Compute the value of $$\sin^2\left(\frac{\pi}{7}\right) + \sin^2\left(\frac{3\pi}{7}\right) + \sin^2\left(\frac{5\pi}{7}\right).$$ Your answer should not involve any trigonometric functions. [i]Proposed by Howard Halim[/i]

Find all positive integers $n$ for which $$S(n^2)+S(n)^2=n$$ where $S(m)$ denotes the sum of digits of $m$.

Find all functions $f : \mathbb{R} \to \mathbb{R} $ which verify the relation \[(x-2)f(y)+f(y+2f(x))= f(x+yf(x)), \qquad \forall x,y \in \mathbb R.\]

A broken line consists of $31$ segments. It has no self intersections, and its start and end points are distinct. All segments are extended to become straight lines. Find the least possible number of straight lines.

A polygon can be divided into 100 rectangles, but not into 99. Prove that it cannot be divided into 100 triangles. [i]A. Shapovalov[/i]

Determine whether there exists a polynomial $f(x_1, x_2)$ with two variables, with integer coefficients, and two points $A=(a_1, a_2)$ and $B=(b_1, b_2)$ in the plane, satisfying the following conditions: (i) $A$ is an integer point (i.e $a_1$ and $a_2$ are integers); (ii) $|a_1-b_1|+|a_2-b_2|=2010$; (iii) $f(n_1, n_2)>f(a_1, a_2)$ for all integer points $(n_1, n_2)$ in the plane other than $A$; (iv) $f(x_1, x_2)>f(b_1, b_2)$ for all integer points $(x_1, x_2)$ in the plane other than $B$. [i]Massimo Gobbino, Italy[/i]

For positive integers $n, k, r$, denote by $A(n, k, r)$ the number of integer tuples $(x_1, x_2, \ldots, x_k)$ satisfying the following conditions. [list] [*] $x_1 \ge x_2 \ge \cdots \ge x_k \ge 0$ [*] $x_1+x_2+ \cdots +x_k = n$ [*] $x_1-x_k \le r$ [/list] For all positive integers $s, t \ge 2$, prove that $$A(st, s, t) = A(s(t-1), s, t) = A((s-1)t, s, t).$$

Find, with proof, the number of positive integers whose base-$n$ representation consists of distinct digits with the property that, except for the leftmost digit, every digit differs by $\pm 1$ from some digit further to the left. (Your answer should be an explicit function of $n$ in simplest form.)

[b]p1.[/b] How many two-digit primes have a units digit of $3$? [b]p2.[/b] How many ways can you arrange the letters $A$, $R$, and $T$ such that it makes a three letter combination? Each letter is used once. [b]p3.[/b] Hanna and Kevin are running a $100$ meter race. If Hanna takes $20$ seconds to finish the race and Kevin runs $15$ meters per second faster than Hanna, by how many seconds does Kevin finish before Hanna? [b]p4.[/b] It takes an ant $3$ minutes to travel a $120^o$ arc of a circle with radius $2$. How long (in minutes) would it take the ant to travel the entirety of a circle with radius $2022$? [b]p5.[/b] Let $\vartriangle ABC$ be a triangle with angle bisector $AD$. Given $AB = 4$, $AD = 2\sqrt2$, $AC = 4$, find the area of $\vartriangle ABC$. [b]p6.[/b] What is the coefficient of $x^5y^2$ in the expansion of $(x + 2y + 4)^8$? [b]p7.[/b] Find the least positive integer $x$ such that $\sqrt{20475x}$ is an integer. [b]p8.[/b] What is the value of $k^2$ if $\frac{x^5 + 3x^4 + 10x^2 + 8x + k}{x^3 + 2x + 4}$ has a remainder of $2$? [b]p9.[/b] Let $ABCD$ be a square with side length $4$. Let $M$, $N$, and $P$ be the midpoints of $\overline{AB}$, $\overline{BC}$ and $\overline{CD}$, respectively. The area of the intersection between $\vartriangle DMN$ and $\vartriangle ANP$ can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$. [b]p10.[/b] Let $x$ be all the powers of two from $2^1$ to $2^{2023}$ concatenated, or attached, end to end ($x = 2481632...$). Let y be the product of all the powers of two from $2^1$ to $2^{2023}$ ($y = 2 \cdot 4 \cdot 8 \cdot 16 \cdot 32... $ ). Let 2a be the largest power of two that divides $x$ and $2^b$ be the largest power of two that divides $y$. Compute $\frac{b}{a}$ . [b]p11.[/b] Larry is making a s’more. He has to have one graham cracker on the top and one on the bottom, with eight layers in between. Each layer can made out of chocolate, more graham crackers, or marshmallows. If graham crackers cannot be placed next to each other, how many ways can he make this s’more? [b]p12.[/b] Let $ABC$ be a triangle with $AB = 3$, $BC = 4$, $AC = 5$. Circle $O$ is centered at $B$ and has radius $\frac{8\sqrt{3}}{5}$ . The area inside the triangle but not inside the circle can be written as $\frac{a-b\sqrt{c}-d\pi}{e}$ , where $gcd(a, b, d, e) =1$ and $c$ is squarefree. Find $a + b + c + d + e$. [b]p13.[/b] Let $F(x)$ be a quadratic polynomial. Given that $F(x^2 - x) = F (2F(x) - 1)$ for all $x$, the sum of all possible values of $F(2022)$ can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$. [b]p14.[/b] Find the sum of all positive integers $n$ such that $6\phi (n) = \phi (5n)+8$, where $\phi$ is Euler’s totient function. Note: Euler’s totient $(\phi)$ is a function where $\phi (n)$ is the number of positive integers less than and relatively prime to $n$. For example, $\phi (4) = 2$ since only $1$, $3$ are the numbers less than and relatively prime to $4$. [b]p15.[/b] Three numbers $x$, $y$, and $z$ are chosen at random from the interval $[0, 1]$. The probability that there exists an obtuse triangle with side lengths $x$, $y$, and $z$ can be written in the form $\frac{a\pi-b}{c}$ , where $a$, $b$, $c$ are positive integers with $gcd(a, b, c) = 1$. Find $a + b + c$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the number whose reciprocal is the sum of the reciprocal of $9 + 15i$ and the reciprocal of $9-15i$ .

