Evaluate $\dfrac{d}{dx}\left(\sin x - \dfrac{4}{3}\sin^3 x\right)$ when $x=15$.

Let $ABC$ be a triangle with midpoint $M$ of $BC$. A point $X$ is called [i]immaculate[/i] if the perpendicular line from $X$ to line $MX$ intersects lines $AB$ and $AC$ at two points that are equidistant from $M$. Suppose $U, V, W$ are three immaculate points on the circumcircle of triangle $ABC$. Prove that $M$ is the incentre of $\triangle UVW$. [i]Proposed by Pranjal Srivastava and Rohan Goyal[/i]

Let $\Gamma$ be the circumcircle of the acute triangle $ABC$. The angle bisector of angle $ABC$ intersects $AC$ in the point $B_1$ and the short arc $AC$ of $\Gamma$ in the point $P$. The line through $B_1$ perpendicular to $BC$ intersects the short arc $BC$ of $\Gamma$ in $K$. The line through $B$ perpendicular to $AK$ intersects $AC$ in $L$. Prove that $K, L$ and $P$ lie on a line.

The numbers in the sequence 101, 104, 109, 116, $\dots$ are of the form $a_n = 100 + n^2$, where $n = 1$, 2, 3, $\dots$. For each $n$, let $d_n$ be the greatest common divisor of $a_n$ and $a_{n + 1}$. Find the maximum value of $d_n$ as $n$ ranges through the positive integers.

Prove that $4x^3-3x+1=2y^2$ has at least $31$ solutions in positive integers $x,y$ with $x\le 1980$. [i] Variant: [/i] Prove the equation $4x^3-3x+1=2y^2$ has infinitely many solutions in positive integers x,y.

Let $P$ be a point on the median $CM$ of a triangle $ABC$ with $AC\ne BC$ and the acute angle $\gamma$ at $C$, such that the bisectors of $\angle PAC$ and $\angle PBC$ intersect at a point $Q$ on the median $CM$. Determine $\angle APB$ and $\angle AQB$.

Consider the square with vertices $(0, 0),(1, 0),(1, 1),(0, 1).$ The line segments from $(t, 0)$ to $(0, 1 - t)$ are drawn for $0 \le t \le 1.$ The set of points inside the square but not on one of these line segments has area $\tfrac{m}{n}$ for coprime positive integers $m$ and $n.$ Find $m + n.$

In $\triangle ABC$ $\angle B$ is obtuse and $AB \ne BC$. Let $O$ is the circumcenter and $\omega$ is the circumcircle of this triangle. $N$ is the midpoint of arc $ABC$. The circumcircle of $\triangle BON$ intersects $AC$ on points $X$ and $Y$. Let $BX \cap \omega = P \ne B$ and $BY \cap \omega = Q \ne B$. Prove that $P, Q$ and reflection of $N$ with respect to line $AC$ are collinear.

Let $AB$ be diameter of a circle $\omega$ and let $C$ be a point on $\omega$, different from $A$ and $B$. The perpendicular from $C$ intersects $AB$ at $D$ and $\omega$ at $E(\neq C)$. The circle with centre at $C$ and radius $CD$ intersects $\omega$ at $P$ and $Q$. If the perimeter of the triangle $PEQ$ is $24$, find the length of the side $PQ$

Let $ABCD$ be a cyclic quadrilateral (a quadrilateral which can be inscribed in a circle). Let $E$ and $F$ be variable points on the sides $AB$ and $CD$, respectively, such that $\frac{AE}{EB} = \frac{C}{FD}$. Let $P$ be the point on the segment $EF$ such that $\frac{PE}{PF} = \frac{AB}{CD}$. Prove that the ratio between the areas of triangle $APD$ and $BPC$ does not depend on the choice of $E$ and $F$.

A pack of $2003$ circus flees are deployed by their circus trainer, named Gogu, on a sufficiently large table, in such a way that they are not all lying on the same line. He now wants to get his Ph.D. in fleas training, so Gogu has thought the fleas the following trick: we chooses two fleas and tells one of them to jump over the other one, such that any flea jumps as far as twice the initial distance between him and the flea over which he is jumping. The Ph.D. circus committee has but only one task to Gogu: he has to make the his flees gather around on the table such that every flea represents a vertex of a convex polygon. Can Gogu get his Ph.D., no matter of how the fleas were deployed?

