A $T$-tetromino is a non-convex as well as non-rotationally symmetrical tetromino, which has a maximum number of outside corners (popularly also "Tetris Stone "called). Find all natural numbers $n$ for which, a $n \times n$ chessboard is found that can be covered only with such $T$-tetrominos.

Find the number of ways to tile a $2 \times 2 \times 2 \times 2$ four dimensional hypercube with $2 \times 1 \times 1 \times 1$ blocks, with reflections and rotations of the large hypercube distinct.

Does there exist a natural number that can be represented as the product of two numeric palindromes in more than $100{}$ ways?

Let \[a=\dfrac{1^2}1+\dfrac{2^2}3+\dfrac{3^2}5+\cdots+\dfrac{1001^2}{2001}\] and \[b=\dfrac{1^2}3+\dfrac{2^2}5+\dfrac{3^2}7+\cdots+\dfrac{1001^2}{2003}.\] Find the integer closest to $a-b$. $\textbf{(A) }500\qquad\textbf{(B) }501\qquad\textbf{(C) }999\qquad\textbf{(D) }1000\qquad\textbf{(E) }1001$

Let $S$ be a set of nonnegative integers such that [list] [*] there exist two elements $a$ and $b$ in $S$ such that $a,b>1$ and $\gcd(a,b)=1$; and [*] for any (not necessarily distinct) element $x$ and nonzero element $y$ in $S$, both $xy$ and the remainder when $x$ is divided by $y$ are in $S$. [/list] Prove that $S$ contains every nonnegative integer. [i]Jacob Paltrowitz[/i]

Four couples are to be seated in a row. If it is required that each woman may only sit next to her husband or another woman, how many different possible seating arrangements are there?

$n>2$ numbers, $ x_1, x_2, ..., x_n$ are odd . Prove that $4$ divides $$ x_1x_2+x_2x_3+...+x_{n-1}x_n+x_nx_1 -n.$$

Let $x_1,\ldots,x_n$ be given real numbers with $0<m\le x_i\le M$ for each $i\in\{1,\ldots,n\}$. Let $X$ be the discrete random variable uniformly distributed on $\{x_1,\ldots,x_n\}$. The mean $\mu$ and the variance $\sigma^2$ of $X$ are defined as $$\mu(X)=\frac{x_1+\ldots+x_n}n\text{ and }\sigma^2(X)=\frac{(x_1-\mu(X))^2+\ldots+(x_n-\mu(X))^2}n.$$ By $X^2$ denote the discrete random variable uniformly distributed on $\{x_1^2,\ldots,x_n^2\}$. Prove that $$\sigma^2(X)\ge\left(\frac m{2M^2}\right)^2\sigma^2(X^2).$$

Let $G$ be a group and $n \ge 2$ be an integer. Let $H_1, H_2$ be $2$ subgroups of $G$ that satisfy $$[G: H_1] = [G: H_2] = n \text{ and } [G: (H_1 \cap H_2)] = n(n-1).$$ Prove that $H_1, H_2$ are conjugate in $G.$ Official definitions: $[G:H]$ denotes the index of the subgroup of $H,$ i.e. the number of distinct left cosets $xH$ of $H$ in $G.$ The subgroups $H_1, H_2$ are conjugate if there exists $g \in G$ such that $g^{-1} H_1 g = H_2.$

Let $k$ be a positive integer. Two players $A$ and $B$ play a game on an infinite grid of regular hexagons. Initially all the grid cells are empty. Then the players alternately take turns with $A$ moving first. In his move, $A$ may choose two adjacent hexagons in the grid which are empty and place a counter in both of them. In his move, $B$ may choose any counter on the board and remove it. If at any time there are $k$ consecutive grid cells in a line all of which contain a counter, $A$ wins. Find the minimum value of $k$ for which $A$ cannot win in a finite number of moves, or prove that no such minimum value exists.

In a given pentagon $ABCDE$, triangles $ABC$, $BCD$, $CDE$, $DEA$ and $EAB$ all have the same area. The lines $AC$ and $AD$ intersect $BE$ at points $M$ and $N$. Prove that $BM = EN$.

Let $O$ be a circumcenter of acute triangle $ABC$ and let $O_1$ and $O_2$ be circumcenters of triangles $OAB$ and $OAC$, respectively. Circumcircles of triangles $OAB$ and $OAC$ intersect side $BC$ in points $D$ ($D \neq B$) and $E$ ($E \neq C$), respectively. Perpendicular bisector of side $BC$ intersects side $AC$ in point $F$($F \neq A$). Prove that circumcenter of triangle $ADE$ lies on $AC$ iff $F$ lies on line $O_1O_2$

Let $n>100$ be an integer. The numbers $1,2 \ldots, 4n$ are split into $n$ groups of $4$. Prove that there are at least $\frac{(n-6)^2}{2}$ quadruples $(a, b, c, d)$ such that they are all in different groups, $a<b<c<d$ and $c-b \leq |ad-bc|\leq d-a$.

