Let $A$ be a fixed point on a circle, $B$ and$ C$ be arbitrary points on the circle different from $A$ and at different distances. The bisector of the angle $\angle BAC$ intersects the chord $BC$ and the circle at points $K$ and $P$, $D$ is the projection of $A$ onto the straight line $BC$. A circle passing through points $K$, $P$ and $D$ intersects the straight line $AD$ for the second time at point $M$. Find the locus of points $M$.

In a school there are 2008 students. Students are members of certain committees. A committee has at most 1004 members and every two students join a common committee. (a) Determine the smallest possible number of committees in the school. (b) If it is further required that the union of any two committees consists of at most 1800 students, will your answer in (a) still hold?

We have a number of towns, with bus routes between some of them (each bus route goes from a town to another town without any stops). It is known that you can get from any town to any other by bus (possibly changing buses several times). Mr. Ivanov bought one ticket for each of the bus routes (a ticket allows single travel in either direction, but not returning on the same route). Mr. Petrov bought n tickets for each of the bus routes. Both Ivanov and Petrov started at town A. Ivanov used up all his tickets without buying any new ones and finished his travel at town B. Petrov, after using some of his tickets, got stuck at town X: he can not leave it without buying a new ticket. Prove that X is either A or B.

Let $ABCD$ be a tetrahedron such that $AD \perp BD$, $BD \perp CD$, $CD \perp AD$ and $AD=10$, $BD=15$, $CD=20$. Let $E$ and $F$ be points such that $E$ lies on $BC$, $DE \perp BC$, and $ADEF$ is a rectangle. If $S$ is the solid consisting of all points inside $ABCD$ but outside $FBCD$, compute the volume of $S$. [i]2015 CCA Math Bonanza Individual Round #13[/i]

Positive integer divisors $a$ and $b$ of $n$ are called [i]complementary[/i] if $ab=n$. Given that $N$ has a pair of complementary divisors that differ by $20$ and a pair of complementary divisors that differ by $23$, find the sum of the digits of $N$. $\textbf{(A) } 11 \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 15 \qquad \textbf{(D) } 17 \qquad \textbf{(E) } 19$

Hapi, the god of the annual flooding of the Nile is preparing for this year’s flooding. The shape of the channel of the Nile can be described by the function $y = \frac{-1000}{ x^2+100}$ where the $x$ and $y$ coordinates are in metres. The depth of the river is $5$ metres now. Hapi plans to increase the water level by $3$ metres. How many metres wide will the river be after the flooding? The depth of the river is always measured at its deepest point. [img]https://cdn.artofproblemsolving.com/attachments/8/3/4e1d277e5cacf64bf82c110d521747592b928e.png[/img]

Prove that for any odd prime number $ p,$ the number of positive integer $ n$ satisfying $ p|n! \plus{} 1$ is less than or equal to $ cp^\frac{2}{3}.$ where $ c$ is a constant independent of $ p.$

In a $100$ by $100$ grid, there is a spider and $100$ bugs. Each time, the spider can walk up, down, left or right, and the spider aims to visit all the squares with bugs to eat them all. The spider begins from the top-left corner. Show that no matter where the bugs are, the spider can always eat them all within $2000$ steps.

Define $ x \heartsuit y$ to be $ |x\minus{}y|$ for all real numbers $ x$ and $ y$. Which of the following statements is [b]not[/b] true? $\textbf{(A)}\ x \heartsuit y \equal{} y \heartsuit x \text{ for all } x \text{ and } y$ $\textbf{(B)}\ 2(x \heartsuit y) \equal{} (2x) \heartsuit (2y) \text{ for all } x \text{ and } y$ $\textbf{(C)}\ x \heartsuit 0 \equal{} x \text{ for all } x$ $\textbf{(D)}\ x \heartsuit x \equal{} 0 \text{ for all } x$ $\textbf{(E)}\ x \heartsuit y > 0 \text{ if } x \ne y$

Find the number of rational numbers $r$, $0<r<1$, such that when $r$ is written as a fraction in lowest terms, the numerator and denominator have a sum of $1000$.

