Let $A$ be a complex $n \times n$ Matrix for $n >1$. Let $A^{H}$ be the conjugate transpose of $A$. Prove that $A\cdot A^{H} =I_{n}$ if and only if $A=S\cdot (S^{H})^{-1}$ for some complex Matrix $S$.

If $\dfrac{a+b}{b+c}=\dfrac{c+d}{d+a}$, then: $\textbf{(A)}\ a\text{ must equal }c \qquad \textbf{(B)}\ a+b+c+d\text{ must equal zero }\qquad$ $ \textbf{(C)}\ \text{either }a=c\text{ or }a+b+c+d=0,\text{ or both} \qquad$ $ \textbf{(D)}\ a+b+c+d\neq 0\text{ if }a=c \qquad \textbf{(E)}\ a(b+c+d)=c(a+b+d)$

The graph of $y^2+2xy+40|x|=400$ partitions the plane into several regions. What is the area of the bounded region?

The numbers $S_1=2^2, S_2=2^4,\ldots, S_n=2^{2n}$ are given. A rectangle $OABC$ is constructed on the Cartesian plane according to these numbers. For this, starting from the point $O$ the points $A_1,A_2,\ldots,A_n$ are consistently marked along the axis $Ox$, and the points $C_1,C_2,\ldots,C_n$ are consistently marked along the axis $Oy$ in such a way that for all $k$ from $1$ to $n$ the lengths of the segments $A_{k-1}A_k=x_k$ and $C_{k-1}C_k=y_k$ are positive integers (let $A_0=C_0=O$, $A_n=A$, and $C_n=C$) and $x_k\cdot y_k=S_k$. [b]a)[/b] Find the maximal possible value of the area of the rctangle $OABC$ and all pairs of sets $(x_1,x_2,\ldots,x_k)$ and $(y_1,y_2,\ldots,y_k)$ at which this maximal area is achieved. [b]b)[/b] Find the minimal possible value of the area of the rctangle $OABC$ and all pairs of sets $(x_1,x_2,\ldots,x_k)$ and $(y_1,y_2,\ldots,y_k)$ at which this minimal area is achieved. [i](E. Manzhulina, B. Rublyov)[/i]

[color=darkred]The altitude $[BH]$ dropped onto the hypotenuse of a triangle $ABC$ intersects the bisectors $[AD]$ and $[CE]$ at $Q$ and $P$ respectively. Prove that the line passing through the midpoints of the segments $[QD]$ and $[PE]$ is parallel to the line $AC$ .[/color]

In acute-angled triangle $ABC$ altitudes $BE,CF$ meet at $H$. A perpendicular line is drawn from $H$ to $EF$ and intersects the arc $BC$ of circumcircle of $ABC$ (that doesn’t contain $A$) at $K$. If $AK,BC$ meet at $P$, prove that $PK=PH$.

Let $P$ be a $2019-$gon, such that no three of its diagonals concur at an internal point. We will call each internal intersection point of diagonals of $P$ a knot. What is the greatest number of knots one can choose, such that there doesn't exist a cycle of chosen knots? ( Every two adjacent knots in a cycle must be on the same diagonal and on every diagonal there are at most two knots from a cycle.)

A given polynomial $P(t) = t^3 + at^2 + bt + c$ has $3$ distinct real roots. If the equation $(x^2 +x+2013)^3 +a(x^2 +x+2013)^2 + b(x^2 + x + 2013) + c = 0$ has no real roots, prove that $P(2013) >\frac{1}{64}$

Define a function on the positive integers recursively by $f(1) = 2$, $f(n) = f(n-1) + 1$ if $n$ is even, and $f(n) = f(n-2) + 2$ if $n$ is odd and greater than $1$. What is $f(2017)$? $\textbf{(A) } 2017 \qquad \textbf{(B) } 2018 \qquad \textbf{(C) } 4034 \qquad \textbf{(D) } 4035 \qquad \textbf{(E) } 4036$

