In a triangle ABC with circumcentre O and centroid M, lines OM and AM are perpendicular. Let AM intersect the circumcircle of ABC again at A′. Let lines BA′ and AC intersect at D and let lines CA′ and AB intersect at E. Prove that the circumcentre of triangle ADE lies on the circumcircle of ABC.

Let $ABC$ be a triangle inscribed in a circle with center $O$. Let $A_1$ be a point on arc $BC$ that does not contain $ A$ such that the line perpendicular to $OA$ at $A_1$ intersects the lines $AB$ and $AC$ at two points and the line segment joining those two points has as midpoint $A_1$. Points $B_1$, $C_1$ are determined similarly. Prove that the lines $AA_1$, $BB_1$, $CC_1$ are concurrent.

If $f(x)=\log \left(\frac{1+x}{1-x}\right)$ for $-1<x<1$, then $f\left(\frac{3x+x^3}{1+3x^2}\right)$ in terms of $f(x)$ is $\textbf{(A) }-f(x)\qquad\textbf{(B) }2f(x)\qquad\textbf{(C) }3f(x)\qquad$ $\textbf{(D) }\left[f(x)\right]^2\qquad \textbf{(E) }[f(x)]^3-f(x)$

In triangle $ABC$ lines $CE$ and $AD$ are drawn so that $\dfrac{CD}{DB}=\dfrac{3}{1}$ and $\dfrac{AE}{EB}=\dfrac{3}{2}$. Let $r=\dfrac{CP}{PE}$ where $P$ is the intersection point of $CE$ and $AD$. Then $r$ equals: [asy] size(8cm); pair A = (0, 0), B = (9, 0), C = (3, 6); pair D = (7.5, 1.5), E = (6.5, 0); pair P = intersectionpoints(A--D, C--E)[0]; draw(A--B--C--cycle); draw(A--D); draw(C--E); label("$A$", A, SW); label("$B$", B, SE); label("$C$", C, N); label("$D$", D, NE); label("$E$", E, S); label("$P$", P, S); //Credit to MSTang for the asymptote [/asy] $\textbf{(A)}\ 3 \qquad \textbf{(B)}\ \dfrac{3}{2}\qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ \dfrac{5}{2}$

Evaluate $ \int_0^{\frac{\pi}{6}} \frac{\sqrt{1\plus{}\sin x}}{\cos x}\ dx$.

How many subsets of $\{1,2,3,4,\dots,12\}$ contain exactly one prime number?

The Fibonacci sequence is given by equalities $$F_1=F_2=1, F_{k+2}=F_k+F_{k+1}, k\in N$$. a) Prove that for every $m \ge 0$, the area of ​​the triangle $A_1A_2A_3$ with vertices $A_1(F_{m+1},F_{m+2})$, $A_2 (F_{m+3},F_{m+4})$, $A_3 (F_{m+5},F_{m+6})$ is equal to $0.5$. b) Prove that for every $m \ge 0$ the quadrangle $A_1A_2A_4$ with vertices $A_1(F_{m+1},F_{m+2})$, $A_2 (F_{m+3},F_{m+4})$, $A_3 (F_{m+5},F_{m+6})$, $A_4 (F_{m+7},F_{m+8})$ is a trapezoid, whose area is equal to $2.5$. c) Prove that the area of ​​the polygon $A_1A_2...A_n$ , $n \ge3$ with vertices does not depend on the choice of numbers $m \ge 0$, and find this area.

The number $123$ is shown on the screen of a computer. Each minute the computer adds $102$ to the number on the screen. The computer expert Misha may change the order of digits in the number on the screen whenever he wishes. Can he ensure that no four-digit number ever appears on the screen? (F.L. Nazarov, Leningrad)

Show that there exists a positive integer $N$ such that the decimal representation of $2000^N$ starts with the digits $200120012001.$

Baron Munсhausen discovered the following theorem: "For any positive integers $a$ and $b$ there exists a positive integer $n$ such that $an$ is a perfect cube, while $bn$ is a perfect fifth power". Determine if the statement of Baron’s theorem is correct.

[color=darkred]Let $A\, ,\, B\in\mathcal{M}_2(\mathbb{C})$ so that : $A^2+B^2=2AB$ . [b]a)[/b] Prove that : $AB=BA$ . [b]b)[/b] Prove that : $\text{tr}\, (A)=\text{tr}\, (B)$ .[/color]

Find all pairs $(m, n)$ of positive integers for which $4 (mn +1)$ is divisible by $(m + n)^2$.

