Five points are chosen on the sphere, no three of them lying on a great circle (a great circle is the intersection of the sphere with some plane passing through the sphere’s centre). Two great circles not containing any of the chosen points are called equivalent if one of them can be moved to the other without passing through any chosen points. (a) How many nonequivalent great circles not containing any chosen points can be drawn on the sphere? (b) Answer the same problem, but with $n$ chosen points.

Five equally skilled tennis players named Allen, Bob, Catheryn, David, and Evan play in a round robin tournament, such that each pair of people play exactly once, and there are no ties. In each of the ten games, the two players both have a 50% chance of winning, and the results of the games are independent. Compute the probability that there exist four distinct players $P_1$, $P_2$, $P_3$, $P_4$ such that $P_i$ beats $P_{i+1}$ for $i=1, 2, 3, 4$. (We denote $P_5=P_1$).

The Bank of Bath issues coins with an $H$ on one side and a $T$ on the other. Harry has $n$ of these coins arranged in a line from left to right. He repeatedly performs the following operation: if there are exactly $k>0$ coins showing $H$, then he turns over the $k$th coin from the left; otherwise, all coins show $T$ and he stops. For example, if $n=3$ the process starting with the configuration $THT$ would be $THT \to HHT \to HTT \to TTT$, which stops after three operations. (a) Show that, for each initial configuration, Harry stops after a finite number of operations. (b) For each initial configuration $C$, let $L(C)$ be the number of operations before Harry stops. For example, $L(THT) = 3$ and $L(TTT) = 0$. Determine the average value of $L(C)$ over all $2^n$ possible initial configurations $C$. [i]Proposed by David Altizio, USA[/i]

Non-degenerate triangle $ABC$ has $AB=20$, $AC=17$, and $BC=n$, an integer. How many possible values of $n$ are there? [i]2017 CCA Math Bonanza Lightning Round #2.2[/i]

Isabella is making sushi. She slices a piece of salmon into the shape of a solid triangular prism. The prism is $2$ cm thick, and its triangular faces have side lengths $7$ cm, $ 24$cm, and $25$ cm. Find the volume of this piece of salmon in cm$^3$. [i]Proposed by Isabella Li[/i]

Jean made a piece of stained glass art in the shape of two mountains, as shown in the figure below. One mountain peak is $8$ feet high and the other peak is $12$ feet high. Each peak forms a $90^\circ$ angle, and the straight sides of the mountains form $45^\circ$ with the ground. The artwork has an area of $183$ square feet. The sides of the mountains meet at an intersection point near the center of the artwork, $h$ feet above the ground. What is the value of $h$? [asy] unitsize(.3cm); filldraw((0,0)--(8,8)--(11,5)--(18,12)--(30,0)--cycle,gray(0.7),linewidth(1)); draw((-1,0)--(-1,8),linewidth(.75)); draw((-1.4,0)--(-.6,0),linewidth(.75)); draw((-1.4,8)--(-.6,8),linewidth(.75)); label("$8$",(-1,4),W); label("$12$",(31,6),E); draw((-1,8)--(8,8),dashed); draw((31,0)--(31,12),linewidth(.75)); draw((30.6,0)--(31.4,0),linewidth(.75)); draw((30.6,12)--(31.4,12),linewidth(.75)); draw((31,12)--(18,12),dashed); label("$45^{\circ}$",(.75,0),NE,fontsize(10pt)); label("$45^{\circ}$",(29.25,0),NW,fontsize(10pt)); draw((8,8)--(7.5,7.5)--(8,7)--(8.5,7.5)--cycle); draw((18,12)--(17.5,11.5)--(18,11)--(18.5,11.5)--cycle); draw((11,5)--(11,0),dashed); label("$h$",(11,2.5),E); [/asy] $\textbf{(A)}~4 \qquad \textbf{(B)}~5 \qquad \textbf{(C)}~4 \sqrt{2} \qquad \textbf{(D)}~6 \qquad \textbf{(E)}~5 \sqrt{2}$

The line joining the midpoints of the diagonals of a trapezoid has length $3$. If the longer base is $97$, then the shorter base is: $ \textbf{(A)}\ 94 \qquad\textbf{(B)}\ 92\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 89 $

Assume that $k$ and $n$ are two positive integers. Prove that there exist positive integers $m_1 , \dots , m_k$ such that \[1+\frac{2^k-1}{n}=\left(1+\frac1{m_1}\right)\cdots \left(1+\frac1{m_k}\right).\] [i]Proposed by Japan[/i]

The logarithm of $ 27\sqrt[4]{9}\sqrt[3]{9}$ to the base $ 3$ is: $ \textbf{(A)}\ 8\frac{1}{2} \qquad\textbf{(B)}\ 4\frac{1}{6} \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{none of these}$

Let $1 < a_1 < a_2 < \cdots < a_N$ be natural numbers. It is known that for any $1\leqslant i\leqslant N$ the product of all these numbers except $a_i$ increased by one, is a multiple of $a_i$. Prove that $a_1\leqslant N$.