Let $n$ be a integer and let $n=d_1>d_2>\cdots>d_k=1$ its positive divisors. a) Prove that $$d_1-d_2+d_3-\cdots+(-1)^{k-1}d_k=n-1$$ iff $n$ is prime or $n=4$. b) Determine the three positive integers such that $$d_1-d_2+d_3-...+(-1)^{k-1}d_k=n-4.$$

Let $A'$ and $B'$ be feet of altitudes from $A$ and $B$, respectively, in acute-angled triangle $ABC$ ($AC\not = BC$). Circle $k$ contains points $A'$ and $B'$ and touches segment $AB$ in $D$. If triangles $ADA'$ and $BDB'$ have the same area, prove that \[\angle A'DB'= \angle ACB.\]

[b]p1.[/b] Let $A(S)$ denote the average value of a set $S$. Let $T$ be the set of all subsets of the set $\{1, 2, 3, 4, ... , 2012\}$, and let $R$ be $\{A(K)|K \in T \}$. Compute $A(R)$. [b]p2.[/b] Consider the minute and hour hands of the Campanile, our clock tower. During one single day ($12:00$ AM - $12:00$ AM), how many times will the minute and hour hands form a right-angle at the center of the clock face? [b]p3.[/b] In a regular deck of $52$ face-down cards, Billy flips $18$ face-up and shuffles them back into the deck. Before giving the deck to Penny, Billy tells her how many cards he has flipped over, and blindfolds her so she can’t see the deck and determine where the face-up cards are. Once Penny is given the deck, it is her job to split the deck into two piles so that both piles contain the same number of face-up cards. Assuming that she knows how to do this, how many cards should be in each pile when he is done? [b]p4.[/b] The roots of the equation $x^3 + ax^2 + bx + c = 0$ are three consecutive integers. Find the maximum value of $\frac{a^2}{b+1}$. [b]p5.[/b] Oski has a bag initially filled with one blue ball and one gold ball. He plays the following game: first, he removes a ball from the bag. If the ball is blue, he will put another blue ball in the bag with probability $\frac{1}{437}$ and a gold ball in the bag the rest of the time. If the ball is gold, he will put another gold ball in the bag with probability $\frac{1}{437}$ and a blue ball in the bag the rest of the time. In both cases, he will put the ball he drew back into the bag. Calculate the expected number of blue balls after $525600$ iterations of this game. [b]p6.[/b] Circles $A$ and $B$ intersect at points $C$ and $D$. Line $AC$ and circle $B$ meet at $E$, line $BD$ and circle $A$ meet at $F$, and lines $EF$ and $AB$ meet at $G$. If $AB = 10$, $EF = 4$, $FG = 8$, find $BG$. PS. You had better use hide for answers.

Consider a $2018 \times 2019$ board with integers in each unit square. Two unit squares are said to be neighbours if they share a common edge. In each turn, you choose some unit squares. Then for each chosen unit square the average of all its neighbours is calculated. Finally, after these calculations are done, the number in each chosen unit square is replaced by the corresponding average. Is it always possible to make the numbers in all squares become the same after finitely many turns?

Determine the largest positive integer $n$ such that the following statement is true: There exists $n$ real polynomials, $P_1(x),\ldots,P_n(x)$ such that the sum of any two of them have no real roots but the sum of any three does.

Find the minimum positive value of $ 1*2*3*4*...*2020*2021*2022$ where you can replace $*$ as $+$ or $-$

Let $N{}$ be a fixed positive integer, $S{}$ be the set $\{1, 2,\ldots , N\}$ and $\mathcal{F}$ be the set of functions $f:S\to S$ such that $f(i)\geqslant i$ for all $i\in S.$ For each $f\in\mathcal{F}$ let $P_f$ be the unique polynomial of degree less than $N{}$ satisfying $P_f(i) = f(i)$ for all $i\in S.$ If $f{}$ is chosen uniformly at random from $\mathcal{F}$ determine the expected value of $P_f'(0)$ where\[P_f'(0)=\frac{\mathrm{d}P_f(x)}{\mathrm{d}x}\bigg\vert_{x=0}.\][i]Proposed by Ishan Nath[/i]