Find the number of ways to split the numbers from $1$ to $12$ into $4$ non-intersecting sets of size $3$ such that each set has sum divisible by $3$.

Prove that there exist infinitely many positive integers $ n$ such that $ n!$ is not divisible by $ n^2\plus{}1$.

Natural numbers $a, b, c, d$ are given such that $c>b$. Prove that if $a + b + c + d = ab-cd$, then $a + c$ is a composite number.

Let $V$ be a point in the exterior of a circle of center $O$, and let $T_1,T_2$ be the points where the tangents from $V$ touch the circle. Let $T$ be an arbitrary point on the small arc $T_1T_2$. The tangent in $T$ at the circle intersects the line $VT_1$ in $A$, and the lines $TT_1$ and $VT_2$ intersect in $B$. We denote by $M$ the intersection of the lines $TT_1$ and $AT_2$. Prove that the lines $OM$ and $AB$ are perpendicular.

Let $S$ be the set of the first $2018$ positive integers, and let $T$ be the set of all distinct numbers of the form $ab$, where $a$ and $b$ are distinct members of $S$. What is the $2018$th smallest member of $T$? [i]2018 CCA Math Bonanza Lightning Round #2.1[/i]

Show that it is possible to express $1$ as a sum of $6$, and as a sum of $9$ reciprocals of odd positive integers. Generalize the problem.

Fill each cell with an integer from $1$-$7$ so each number appears exactly once in each row and column. In each ``cage" of three cells, the three numbers must be valid lengths for the sides of a non-degenerate triangle. Additionally, if a cage has an ``A", the triangle must be acute, and if the cage has an ``R", the triangle must be right. [asy] for(int i = 0; i < 8; ++i){ draw((0,i) -- (7,i)^^(i,0)--(i,7), gray(0.7)); } draw((2.1,6.1) -- (4.9, 6.1)--(4.9, 6.9) -- (2.1,6.9)--cycle); draw((5.1,6.1) -- (6.1, 6.1) -- (6.1, 5.1) -- (6.9, 5.1) -- (6.9, 6.9) --(5.1, 6.9) -- cycle); label(scale(0.5)*"R", (5.1, 6.9), SE); draw((1.1, 5.9) -- (1.1, 4.1) -- (2.9, 4.1)-- (2.9, 4.9) -- (1.9, 4.9) -- (1.9, 5.9) -- cycle); draw((3.1, 3.1) -- (3.9, 3.1) -- (3.9, 5.9) -- (3.1, 5.9) -- cycle); draw(shift((3,0))*((1.1, 5.9) -- (1.1, 4.1) -- (2.9, 4.1)-- (2.9, 4.9) -- (1.9, 4.9) -- (1.9, 5.9) -- cycle)); draw(shift((3,-1))*((3.1, 3.1) -- (3.9, 3.1) -- (3.9, 5.9) -- (3.1, 5.9) -- cycle)); label(scale(0.5)*"A", (6.1, 4.9), SE); draw(shift((2,-2))*((3.1, 3.1) -- (3.9, 3.1) -- (3.9, 5.9) -- (3.1, 5.9) -- cycle)); draw((3.1, 2.1) -- (4.9, 2.1) -- (4.9, 3.9) -- (4.1, 3.9) -- (4.1, 2.9) -- (3.1, 2.9) -- cycle); label(scale(0.5)*"R", (4.1, 3.9), SE); draw((0.1, 2.1) -- (0.1, 3.9) -- (1.9, 3.9) -- (1.9, 3.1) -- (0.9, 3.1) -- (0.9, 2.1) -- cycle); draw(shift((0, -3))*((1.1, 5.9) -- (1.1, 4.1) -- (2.9, 4.1)-- (2.9, 4.9) -- (1.9, 4.9) -- (1.9, 5.9) -- cycle)); label(scale(0.5)*"R", (1.1, 2.9), SE); draw(shift((-2, -6)) * ((2.1,6.1) -- (4.9, 6.1)--(4.9, 6.9) -- (2.1,6.9)--cycle)); label(scale(0.5)*"A", (0.1, 0.9), SE); draw(shift((0,-2))*((3.1, 2.1) -- (4.9, 2.1) -- (4.9, 3.9) -- (4.1, 3.9) -- (4.1, 2.9) -- (3.1, 2.9) -- cycle)); label(scale(0.5)*"A", (4.1, 1.9), SE); draw(shift((2,-2))*((3.1, 2.1) -- (4.9, 2.1) -- (4.9, 3.9) -- (4.1, 3.9) -- (4.1, 2.9) -- (3.1, 2.9) -- cycle)); [/asy]

Let $a,b,c,d$ be positive integers such that $ad \neq bc$ and $gcd(a,b,c,d)=1$. Let $S$ be the set of values attained by $\gcd(an+b,cn+d)$ as $n$ runs through the positive integers. Show that $S$ is the set of all positive divisors of some positive integer.

Prove the inequality $$ \frac{y^2-x^2}{2x^2+1}+\frac{z^2-y^2}{2y^2+1}+\frac{x^2-z^2}{2z^2+1} \geq 0$$ where $x$, $y$ and $z$ are real numbers

Consider an $n\times{n}$ grid formed by $n^2$ unit squares. We define the centre of a unit square as the intersection of its diagonals. Find the smallest integer $m$ such that, choosing any $m$ unit squares in the grid, we always get four unit squares among them whose centres are vertices of a parallelogram.