Prove that for every natural number $n$, and for every real number $x \neq \frac{k\pi}{2^t}$ ($t=0,1, \dots, n$; $k$ any integer) \[ \frac{1}{\sin{2x}}+\frac{1}{\sin{4x}}+\dots+\frac{1}{\sin{2^nx}}=\cot{x}-\cot{2^nx} \]

For an integer $n\ge 2$ find all $a_1,a_2,\cdots ,a_n, b_1,b_2,\cdots , b_n$ so that (a) $0\le a_1\le a_2\le \cdots \le a_n\le 1\le b_1\le b_2\le \cdots \le b_n;$ (b) $\sum_{k=1}^n (a_k+b_k)=2n;$ (c) $\sum_{k=1}^n (a_k^2+b_k^2)=n^2+3n.$

Four mathletes and two coaches sit at a circular table. How many distinct arrangements are there of these six people if the two coaches sit opposite each other?

Prove that for all real numbers $a, b, c > 1$ the inequality \[a(b^2 + c) + b(c^2 + a) + c(a^2 + b) \ge a^2 + b^2 + c^2 + 3abc\] holds. When does equality hold? Proposed by [i]Ilija Jovcevski[/i]

Find the largest value of the expression $\frac{p}{R}\left( 1- \frac{r}{3R}\right)$ , where $p,R, r$ is, respectively, the perimeter, the radius of the circumscribed circle and the radius of the inscribed circle of a triangle.

The last digit of the number $x^2 +xy+y^2$ is zero (where $x$ and $y$ are positive integers). Prove that two last digits of this numbers are zeros.

In his spare time, Thomas likes making rectangular windows. He builds windows by taking four $30\text{ cm}\times20\text{ cm}$ rectangles of glass and arranging them in a larger rectangle in wood. The window has an $x\text{ cm}$ wide strip of wood between adjacent glass pieces and an $x\text{ cm}$ wide strip of wood between each glass piece and the adjacent edge of the window. Given that the total area of the glass is equivalent to the total area of the wood, what is $x$? [center]<see attached>[/center]

Let $ n \geq 2$ be an integer, and let $ X$ be a connected Hausdorff space such that every point of $ X$ has a neighborhood homeomorphic to the Euclidean space $ \mathbb{R}^n$. Suppose that any discrete (not necessarily closed ) subspace $ D$ of $ X$ can be covered by a family of pairwise disjoint, open sets of $ X$ so that each of these open sets contains precisely one element of $ D$. Prove that $ X$ is a union of at most $ \aleph_1$ compact subspaces. [i]Z. Balogh[/i]

Let $ABC$ be an isosceles triangle with $AC=BC$, whose incentre is $I$. Let $P$ be a point on the circumcircle of the triangle $AIB$ lying inside the triangle $ABC$. The lines through $P$ parallel to $CA$ and $CB$ meet $AB$ at $D$ and $E$, respectively. The line through $P$ parallel to $AB$ meets $CA$ and $CB$ at $F$ and $G$, respectively. Prove that the lines $DF$ and $EG$ intersect on the circumcircle of the triangle $ABC$. [i]Proposed by Hojoo Lee, Korea[/i]

Find all natural numbers that can be expressed in a unique way as a sum of five or less perfect squares.

A number is called \textit{super odd} if it is an odd number divisible by the square of an odd prime. For example, $2023$ is a \textit{super odd} number because it is odd and divisible by $17^2$. Find the sum of all \textit{super odd} numbers from $1$ to $100$ inclusive. [i]Proposed by Andy Xu[/i]

$A$ and $B$ are two identical convex polygons, each with an area of $2015$. The polygon $A$ is divided into polygons $A_1, A_2,...,A_{2015}$, while $B$ is divided into polygons $B_1, B_2,...,B_{2015}$. Each of these smaller polygons has a positive area. Furthermore, $A_1, A_2,...,A_{2015}$ and $B_1, B_2,...,B_{2015}$ are colored in $2015$ distinct colors, such that $A_i$ and $A_j$ are differently colored for every distinct $i$ and $j$ and $B_i$ and $B_j$ are also differently colored for every distinct $i$ and $j$. After $A$ and $B$ overlap, we calculate the sum of the areas with the same colors. Prove that we can color the polygons such that this sum is at least $1$.

A deck of $2n$ cards numbered from $1$ to $2n$ is shuffled and n cards are dealt to $A$ and $B$. $A$ and $B$ alternately discard a card face up, starting with $A$. The game when the sum of the discards is first divisible by $2n + 1$, and the last person to discard wins. What is the probability that $A$ wins if neither player makes a mistake?