Define $f(x)=\max \{\sin x, \cos x\} .$ Find at how many points in $(-2 \pi, 2 \pi), f(x)$ is not differentiable? [list=1] [*] 0 [*] 2 [*] 4 [*] $\infty$ [/list]

Let $A\in M_4(C)$ be a non-zero matrix. $a)$ If $\text{rank}(A)=r<4$, prove the existence of two invertible matrices $U,V\in M_4(C)$, such that: \[UAV=\begin{pmatrix}I_r&0\\0&0\end{pmatrix}\] where $I_r$ is the $r$-unit matrix. $b)$ Show that if $A$ and $A^2$ have the same rank $k$, then the matrix $A^n$ has rank $k$, for any $n\ge 3$.

Let $a,b,c\in \left( 0,\infty \right)$.Prove the inequality $\frac{a-\sqrt{bc}}{a+2\left( b+c \right)}+\frac{b-\sqrt{ca}}{b+2\left( c+a \right)}+\frac{c-\sqrt{ab}}{c+2\left( a+b \right)}\ge 0.$

How many sequences $ a_1,a_2,...,a{}_2{}_0{}_0{}_8$ are there such that each of the numbers $ 1,2,...,2008$ occurs once in the sequence, and $ i \in (a_1,a_2,...,a_i)$ for each $ i$ such that $ 2\le i \le2008$?

1007 distinct potatoes are chosen independently and randomly from a box of 2016 potatoes numbered $1, 2, \dots, 2016$, with $p$ being the smallest chosen potato. Then, potatoes are drawn one at a time from the remaining 1009 until the first one with value $q < p$ is drawn. If no such $q$ exists, let $S = 1$. Otherwise, let $S = pq$. Then given that the expected value of $S$ can be expressed as simplified fraction $\tfrac{m}{n}$, find $m+n$.

Let $P$ and $Q$ be two distinct points in the plane. Let us denote by $m(PQ)$ the segment bisector of $PQ$. Let $S$ be a finite subset of the plane, with more than one element, that satisfies the following properties: (i) If $P$ and $Q$ are in $S$, then $m(PQ)$ intersects $S$. (ii) If $P_1Q_1, P_2Q_2, P_3Q_3$ are three diferent segments such that its endpoints are points of $S$, then, there is non point in $S$ such that it intersects the three lines $m(P_1Q_1)$, $m(P_2Q_2)$, and $m(P_3Q_3)$. Find the number of points that $S$ may contain.

Let $n\geqslant 3$ be a positive integer and $\mathcal F$ be a family of at most $n$ distinct subsets of the set $\{1,2,\ldots,n\}$ with the following property: we can consider $n$ distinct points in the plane, labelled $1,2,\ldots,n$ and draw segments connecting these points such that points $i$ and $j$ are connected if and only if $i{}$ belongs to $j$ subsets in $\mathcal F$ for any $i\neq j.$ Determine the maximal value that the sum of the cardinalities of the subsets in $\mathcal{F}$ can take.

Let $H$ be the orthocenter of a triangle $ABC$, and $M$, $N$ be the midpoints of segments $BC$, $AH$ respectively. The perpendicular from $N$ to $MH$ meets $BC$ at point $A^{\prime}$. Points $B^{\prime}$ and $C^{\prime}$ are defined similarly. Prove that $A^{\prime}$, $B^{\prime}$, $C^{\prime}$ are collinear. Proposed by: F.Ivlev

If $n=p_1^{a_1},p_2^{a_2}...p_s^{a_s}$ then $\phi (n)=n \left(1- \frac{1}{p_1}\right)\left(1 - \frac{1}{p_2}\right)...\left(1- \frac{1}{p_s}\right)$. Find the smallest positive integer $n$ such that $\phi (n)=\frac{2^5}{47}n.$

Six congruent isosceles triangles have been put together as described in the picture below. Prove that points M, F, C lie on one line. [img]https://i.imgur.com/1LU5Zmb.png[/img]

Some friends formed $6$ football teams, and decided to hold a tournament where each team faces each other exactly once in a match. In each match, whoever wins gets $3$ points, whoever loses gets no points, and if the two teams draw, each gets $1$ point. At the end of the tournament, it was found that the teams' scores were $10$, $9$, $6$, $6$, $4$ and $2$ points. Regarding this tournament, answer the following items, justifying your answer in each one. (a) How many matches ended in a draw in the tournament? (b) Determine, for each of the $6$ teams, the number of wins, draws and losses. (c) If we consider only the matches played between the team that scored $9$ points against the two teams that scored $6$ points, and the one played between the two teams that scored $6$ points, explain why among these three matches, there are at least $2$ draws.

In acute-angled triangle $ABC,$ $E,F$ are on the side $BC,$ such that $\angle BAE=\angle CAF,$ and let $M,N$ be the projections of $F$ onto $AB,AC,$ respectively. The line $AE$ intersects $ \odot (ABC) $ at $D$(different from point $A$). Prove that $S_{AMDN}=S_{\triangle ABC}.$

In the triangle $[ABC], D$ is the midpoint of $[AB]$ and $E$ is the trisection point of $[BC]$ closer to $C$. If $\angle ADC= \angle BAE$ , find the measue of $\angle BAC$ .