Find $4$ consecutive positive integers whose product is $1680$.

Points $E$ and $F$ are chosen on the base side $AD$ and the lateral side $AB$ of an isosceles trapezoid $ABCD$, respectively. Quadrilateral $CDEF$ is an isosceles trapezoid as well. Prove that $AE \cdot ED = AF \cdot FB$.

Let $n$ and $k$ be two integers with $n>k\geqslant 1$. There are $2n+1$ students standing in a circle. Each student $S$ has $2k$ [i]neighbors[/i] - namely, the $k$ students closest to $S$ on the left, and the $k$ students closest to $S$ on the right. Suppose that $n+1$ of the students are girls, and the other $n$ are boys. Prove that there is a girl with at least $k$ girls among her neighbors. [i]Proposed by Gurgen Asatryan, Armenia[/i]

Let $a_1, a_2, \ldots, a_{2024}$ be a permutation of $1, 2, \ldots, 2024$. Find the minimum possible value of\[\sum_{i=1} ^{2023} \Big[(a_i+a_{i+1})\Big(\frac{1}{a_i}+\frac{1}{a_{i+1}}\Big)+\frac{1}{a_ia_{i+1}}\Big]\] [i]Proposed by Md. Ashraful Islam Fahim[/i]

Let $ABC$ be an acute-angled triangle and let $D, E$, and $F$ be the feet of altitudes from $A, B$, and $C$ to sides $BC, CA$, and $AB$, respectively. Denote by $\omega_B$ and $\omega_C$ the incircles of triangles $BDF$ and $CDE$, and let these circles be tangent to segments $DF$ and $DE$ at $M$ and $N$, respectively. Let line $MN$ meet circles $\omega_B$ and $\omega_C$ again at $P \ne M$ and $Q \ne N$, respectively. Prove that $MP = NQ$. (Vietnam)

Prove that you can represent an arbitrary number not exceeding $n!$ as a sum of $k$ different numbers ($k\le n$) that are divisors of $n!$.

A circle of radius $ 10$ inches has its center at the vertex $ C$ of an equilateral triangle $ ABC$ and passes through the other two vertices. The side $ AC$ extended through $ C$ intersects the circle at $ D$. The number of degrees of angle $ ADB$ is: $ \textbf{(A)}\ 15 \qquad\textbf{(B)}\ 30 \qquad\textbf{(C)}\ 60 \qquad\textbf{(D)}\ 90 \qquad\textbf{(E)}\ 120$

Hamster is playing a game on an $m \times n$ chessboard. He places a rook anywhere on the board and then moves it around with the restriction that every vertical move must be followed by a horizontal move and every horizontal move must be followed by a vertical move. For what values of $m,n$ is it possible for the rook to visit every square of the chessboard exactly once? A square is only considered visited if the rook was initially placed there or if it ended one of its moves on it. [i]Brian Hamrick.[/i]