For any nonnegative integer $n$, let $S(n)$ be the sum of the digits of $n$. Let $K$ be the number of nonnegative integers $n \le 10^{10}$ that satisfy the equation \[ S(n) = (S(S(n)))^2. \] Find the remainder when $K$ is divided by $1000$.

The number of distinct points in the xy-plane common to the graphs of $(x+y-5)(2x-3y+5)=0$ and $(x-y+1)(3x+2y-12)=0$ is $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad \textbf{(E) }4$

Let a semicircle is given with diameter $AB$ and centre $O$ and let $C$ be a arbitrary point on the segment $OB$. Point $D$ on the semicircle is such that $CD$ is perpendicular to $AB$. A circle with centre $P$ is tangent to the arc $BD$ at $F$ and to the segment $CD$ and $AB$ at $E$ and $G$ respectively. Prove that the triangle $ADG$ is isosceles.

In triangle $ABC$, $\measuredangle CBA=72^\circ$, $E$ is the midpoint of side $AC$, and $D$ is a point on side $BC$ such that $2BD=DC$; $AD$ and $BE$ intersect at $F$. The ratio of the area of triangle $BDF$ to the area of quadrilateral $FDCE$ is [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair B=origin, C=(15,3), D=(5,1), A=7*dir(72)*dir(B--C), E=midpoint(A--C), F=intersectionpoint(A--D, B--E); draw(E--B--A--C--B^^A--D); label("$A$", A, dir(D--A)); label("$B$", B, dir(E--B)); label("$C$", C, dir(0)); label("$D$", D, SE); label("$E$", E, N); label("$F$", F, dir(80));[/asy] $\text{(A)} \ \frac 15 \qquad \text{(B)} \ \frac 14 \qquad \text{(C)} \ \frac 13 \qquad \text{(D)} \ \frac 25 \qquad \text{(E)} \ \text{none of these}$

$(a)$ A plane $\pi$ passes through the vertex $O$ of the regular tetrahedron $OPQR$. We define $p, q, r$ to be the signed distances of $P,Q,R$ from $\pi$ measured along a directed normal to $\pi$. Prove that \[p^2 + q^2 + r^2 + (q - r)^2 + (r - p)^2 + (p - q)^2 = 2a^2,\] where $a$ is the length of an edge of a tetrahedron. $(b)$ Given four parallel planes not all of which are coincident, show that a regular tetrahedron exists with a vertex on each plane. [u]Note:[/u] Part $(b)$ is [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=49&t=60825&start=0]IMO 1972 Problem 6[/url]

Given an integer $d \ge 2$ and a circle $\omega$. Hansel drew a finite number of chords of circle $\omega$. The following condition is fulfilled: each end of each chord drawn is at least an end of $d$ different drawn chords. Prove that there is a drawn chord which intersects at least $\tfrac{d^2}{4}$ other drawn chords. Here we assume that the chords with a common end intersect. Note: Proof that a certain drawn chord crosses at least $\tfrac{d^2}{8}$ other drawn chords will be awarded two points.