Let $S=\{1, 2, 3, \dots 2021\}$ and $f:S \to S$ be a function such that $f^{(n)}(n)=n$ for each $n \in S$. Find all possible values for $f(2021)$. (Here, $f^{(n)}(n) = \underbrace{f(f(f\dots f(}_{n \text{ times} }n)))\dots))$.) [i]Authored by Viktor Simjanoski[/i]

A game starts with four heaps of beans, containing 3, 4, 5 and 6 beans. The two players move alternately. A move consists of taking [list] (a) $\text{either}$ one bean from a heap, provided at least two beans are left behind in that heap, (b) $\text{or}$ a complete heap of two or three beans.[/list] The player who takes the last heap wins. To win the game, do you want to move first or second? Give a winning strategy.

$f(n+f(n))=f(n)$ for every $n \in \mathbb{N}$. a)Prove, that if $f(n)$ is finite, then $f$ is periodic. b) Give example nonperiodic function. PS. $0 \not \in \mathbb{N}$

For which positive integers $n$, one can find real numbers $x_1,x_2,\cdots ,x_n$ such that $$\dfrac{x_1^2+x_2^2+\cdots+x_n^2}{\left(x_1+2x_2+\cdots+nx_n\right)^2}=\dfrac{27}{4n(n+1)(2n+1)}$$ and $i\leq x_i\leq 2i$ for all $i=1,2,\cdots ,n$ ?

The sum $\sqrt[3] {5+2\sqrt{13}}+\sqrt[3]{5-2\sqrt{13}}$ equals $\text{(A)} \ \frac 32 \qquad \text{(B)} \ \frac{\sqrt[3]{65}}{4} \qquad \text{(C)} \ \frac{1+\sqrt[6]{13}}{2} \qquad \text{(D)} \ \sqrt[3]{2} \qquad \text{(E)} \ \text{none of these}$

Let $[ABCD]$ be a parallelogram and $P$ a point between $C$ and $D$. The line parallel to $AD$ that passes through $P$ intersects the diagonal $AC$ in $Q$. Knowing that the area of $[PBQ]$ is $2$ and the area of $[ABP]$ is $6$, determine the area of $[PBC]$. [img]https://cdn.artofproblemsolving.com/attachments/0/8/664a00020065b7ad6300a062613fca4650b8d0.png[/img]

On an initially colourless plane three points are chosen and marked in red, blue and yellow. At each step two points marked in different colours are chosen. Then one more point is painted in the third colour so that these three points form a regular triangle with the vertices coloured clockwise in ''red, blue, yellow". A point already marked may be marked again so that it may have several colours. Prove that for any number of moves all the points containing the same colour lie on the same line.

The largest of the following integers which divides each of the numbers of the sequence $ 1^5 \minus{} 1,\, 2^5 \minus{} 2,\, 3^5 \minus{} 3,\, \cdots, n^5 \minus{} n, \cdots$ is: $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 60 \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ 120\qquad \textbf{(E)}\ 30$

In a cyclic quadrilateral $ABCD$ we have $|AD| > |BC|$ and the vertices $C$ and $D$ lie on the shorter arc $AB$ of the circumcircle. Rays $AD$ and $BC$ intersect at point $K$, diagonals $AC$ and $BD$ intersect at point $P$. Line $KP$ intersects the side $AB$ at point $L$. Prove that $\angle ALK$ is acute.

For a fixed positive integer $k$, prove that there exist infinitely many primes $p$ such that there is an integer $w$, where $w^2-1$ is not divisible by $p$, and the order of $w$ in modulus $p$ is the same as the order of $w$ in modulus $p^k$. [i]Author: James Rickards[/i]

For any real number $a$, find all real numbers $x$ that satisfy the following equation. $$(2x + 1)^4 + ax(x + 1) - \frac{x}{2}= 0$$

A [i]triangulation[/i] of a polygon is a subdivision of the polygon into triangles meeting edge to edge, with the property that the set of triangle vertices coincides with the set of vertices of the polygon. Adam randomly selects a triangulation of a regular $180$-gon. Then, Bob selects one of the $178$ triangles in this triangulation. The expected number of $1^\circ$ angles in this triangle can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$. [i]Proposed by Lewis Chen[/i]

Ryan is messing with Brice’s coin. He weights the coin such that it comes up on one side twice as frequently as the other, and he chooses whether to weight heads or tails more with equal probability. Brice flips his modified coin twice and it lands up heads both times. The probability that the coin lands up heads on the next flip can be expressed in the form $\tfrac{p}{q}$ for positive integers $p, q$ satisfying $\gcd(p, q) = 1$, what is $p + q$?

Let $H$ a regular hexagon and let $P$ a point in the plane of $H$. Let $V(P)$ the sum of the distances from $P$ to the vertices of $H$ and let $L(P)$ the sum of the distances from $P$ to the edges of $H$. a) Find all points $P$ so that $L(P)$ is minimun b) Find all points $P$ so that $V(P)$ is minimun