We have a half-circle with endpoints $A$ and $B$ and center $S$. The points $C$ and $D$ lie on the half-circle such that $ \angle BAC \equal{} 20^\circ$ and the lines $ AC$ and $ SD$ are perpendicular to each other. What is the angle between the lines $ AC$ and $ BD$? [asy] size(8cm); pair A = (-1, 0), B = (1, 0), S = (0, 0), C = (sqrt(3)/2, 1/2); path circ = arc(S, 1, 0, 180); pair P = foot(S, A, C); pair D = intersectionpoints(circ, S--(7*(P-S)+S))[0]; draw(circ); draw(A--C--B--cycle); draw(S--D--B); dot(A); dot(B); dot(S); dot(C); dot(D); label("$A$", A, SW); label("$B$", B, SE); label("$S$", S, SW); label("$D$", D, NW); label("$C$", C, NE); markscalefactor *= 0.5; draw(rightanglemark(A, P, D)); draw(anglemark(S, A, C)); label("$20^\circ$", A + (0.3, 0.05), E);[/asy] A. $ 45^\circ$ B. $ 55^\circ$ C. $ 60^\circ$ D. $ 67 \frac{1}{2}^\circ$ E. $ 72^\circ$

Equilateral triangle $ABC$ has side lengths of $24$. Points $D$, $E$, and $F$ lies on sides $BC$, $CA$, $AB$ such that ${AD}\perp{BC}$, ${DE}\perp{AC}$, and ${EF}\perp{AB}$. $G$ is the intersection of $AD$ and $EF$. Find the area of quadrilateral $BFGD$

Find all pairs $(a,b)$ of positive integers that satisfy the equation \[a^{a^{a}}= b^{b}.\]

If $ 10^{2y} \equal{} 25$, then $ 10^{ \minus{} y}$ equals: $ \textbf{(A)}\ \minus{} \frac {1}{5} \qquad\textbf{(B)}\ \frac {1}{625} \qquad\textbf{(C)}\ \frac {1}{50} \qquad\textbf{(D)}\ \frac {1}{25} \qquad\textbf{(E)}\ \frac {1}{5}$

Is it possible to cover a circle of area $1$ with finitely many equilateral triangles whose areas sum to $1.01$, all pointing in the same direction? [i]Proposed by Ethan Tan[/i]

Quadrilaterals may be obtained from an octagon by cutting along its diagonals (in $8$ different ways) . Can it happen that among these $8$ quadrilaterals (a) four (b ) five possess an inscribed circle? (P. M . Sedrakyan , Yerevan)

Let $S$ be a set of in $3-$ space such that each of the points in $S$ has integer coordinates $(x,y,z)$ with $1 \le x,y,z \le n $. Suppose the pairwise distances between these points are all distinct. Prove that $$|S| \le min \{(n+2)\sqrt{\frac{n}{3}},n\sqrt{6} \}$$

Prove that there exist infinitely many integers $n$ such that $n^2+1$ is squarefree.

If $a,b,c$ are positive real numbers such that $a+b+c = 1$, prove that \[ (1+a)(1+b)(1+c) \geq 8 (1-a)(1-b)(1-c) . \]

Let $O(0,0)$, $A(\tfrac{1}{2},0)$, and $B(0, \tfrac{\sqrt{3}}{2})$ be points in the coordinate plane. Let $\mathcal{F}$ be the family of segments $\overline{PQ}$ of unit length lying in the first quadrant with $P$ on the $x$-axis and $Q$ on the $y$-axis. There is a unique point $C$ on $\overline{AB}$, distinct from $A$ and $B$, that does not belong to any segment from $\mathcal{F}$ other than $\overline{AB}$. Then $OC^2 = \tfrac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p + q$.

In rectangle $ABCD$, $AB = 3$ and $BC = 4$. If the feet of the perpendiculars from $B$ and $D$ to $AC$ are $X$ and $Y$ , the length of $X Y$ can be expressed in the form m/n , where m and n are relatively prime positive integers. Find $m +n$.

Prove that the product of five consecutive positive integers cannot be the square of an integer.

Let $ A, B, D, E, F, C $ be six points lie on a circle (in order) satisfy $ AB=AC $ . Let $ P=AD \cap BE, R=AF \cap CE, Q=BF \cap CD, S=AD \cap BF, T=AF \cap CD $ . Let $ K $ be a point lie on $ ST $ satisfy $ \angle QKS=\angle ECA $ . Prove that $ \frac{SK}{KT}=\frac{PQ}{QR} $

Let a right isosceles triangle $APQ$ with the hypotenuse $AP$ be given in plane. Construct such a square $ABCD$ that the lines $BC, CD$ contain points $P, Q,$ respectively. Compute the length of side $AB = b$ in terms of $AQ=a$.

The sequence of polynomials $\left( P_{n}(X)\right)_{n\in Z_{>0}}$ is defined as follows: $P_{1}(X)=2X$ $P_{2}(X)=2(X^2+1)$ $P_{n+2}(X)=2X\cdot P_{n+1}(X)-(X^2-1)P_{n}(X)$, for all positive integers $n$. Find all $n$ for which $X^2+1\mid P_{n}(X)$