There are $50$ male and $50$ female members registered in the QED-DB, who are also there are numbered from $1$ to $100$. In $100$ rounds, Andreas chooses at random one member for the seminar in Bad Tolz, whereupon Katharina already has two each time selected QED members of different sexes may or may not be paired up. Of course QED members cannot be coupled multiple times, ignoring relationships from the time before but both conscientiously. The stability of a relationship between two QED members is the amount of the difference between their numbers in DB and the sum of all stabilities is the promotion of young talent in the QED. What is the greatest possible demand of offspring guaranteed to achieve orgasm? [hide=original wording]In der QED-DB sind 50 m¨annliche und 50 weibliche Mitglieder eingetragen, welche dort mit den Zahlen von 1 bis 100 durchnummeriert sind. In 100 Runden w¨ahlt Andreas jeweils zuf¨allig ein Mitglied fu¨r das Seminar in Bad T¨olz aus, woraufhin jedes Mal Katharina zwei bereits ausgew¨ahlte QEDler unterschiedlichen Geschlechts verkuppeln darf, aber nicht muss. Natu¨rlich k¨onnen QEDler nicht mehrfach verkuppelt werden, Beziehungen aus der Zeit davor ignorieren beide aber gewissenhaft. Die Stabilit¨at einer Beziehung zwischen zwei QEDlern ist der Betrag der Differenz ihrer Zahlen in der DB und die Summe aller Stabilit¨aten ist die Nachwuchsf¨orderung im QED. Was ist die gr¨oßtm¨ogliche Nachwuchsf¨orderung, welche die Orgas garantiert erreichen k¨onnen¿[/hide]

Let the quadratic $P(x)=x^2+5x+1$. Two distinct real numbers $a,b$ satisfy \[P(a+b)=ab\] \[P(ab)=a+b\] Find the sum of all possible values of $a^2$. [i]Proposed by Harry Kim[/i]

On a flat plane in Camelot, King Arthur builds a labyrinth $\mathfrak{L}$ consisting of $n$ walls, each of which is an infinite straight line. No two walls are parallel, and no three walls have a common point. Merlin then paints one side of each wall entirely red and the other side entirely blue. At the intersection of two walls there are four corners: two diagonally opposite corners where a red side and a blue side meet, one corner where two red sides meet, and one corner where two blue sides meet. At each such intersection, there is a two-way door connecting the two diagonally opposite corners at which sides of different colours meet. After Merlin paints the walls, Morgana then places some knights in the labyrinth. The knights can walk through doors, but cannot walk through walls. Let $k(\mathfrak{L})$ be the largest number $k$ such that, no matter how Merlin paints the labyrinth $\mathfrak{L},$ Morgana can always place at least $k$ knights such that no two of them can ever meet. For each $n,$ what are all possible values for $k(\mathfrak{L}),$ where $\mathfrak{L}$ is a labyrinth with $n$ walls?

Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.

A plane intersects a sphere in a circle $C$. The points $A$ and $B$ lie on the sphere on opposite sides of the plane. The line joining $A$ to the center of the sphere is normal to the plane. Another plane $p$ intersects the segment $AB$ and meets $C$ at $P$ and $Q$. Show that $BP\cdot BQ$ is independent of the choice of $p$.

There are $25$ IMO participants attending a party. Every two of them speak to each other in some language, and they use only one language even if they both know some other language as well. Among every three participants there is a person who uses the same language to speak to the other two (in that group of three). Prove that there is an IMO participant who speaks the same language to at least $10$ other participants

Given a positive integer $k$ show that there exists a prime $p$ such that one can choose distinct integers $a_1,a_2\cdots, a_{k+3} \in \{1, 2, \cdots ,p-1\}$ such that p divides $a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i= 1, 2, \cdots, k$. [i]South Africa [/i]

A convex quadrilateral $ABCD$ is given in which the bisectors of the interior angles $\angle ABC$ and $\angle ADC$ have a common point on the diagonal $AC$. Prove that the bisectors of the interior angles $\angle BAD$ and $\angle BCD$ have a common point on the diagonal $BD$.

Which compound contains the highest percentage of nitrogen by mass? $ \textbf{(A)} \text{NH}_2\text{OH} (M=33.0) \qquad\textbf{(B)}\text{NH}_4\text{NO}_2 (M=64.1)\qquad$ $\textbf{(C)}\text{N}_2\text{O}_3 (M=76.0)\qquad\textbf{(D)}\text{NH}_4\text{NH}_2\text{CO}_2 (M=78.1)\qquad $