When flipped, coin A shows heads $\textstyle\frac{1}{3}$ of the time, coin B shows heads $\textstyle\frac{1}{2}$ of the time, and coin C shows heads $\textstyle\frac{2}{3}$ of the time. Anna selects one of the coins at random and flips it four times, yielding three heads and one tail. The probability that Anna flipped coin A can be expressed as $\textstyle\frac{p}{q}$ for relatively prime positive integers $p$ and $q$. Compute $p + q$. [i]Proposed by Eugene Chen[/i]

Let $ABC$ be a triangle with $AB = 15, BC = 17$, $CA = 21$, and incenter $I$. If the circumcircle of triangle $IBC$ intersects side $AC$ again at $P$, find $CP$.

Let $N_6$ be the answer to problem 6. Given positive integers $n$ and $a$, the $n$[i]th tetration of[/i] $a$ is defined as \[ ^{n}a=\underbrace{a^{a^{\cdot^{\cdot^{\cdot^a}}}}}_{n \text{ times}}. \] For example, $^{4}2=2^{2^{2^2}}=2^{2^4}=2^{16}=65536$. Compute the units digit of $^{2024}N_6$.

Each of two regular polyhedrons $P$ and $Q$ was divided by the plane into two parts. One part of $P$ was attached to one part of $Q$ along the dividing plane and formed a regular polyhedron not equal to $P$ and $Q$. How many faces can it have?

A permutation $a_1,a_2,\cdots ,a_6$ of numbers $1,2,\cdots ,6$ can be transformed to $1,2,\cdots,6$ by transposing two numbers exactly four times. Find the number of such permutations.

Let $ P(x)$ be a polynomial with integer coefficients such that \[ P(m_1) \equal{} P(m_2) \equal{} P(m_3) \equal{} P(m_4) \equal{} 7\] for given distinct integers $ m_1,m_2,m_3,$ and $ m_4.$ Show that there is no integer m such that $ P(m) \equal{} 14.$

The incircle of a non-isosceles triangle $ABC$ has centre $I$ and is tangent to $BC$ and $CA$ in $D$ and $E$, respectively. Let $H$ be the orthocentre of $ABI$, let $K$ be the intersection of $AI$ and $BH$ and let $L$ be the intersection of $BI$ and $AH$. Show that the circumcircles of $DKH$ and $ELH$ intersect on the incircle of $ABC$.

For any positive integer $n$, we define $S(n)$ to be the sum of its digits in the decimal representation. Prove that for any positive integer $m$, there exists a positive integer $n$ such that $S(n)-S(n^2)>m$.

Let be a square $ ABCD, $ a point $ E $ on $ AB, $ a point $ N $ on $ CD, $ points $ F,M $ on $ BC, $ name $ P $ the intersection of $ AN $ with $ DE, $ and name $ Q $ the intersection of $ AM $ with $ EF. $ If the triangles $ AMN $ and $ DEF $ are equilateral, prove that $ PQ=FM. $

Let $k$ be a fixed real number. Find all functions $f: R \to R$ such that $f(x)+ (f(y))^2 = kf(x + y^2)$ for all real numbers $x$ and $y$.

$10$ integers are written in a row. A second row of $10$ integers is formed as follows: the integer written under each integer $A$ of the first row is equal to the total number of integers that stand to the right side of $A$ (in the first row) and are strictly greater than A. A third row is formed by the same way under the second one, and so on. (a) Prove that after several steps a “zero row” (i.e. a row consisting entirely of zeros) appears. (b) What is the maximal possible number of non-zero rows (i.e. rows in which at least one entry is not zero)? (S Tokarev)

Consider a triangle $ABC$ in which $AB<BC<CA$. The excircles touch the sides $BC, CA,$ and $AB$ at points $A_1, B_1$ and $C_1$ respectively. A circle is drawn through the points $A_1, B_1$ and $C_1$ which intersects the sides $BC, CA$ and $AB$ for the second time at the points $A_2, B_2$ and $C_2$ respectively. On which side of the triangle can lie the largest of the segments $A_1A_2, B_1B_2$ and $C_1C_2$? [i]Proposed by I. Weinstein[/i]

Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen? $ \textbf{(A)}\ 60 \qquad\textbf{(B)}\ 170 \qquad\textbf{(C)}\ 290 \qquad\textbf{(D)}\ 320 \qquad\textbf{(E)}\ 660 $

There is one nonzero digit in every cell of $2017\times 2017 $ table. On the board we writes $4034$ numbers that are rows and columns of table. It is known, that $4033$ numbers are divisible by prime $p$ and last is not divisible by $p$. Find all possible values of $p$. [hide=Example]Example for $2\times2$. If table is |1|4| |3|7|. Then numbers on board are $14,37,13,47$[/hide]

Let $P(x)$ be a polynomial with integer coefficients. Prove that the polynomial $Q(x) = P(x^4)P(x^3)P(x^2)P(x)+1$ has no integer roots.