Prove that there are no perfect squares in the array below: \[\begin{array}{cccc}11&111&1111&...\\22&222&2222&...\\33&333&3333&...\\44&444&4444&...\\55&555&5555&... \\66&666&6666&...\\77&777&7777&...\\88&888&8888&...\\99&999&9999&...\end{array}\]

Let's say a positive integer $ n$ is [i]atresvido[/i] if the set of its divisors (including 1 and $ n$) can be split in in 3 subsets such that the sum of the elements of each is the same. Determine the least number of divisors an atresvido number can have.

Let $ ABCD $ be a convex quadrilateral. Let angle bisectors of $ \angle B $ and $ \angle C $ intersect at $ E $. Let $ AB $ intersect $ CD $ at $ F $. Prove that if $ AB+CD=BC $, then $A,D,E,F$ is cyclic.

As shown in the figure below, $ABCD$ is a trapezoid, $AB \parallel CD$. The sides $DA$, $AB$, $BC$ are tangent to $\odot O_1$ and $AB$ touches $\odot O_1$ at $P$. The sides $BC$, $CD$, $DA$ are tangent to $\odot O_2$, and $CD$ touches $\odot O_2$ at $Q$. Prove that the lines $AC$, $BD$, $PQ$ meet at the same point. [asy] size(200); defaultpen(linewidth(0.8)+fontsize(10pt)); pair A=origin,B=(1,-7),C=(30,-15),D=(26,6); pair bisA=bisectorpoint(B,A,D),bisB=bisectorpoint(A,B,C),bisC=bisectorpoint(B,C,D),bisD=bisectorpoint(C,D,A); path bA=A--(bisA+100*(bisA-A)),bB=B--(bisB+100*(bisB-B)),bC=C--(bisC+100*(bisC-C)),bD=D--(bisD+100*(bisD-D)); pair O1=intersectionpoint(bA,bB),O2=intersectionpoint(bC,bD); dot(O1^^O2,linewidth(2)); pair h1=foot(O1,A,B),h2=foot(O2,C,D); real r1=abs(O1-h1),r2=abs(O2-h2); draw(circle(O1,r1)^^circle(O2,r2)); draw(A--B--C--D--cycle); draw(A--C^^B--D^^h1--h2); label("$A$",A,NW); label("$B$",B,SW); label("$C$",C,dir(350)); label("$D$",D,dir(350)); label("$P$",h1,dir(200)); label("$Q$",h2,dir(350)); label("$O_1$",O1,dir(150)); label("$O_2$",O2,dir(300)); [/asy]

Let $n$ be a positive integer and $N=n^{2021}$. There are $2021$ concentric circles centered at $O$, and $N$ equally-spaced rays are emitted from point $O$. Among the $2021N$ intersections of the circles and the rays, some are painted red while the others remain unpainted. It is known that, no matter how one intersection point from each circle is chosen, there is an angle $\theta$ such that after a rotation of $\theta$ with respect to $O$, all chosen points are moved to red points. Prove that the minimum number of red points is $2021n^{2020}$. [I]Proposed by usjl.[/i]

Find the value of $x$ where the graph of $$y=\log_3(\sqrt{x^2+729}+x)-2\log_3(\sqrt{x^2+729}-x)$$ crosses the $x$-axis.

In triangle $ABC$ with $AB=8$ and $AC=10$, the incenter $I$ is reflected across side $AB$ to point $X$ and across side $AC$ to point $Y$. Given that segment $XY$ bisects $AI$, compute $BC^2$. (The incenter is the center of the inscribed circle of triangle .)

Given positive integers $k, m, n$ such that $1 \leq k \leq m \leq n$. Evaluate \[\sum^{n}_{i=0} \frac{(-1)^i}{n+k+i} \cdot \frac{(m+n+i)!}{i!(n-i)!(m+i)!}.\]

Consider those functions $ f$ that satisfy $ f(x \plus{} 4) \plus{} f(x \minus{} 4) \equal{} f(x)$ for all real $ x$. Any such function is periodic, and there is a least common positive period $ p$ for all of them. Find $ p$. $ \textbf{(A)}\ 8\qquad \textbf{(B)}\ 12\qquad \textbf{(C)}\ 16\qquad \textbf{(D)}\ 24\qquad \textbf{(E)}\ 32$

Solve the given system in prime numbers \begin{align*} x^2+yu = (x+u)^v \end{align*} \begin{align*} x^2+yz=u^4 \end{align*}

Let $a>0,b>0,c>0$ and $a+b+c=1$. Show the inequality $$\frac{a^4+b^4}{a^2+b^2}+\frac{b^3+c^3}{b+c} + \frac{2a^2+b^2+2c^2}{2} \geq \frac{1}{2}$$

Given a finite set $B$ of points in space, any two distances between the points of this set are different. Each point of the set $B$ is connected by a line segment to the closest point of the set $B$. This way we will get a set of sections, one of which (any chosen one) we paint red, all the remaining sections we paint green. Prove that there are two points of the set $B$ that cannot be connected by a line composed of green segments.