Find the exact value of $ E\equal{}\displaystyle\int_0^{\frac\pi2}\cos^{1003}x\text{d}x\cdot\int_0^{\frac\pi2}\cos^{1004}x\text{d}x\cdot$.

[b]3.[/b] Let $k$ and $K$ be concentric circles on the plane, and let $k$ be contained inside $K$. Assume that $k$ is covered by a finite system of convex angular domains with vertices on $K$. Prove that the sum of the angles of the domains is not less than the angle under which $k$ can be seen from a point of $K$. ([b]G.38[/b]) [Zs.. Páles]

Let $ABC$ be an acute-angled triangle with $AB \neq BC$, $M$ the midpoint of $AC$, $N$ the point where the median $BM$ meets again the circumcircle of $\triangle ABC$, $H$ the orthocentre of $\triangle ABC$, $D$ the point on the circumcircle for which $\angle BDH = 90^{\circ}$, and $K$ the point that makes $ANCK$ a parallelogram. Prove the lines $AC$, $KH$, $BD$ are concurrent. (I. Nagel)

Two circles $\omega_1$ and $\omega_2$ are externally tangent at a point $A$. Let $\ell$ be a line tangent to $\omega_1$ at $B\neq A$ and $\omega_2$ at $C\neq A$. Let $BX$ and $CY$ be diameters in $\omega_1$ and $\omega_2$ respectively. Suppose points $P$ and $Q$ lies on $\omega_2$ such that $XP$ and $XQ$ are tangent to $\omega_2$, and points $R$ and $S$ lies on $\omega_1$ such that $YR$ and $YS$ are tangent to $\omega_1$. a) Prove that the points $P$, $Q$, $R$, $S$ lie on a circle $\Gamma$. b) Prove that the four segments $XP$, $XQ$, $YR$, $YS$ determine a quadrilateral with an incircle $\gamma$, and its radius is $\displaystyle\frac{1}{\sqrt{5}}$ times the radius of $\Gamma$. [i]Proposed by Ivan Chan Kai Chin[/i]

Points $A_1,A_2,A_3,A_4$ are not concyclic, the same for points $B_1,B_2,B_3,B_4$. For all $i, j, k$ the circumradii of triangles $A_iA_jA_k$ and $B_iB_jB_k$ are equal. Can we assert that $A_iA_j=B_iB_j$ for all $i, j$'?

Compute the sum of $x^2+y^2$ over all four ordered pairs $(x,y)$ of real numbers satisfying $x=y^2-20$ and $y=x^2+x-21$. [i]2021 CCA Math Bonanza Lightning Round #3.4[/i]

Let $n \ge 4$ and $k$ be positive integers. We consider $n$ lines in the plane between which there are not two parallel nor three concurrent. In each of the $\frac{n(n-1)}{2}$ points of intersection of these lines, $k$ coins are placed. Ana and Beto play the following game in turns: each player, in turn, chooses one of those points that does not share one of the $n$ lines with the point chosen immediately before by the other player, and removes a coin from that point. Ana starts and can choose any point. The player who cannot make his move loses. Determine based on $n$ and $k$ who has a winning strategy.

Integers $a$, $b$, and $c$ satisfy $ab + c = 100$, $bc + a = 87$, and $ca + b = 60$. What is $ab + bc + ca$? $ \textbf{(A) }212 \qquad \textbf{(B) }247 \qquad \textbf{(C) }258 \qquad \textbf{(D) }276 \qquad \textbf{(E) }284 \qquad $

For which value of $m$, there is no triple of integer $(x,y,z)$ such that $3x^2+4y^2-5z^2=m$? $ \textbf{(A)}\ 16 \qquad\textbf{(B)}\ 14 \qquad\textbf{(C)}\ 12 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 8 $

Kid and Karlsson play a game. Initially they have a square piece of chocolate $2019\times 2019$ grid with $1\times 1$ cells . On every turn Kid divides an arbitrary piece of chololate into three rectanglular pieces by cells, and then Karlsson chooses one of them and eats it. The game finishes when it's impossible to make a legal move. Kid wins if there was made an even number of moves, Karlsson wins if there was made an odd number of moves. Who has the winning strategy? [i] (Д. Ширяев)[/i] [hide=Thanks]Thanks to the user Vlados021 for translating the problem.[/hide]

A function $f(x)$ defined in $x\geq 0$ satisfies $\lim_{x\to\infty} \frac{f(x)}{x}=1$. Find $\int_0^{\infty} \{f(x)-f'(x)\}e^{-x}dx$. [i]1997 Hokkaido University entrance exam/Science[/i]

Assume that $a, b,$ and $c$ are positive real numbers for which $(a+b)(a+c)(b+c) = 1$. Prove that $ab+bc+ca \leq\frac{3 }{4}.$

Let $ABCDEF$ be a regular hexagon and $M\in (DE)$, $N\in(CD)$ such that $m (\widehat {AMN}) = 90^\circ$ and $AN = CM \sqrt {2}$. Find the value of $\frac{DM}{ME}$.