In the net drawn below, in how many ways can one reach the point $3n+1$ starting from the point $1$ so that the labels of the points on the way increase? [asy] import graph; size(12cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.3,xmax=12.32,ymin=-10.66,ymax=6.3; draw((1,2)--(xmax,0*xmax+2)); draw((1,0)--(xmax,0*xmax+0)); draw((0,1)--(1,2)); draw((1,0)--(0,1)); draw((1,2)--(3,0)); draw((1,0)--(3,2)); draw((3,2)--(5,0)); draw((3,0)--(5,2)); draw((5,2)--(7,0)); draw((5,0)--(7,2)); draw((7,2)--(9,0)); draw((7,0)--(9,2)); dot((1,0),linewidth(1pt)+ds); label("2",(0.96,-0.5),NE*lsf); dot((0,1),linewidth(1pt)+ds); label("1",(-0.42,0.9),NE*lsf); dot((1,2),linewidth(1pt)+ds); label("3",(0.98,2.2),NE*lsf); dot((2,1),linewidth(1pt)+ds); label("4",(1.92,1.32),NE*lsf); dot((3,2),linewidth(1pt)+ds); label("6",(2.94,2.2),NE*lsf); dot((4,1),linewidth(1pt)+ds); label("7",(3.94,1.32),NE*lsf); dot((6,1),linewidth(1pt)+ds); label("10",(5.84,1.32),NE*lsf); dot((3,0),linewidth(1pt)+ds); label("5",(2.98,-0.46),NE*lsf); dot((5,2),linewidth(1pt)+ds); label("9",(4.92,2.24),NE*lsf); dot((5,0),linewidth(1pt)+ds); label("8",(4.94,-0.42),NE*lsf); dot((8,1),linewidth(1pt)+ds); label("13",(7.88,1.34),NE*lsf); dot((7,2),linewidth(1pt)+ds); label("12",(6.8,2.26),NE*lsf); dot((7,0),linewidth(1pt)+ds); label("11",(6.88,-0.38),NE*lsf); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]

Pentagon \(ABCDE\) is inscribed in a circle. Its diagonals \(AC\) and \(BD\) intersect at \(F\). The bisectors of \(\angle BAC\) and \(\angle CDB\) intersect at \(G\). Let \(AG\) intersect \(BD\) at \(H\), let \(DG\) intersect \(AC\) at \(I\), and let \(EG\) intersect \(AD\) at \(J\). If \(FHGI\) is cyclic and \[JA \cdot FC \cdot GH = JD \cdot FB \cdot GI,\] prove that \(G\), \(F\) and \(E\) are collinear.

$ABCD$ is a cyclic quadrilateral and $\omega$ its circumcircle. The perpendicular line to $AC$ at $D$ intersects $AC$ at $E$ and $\omega$ at F. Denote by $\ell$ the perpendicular line to $BC$ at $F$. The perpendicular line to $\ell$ at A intersects $\ell$ at $G$ and $\omega$ at $H$. Line $GE$ intersects $FH$ at $I$ and $CD$ at $J$. Prove that points $C, F, I$ and $J$ are concyclic

Let $ABCD$ be a parallelogram. Half-circles $\omega_1,\omega_2,\omega_3$ and $\omega_4$ with diameters $AB,BC,CD$ and $DA$, respectively, are erected on the exterior of $ABCD$. Line $l_1$ is parallel to $BC$ and cuts $\omega_1$ at $X$, segment $AB$ at $P$, segment $CD$ at $R$ and $\omega_3$ at $Z$. Line $l_2$ is parallel to $AB$ and cuts $\omega_2$ at $Y$, segment $BC$ at $Q$, segment $DA$ at $S$ and $\omega_4$ at $W$. If $XP\cdot RZ=YQ\cdot SW$, prove that $PQRS$ is cyclic. [i]Proposed by José Alejandro Reyes González[/i]

For which values of $ z$ does the series $ \sum_{n\equal{}1}^{\infty}c_1c_2\cdots c_n z^n$ converge, provided that $ c_k>0$ and $ \sum_{k\equal{}1}^{\infty} \frac{c_k}{k}<\infty$ ?

At the beginning of the school year, $ 50\%$ of all students in Mr. Well's math class answered "Yes" to the question "Do you love math", and $ 50\%$ answered "No." At the end of the school year, $ 70\%$ answered "Yes" and $ 30\%$ answered "No." Altogether, $ x\%$ of the students gave a different answer at the beginning and end of the school year. What is the difference between the maximum and the minimum possible values of $ x$? $ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 60\qquad\textbf{(E)}\ 80$

Each face and each vertex of a regular tetrahedron is coloured red or blue. How many different ways of colouring are there? (Two tetrahedrons are said to have the same colouring if we can rotate them suitably so that corresponding faces and vertices are of the same colour.