be an arbitrary point inside the face $ABF$. Let the line $EX$ intersect $k$ in $E$ and $Z$, and the plane $ABCD$ in $Y$. Show that $\sphericalangle{EMZ}=\sphericalangle{EYF}$.

Let $\overline{AB}$ be a chord of a circle $\omega$, and let $P$ be a point on the chord $\overline{AB}$. Circle $\omega_1$ passes through $A$ and $P$ and is internally tangent to $\omega$. Circle $\omega_2$ passes through $B$ and $P$ and is internally tangent to $\omega$. Circles $\omega_1$ and $\omega_2$ intersect at points $P$ and $Q$. Line $PQ$ intersects $\omega$ at $X$ and $Y$. Assume that $AP=5$, $PB=3$, $XY=11$, and $PQ^2 = \tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Here’s a screenshot of the problem. If someone could LaTEX a diagram, that would be great!

Find the number $\overline {523abc}$ divisible by $7, 8$ and $9$.

A group $G$ has elements $g,h$ satisfying $ghg=hg^{2}h, g^{3}=1$ and $h^n=1$ for some odd integer $n$. Prove that $h=e$, where $e$ is the identity element.

Let $p$ be a prime number and let $\mathbb{F}_p$ be the finite field with $p$ elements. Consider an automorphism $\tau$ of the polynomial ring $\mathbb{F}_p[x]$ given by \[\tau(f)(x)=f(x+1).\] Let $R$ denote the subring of $\mathbb{F}_p[x]$ consisting of those polynomials $f$ with $\tau(f)=f$. Find a polynomial $g \in \mathbb{F}_p[x]$ such that $\mathbb{F}_p[x]$ is a free module over $R$ with basis $g,\tau(g),\dots,\tau^{p-1}(g)$.

A square with side length $6$ has a circle with radius $2$ inside of it, with the centers of the square and circle vertically aligned. Aarush is standing $4$ units directly above the center of the circle, at point $P.$ What is the area of the region inside the square that he can see? (Assume that Aarush can only see parts of the square along straight lines of sight from $P$ that are unblocked by any other objects.) [center] [img] https://cdn.artofproblemsolving.com/attachments/7/7/ede4444ab82235fc90a27c0b481d320b486cf2.png [/img] [/center]

Let $p$, $q$, and $r$ be primes satisfying \[ pqr = 189999999999999999999999999999999999999999999999999999962. \] Compute $S(p) + S(q) + S(r) - S(pqr)$, where $S(n)$ denote the sum of the decimals digits of $n$. [i]Proposed by Evan Chen[/i]

An integer $n$ will be called [i]good[/i] if we can write \[n=a_1+a_2+\cdots+a_k,\] where $a_1,a_2, \ldots, a_k$ are positive integers (not necessarily distinct) satisfying \[\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=1.\] Given the information that the integers 33 through 73 are good, prove that every integer $\ge 33$ is good.

A prime number $p \ge 5$ is given. Write $\frac13+\frac24+... +\frac{p -3}{p - 1}=\frac{a}{b}$ for natural numbers $a$ and $b$. Show that $p$ divides $a$.

Before Harry Potter died, he decided to bury his wand in one of eight possible locations (uniformly at random). A squad of Death Eaters decided to go hunting for the wand. They know the eight locations but have poor vision, so even if they're at the correct location they only have a $50\%$ chance of seeing the wand. They also get tired easily, so they can only check three different locations a day. At least they have one thing going for them: they're clever. Assuming they strategize optimally, what is the expected number of days it will take for them to find the wand? [i]2019 CCA Math Bonanza Individual Round #15[/i]

Do there exist $2011$ positive integers $a_1 < a_2 < \ldots < a_{2011}$ such that $\gcd(a_i,a_j) = a_j - a_i$ for any $i$, $j$ such that $1 \le i < j \le 2011$?