Let $ a_1,a_2,...,a_N$ be positive real numbers whose sum equals $ 1$. For a natural number $ i$, let $ n_i$ denote the number of $ a_k$ for which $ 2^{1-i} \geq a_k \geq 2^{-i}$ holds. Prove that \[ \sum_{i=1}^{\infty} \sqrt{n_i2^{-i}} \leq 4+\sqrt{\log_2 N}.\] [i]L. Leinder[/i]

A square grid on the Euclidean plane consists of all points $(m,n)$, where $m$ and $n$ are integers. Is it possible to cover all grid points by an infinite family of discs with non-overlapping interiors if each disc in the family has radius at least $5$?

Let $0<c<1$ be a given real number. Determine the least constant $K$ such that the following holds: For all positive real $M$ that is greater than $1$, there exists a strictly increasing sequence $x_0, x_1, \ldots, x_n$ (of arbitrary length) such that $x_0=1, x_n\geq M$ and \[\sum_{i=0}^{n-1}\frac{\left(x_{i+1}-x_i\right)^c}{x_i^{c+1}}\leq K.\] (From 2020 IMOCSL A5. I think this problem is particularly beautiful so I want to make a separate thread for it :D )

Given is a triangle $ABC$ with circumcenter $O$ and orthocenter $H$. If $O_a, O_b, O_c$ denote the circumcenters of $\triangle AOH$, $\triangle BOH$, $\triangle COH$, then prove that $AO_a, BO_b, CO_c$ are concurrent.

Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function. A point $x$ is called a [i]shadow[/i] point if there exists a point $y\in \mathbb{R}$ with $y>x$ such that $f(y)>f(x).$ Let $a<b$ be real numbers and suppose that $\bullet$ all the points of the open interval $I=(a,b)$ are shadow points; $\bullet$ $a$ and $b$ are not shadow points. Prove that a) $f(x)\leq f(b)$ for all $a<x<b;$ b) $f(a)=f(b).$ [i]Proposed by José Luis Díaz-Barrero, Barcelona[/i]

Find a $10$-digit number, in which no digit is zero, that is divisible by the sum of their digits.

A carton contains milk that is $ 2\%$ fat, and amount that is $ 40\%$ less fat than the amount contained in a carton of whole milk. What is the percentage of fat in whole milk? $ \textbf{(A)}\ \frac{12}{5} \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ \frac{10}{3} \qquad \textbf{(D)}\ 38 \qquad \textbf{(E)}\ 42$

Consider an acute angled triangle $PQR$ such that $C,I$ and $O$ are the circumcentre, incentre and orthocentre respectively. Suppose $\angle QCR, \angle QIR$ and $\angle QOR$, measured in degrees, are $\alpha, \beta$ and $\gamma$ respectively. Show that \[\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}>\frac{1}{45}\]

Find all odd integers $n$ for which $\frac{2^{\phi (n)}-1}{n}$ is a perfect square.

It is known that $ k$ and $ n$ are positive integers and \[ k \plus{} 1\leq\sqrt {\frac {n \plus{} 1}{\ln(n \plus{} 1)}}.\] Prove that there exists a polynomial $ P(x)$ of degree $ n$ with coefficients in the set $ \{0,1, \minus{} 1\}$ such that $ (x \minus{} 1)^{k}$ divides $ P(x)$.

Let $q$ be a fixed positive rational number. Call number $x$ [i]charismatic [/i] if there exist a positive integer $n$ and integers $a_1, a_2, . . . , a_n$ such that $x = (q + 1)^{a_1} \cdot (q + 2)^{a_2} ...(q + n)^{a_n}$. a) Prove that $q$ can be chosen in such a way that every positive rational number turns out to be charismatic. b) Is it true for every $q$ that, for every charismatic number $x$, the number $x + 1$ is charismatic, too?

Let $\mathsf{S}_1,\mathsf{S}_2$ be concentric spheres with radii $a,b$ respectively, where $a<b.$ Denote $ABCDA'B'C'D'$ a square cuboid ($ABCD,A'B'C'D$ are the squares and $AA'\parallel BB'\parallel CC'\parallel DD'$) such that $A,B,C,D\in\mathsf{S}_2$ and the plane $A'B'C'D'$ is tangent to $\mathsf{S}_1.$ Finally assume that \[\frac{AB}{AA'}=\frac ab.\] Compute the lengths $AB,AA'.$ How many of such cuboids exist (up to a congruence)?