In a triangle $ ABC,$ let $ D$ and $ E$ be the intersections of the bisectors of $ \angle ABC$ and $ \angle ACB$ with the sides $ AC,AB,$ respectively. Determine the angles $ \angle A,\angle B, \angle C$ if $ \angle BDE \equal{} 24 ^{\circ},$ $ \angle CED \equal{} 18 ^{\circ}.$

Is it true that for integer $n\ge 2$, and given any non-negative reals $\ell_{ij}$, $1\le i<j\le n$, we can find a sequence $0\le a_1,a_2,\ldots,a_n$ such that for all $1\le i<j\le n$ to have $|a_i-a_j|\ge \ell_{ij}$, yet still $\sum_{i=1}^n a_i\le \sum_{1\le i<j\le n}\ell_{ij}$?

Let $ABC$ be a triangle such that in its interior there exists a point $D$ with $\angle DAC = \angle DCA = 30^o$ and $ \angle DBA = 60^o$. Denote $E$ the midpoint of the segment $BC$, and take $F$ on the segment $AC$ so that $AF = 2FC$. Prove that $DE \perp EF$.

Let $ABCD$ be a unit square. Points $M$ and $N$ are the midpoints of sides $AB$ and $BC$ respectively. Let $P$ and $Q$ be the midpoints of line segments $AM$ and $BN$ respectively. Find the reciprocal of the area of the triangle enclosed by the three line segments $PQ$, $MN$, and $DB$. [asy] size(5 cm); pair A=(0,0); pair B=(1,0); pair C=(1,1); pair D=(0,1); pair M=(0.5,0); pair N=(1,0.5); pair P=(0.25,0); pair Q=(1,0.25); draw(A--B--C--D--cycle); draw(M--N); draw(P--Q); draw(B--D); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,NE); label("$D$",D,NW); label("$M$",M,S); label("$N$",N,E); label("$P$",P,S); label("$Q$",Q,E); fill((0.8125,0.1875)--(0.75,0.25)--(0.625,0.125)--cycle, gray);[/asy] [i]2021 CCA Math Bonanza Team Round #4[/i]

What is the greatest value of $\sin x \cos y + \sin y \cos z + \sin z \cos x$, where $x,y,z$ are real numbers? $ \textbf{(A)}\ \sqrt 2 \qquad\textbf{(B)}\ \dfrac 32 \qquad\textbf{(C)}\ \dfrac {\sqrt 3}2 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 3 $

Show that the sum of the (local) maximum and minimum values of the function $\frac{tan(3x)}{tan^3x}$ on the interval $\big(0, \frac{\pi }{2}\big)$ is rational.

Consider two segments of length $a, b \ (a > b)$ and a segment of length $c = \sqrt{ab}$. [b](a)[/b] For what values of $a/b$ can these segments be sides of a triangle ? [b](b)[/b] For what values of $a/b$ is this triangle right-angled, obtuse-angled, or acute-angled ?

Let $\triangle ABC$ be an acute triangle with angles $\alpha, \beta,$ and $\gamma$. Prove that $$\frac{\cos(\beta-\gamma)}{cos\alpha}+\frac{\cos(\gamma-\alpha)}{\cos \beta}+\frac{\cos(\alpha-\beta)}{\cos \gamma} \geq \frac{3}{2}$$

Bob has picked positive integer $1<N<100$. Alice tells him some integer, and Bob replies with the remainder of division of this integer by $N$. What is the smallest number of integers which Alice should tell Bob to determine $N$ for sure?

Two equilateral triangles $ABC$ and $CDE$ have a common vertex (see fig). Find the angle between straight lines $AD$ and $BE$. [img]https://1.bp.blogspot.com/-OWpqpAqR7Zw/Xzj_fyqhbFI/AAAAAAAAMao/5y8vCfC7PegQLIUl9PARquaWypr8_luAgCLcBGAsYHQ/s0/2010%2Boral%2Bmoscow%2Bgeometru%2B8.1.gif[/img]

In triangle $ABC$, the bisectors of the internal angles $AA_1$ , $BB_1$ and $CC_1$ are drawn ($A_1, B_1$, $C_1$ - on the sides of the triangle). It is known that $\angle AA_1C = \angle AC_1B_1$. Find $\angle BCA$.

For a given real number $a$ and a positive integer $n$, prove that: i) there exists exactly one sequence of real numbers $x_0,x_1,\ldots,x_n,x_{n+1}$ such that \[\begin{cases} x_0=x_{n+1}=0,\\ \frac{1}{2}(x_i+x_{i+1})=x_i+x_i^3-a^3,\ i=1,2,\ldots,n.\end{cases}\] ii) the sequence $x_0,x_1,\ldots,x_n,x_{n+1}$ in i) satisfies $|x_i|\le |a|$ where $i=0,1,\ldots,n+1$. [i]Liang Yengde[/i]