If $ x_{k\plus{}1} \equal{} x_k \plus{} \frac12$ for $ k\equal{}1, 2, \dots, n\minus{}1$ and $ x_1\equal{}1,$ find $ x_1 \plus{} x_2 \plus{} \dots \plus{} x_n.$ $ \textbf{(A)}\ \frac{n\plus{}1}{2} \qquad \textbf{(B)}\ \frac{n\plus{}3}{2} \qquad \textbf{(C)}\ \frac{n^2\minus{}1}{2} \qquad \textbf{(D)}\ \frac{n^2\plus{}n}{4} \qquad \textbf{(E)}\ \frac{n^2\plus{}3n}{4}$

Let $n$ be an odd positive integer bigger than $1$. Prove that $3^n+1$ is not divisible with $n$

Let $A\in\mathcal{M}_n(\mathbb{C})$ where $n\geq 2.$ Prove that if $m=|\{\text{rank}(A^k)-\text{rank}(A^{k+1})":k\in\mathbb{N}^*\}|$ then $n+1\geq m(m+1)/2.$

Find positive integer $n$ so that $\tfrac{80-6\sqrt{n}}{n}$ is the reciprocal of $\tfrac{80+6\sqrt{n}}{n}.$

Given is a triangle $ABC$. The points $P, Q$ lie on the extensions of $BC$ beyond $B, C$, respectively, such that $BP=BA$ and $CQ=CA$. Prove that the circumcenter of triangle $APQ$ lies on the angle bisector of $\angle BAC$.

Let $(a_n)_{n=0}^{\infty}$ be a sequence of real numbers such that $a_0=0$ and $$a_{n+1}^3=a_n^2-8\quad \text{for} \quad n=0,1,2,…$$ Prove that the following series is convergent: $$\sum_{n=0}^{\infty}{|a_{n+1}-a_n|}.$$ [i]Proposed by Orif Ibrogimov, National University of Uzbekistan[/i]

There are $256$ players in a tennis tournament who are ranked from $1$ to $256$, with $1$ corresponding to the highest rank and $256$ corresponding to the lowest rank. When two players play a match in the tournament, the player whose rank is higher wins the match with probability $\frac{3}{5}$. In each round of the tournament, the player with the highest rank plays against the player with the second highest rank, the player with the third highest rank plays against the player with the fourth highest rank, and so on. At the end of the round, the players who win proceed to the next round and the players who lose exit the tournament. After eight rounds, there is one player remaining and they are declared the winner. Determine the expected value of the rank of the winner.

Let $a_{i}$ and $b_{i}$ ($i=1,2, \cdots, n$) be rational numbers such that for any real number $x$ there is: \[x^{2}+x+4=\sum_{i=1}^{n}(a_{i}x+b)^{2}\] Find the least possible value of $n$.

Can any triangle be cut into four convex figures: a triangle, a quadrilateral, a pentagon, a hexagon?

Define $A(n)$ as the average of all positive divisors of the positive integer $n$. Find the sum of all solutions to $A(n)=42$. [i] Proposed by Yannick Yao [/i]

The triangle $ABC$ is isosceles with apex at $A{}$ and $M,N,P$ are the midpoints of the sides $BC,CA,AB$ respectively. Let $Q{}$ and $R{}$ be points on the segments $BM$ and $CM$ such that $\angle BAQ =\angle MAR.$ The segment $NP{}$ intersects $AQ,AR$ at $U,V$ respectively. The point $S{}$ is considered on the ray $AQ$ such that $SV$ is the angle bisector of $\angle ASM.$ Similarly, the point $T{}$ lies on the ray $AR$ uch that $TU$ is the angle bisector of $\angle ATM.$ Prove that one of the intersection points of the circles $(NUS)$ and $(PVT)$ lies on the line $AM.$ [i]Proposed by Flavian Georgescu[/i]

Given a triangle $ABC$ inscribed in circle $c(O,R)$ (with center $O$ and radius $R$) with $AB<AC<BC$ and let $BD$ be a diameter of the circle $c$. The perpendicular bisector of $BD$ intersects line $AC$ at point $M$ and line $AB$ at point $N$. Line $ND$ intersects the circle $c$ at point $T$. Let $S$ be the second intersection point of cicumcircles $c_1$ of triangle $OCM$, and $c_2$ of triangle $OAD$. Prove that lines $AD, CT$ and $OS$ pass through the same point.