In rectangle $ABCD$, point $M$ is the midpoint of $AB$ and $P$ is a point on side $BC$. The perpendicular bisector of $MP$ intersects side $DA$ at point $X$. Given that $AB = 33$ and $BC = 56$, find the least possible value of $MX$. [i]Proposed by Michael Tang[/i]

a) We call [i]admissible sequence[/i] a sequence of 4 even digits in which no digits appears more than two times. Find the number of admissible sequences. b) For each integer $ n\geq 2$ we denote $ d_n$ the number of possibilities of completing with even digits an array with $ n$ rows and 4 columns, such that (1) any row is an admissible sequence; (2) the sequence 2, 0, 0, 8 appears exactly ones in the array. Find the values of $ n$ for which the number $ \frac {d_{n\plus{}1}}{d_n}$ is an integer.

Rectangle $ABCD$ has sides $\overline {AB}$ of length 4 and $\overline {CB}$ of length 3. Divide $\overline {AB}$ into 168 congruent segments with points $A=P_0, P_1, \ldots, P_{168}=B$, and divide $\overline {CB}$ into 168 congruent segments with points $C=Q_0, Q_1, \ldots, Q_{168}=B$. For $1 \le k \le 167$, draw the segments $\overline {P_kQ_k}$. Repeat this construction on the sides $\overline {AD}$ and $\overline {CD}$, and then draw the diagonal $\overline {AC}$. Find the sum of the lengths of the 335 parallel segments drawn.

For natural numbers $n$ and $b$, let $V(n, b)$ denote the number of decompositions of $n$ into the product of integers each of which is greater than $b$: for example $$36 = 6\times 6 = 4\times 9 = 3\times 3\times 4 = 3\times 12,$$ i.e. $V(36,2) = 5$. Prove that $V(n, b) < n/b$ for all $n$ and $b$. (N.B. Vasiliev, Moscow)

Find max. numbers $A$ wich is true ineq.: $\frac{x}{\sqrt{y^{2}+z^{2}}}+\frac{y}{\sqrt{x^{2}+z^{2}}}+\frac{z}{\sqrt{x^{2}+y^{2}}}\geq A$ $x,y,z$ are positve reals numberes! :wink:

$1+\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}$ is equal to $\textbf{(A)}\ \dfrac{15}{8} \qquad \textbf{(B)}\ 1\dfrac{3}{14} \qquad \textbf{(C)}\ \dfrac{11}{8} \qquad \textbf{(D)}\ 1\dfrac{3}{4} \qquad \textbf{(E)}\ \dfrac{7}{8}$

What is the maximum number of terms in a geometric progression with common ratio greater than 1 whose entries all come from the set of integers between 100 and 1000 inclusive?

The positive real numbers $x,y,z$ satisfy $x+y+z=1$. Show that \[\sqrt{3xyz}\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{1-x}+\frac{1}{1-y}+\frac{1}{1-z}\right)\geq4+ \frac{4xyz}{(1-x)(1-y)(1-z)}.\]

The circle is divided into four non-overlapping gaps $ AB $, $ BC $, $ CD $ and $ DA $. Prove that the segment joining the midpoints of the arcs $AB$ and $CD$ is perpendicular to the segment joining the midpoints of the arcs $BC$ and $DA$.

How many integers divide either $2018$ or $2019$? Note: $673$ and $1009$ are both prime. [i]2019 CCA Math Bonanza Lightning Round #1.1[/i]

The 20 members of a local tennis club have scheduled exactly 14 two-person games among themselves, with each member playing in at least one game. Prove that within this schedule there must be a set of 6 games with 12 distinct players.

Let $ a_1, a_2, \ldots , a_n$ be distinct positive integers and let $ M$ be a set of $ n \minus{} 1$ positive integers not containing $ s \equal{} a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n.$ A grasshopper is to jump along the real axis, starting at the point $ 0$ and making $ n$ jumps to the right with lengths $ a_1, a_2, \ldots , a_n$ in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in $ M.$ [i]Proposed by Dmitry Khramtsov, Russia[/i]

In triangle $ABC$, $M,N$ and $P$ are midpoints of sides $BC,CA$ and $AB$. Point $K$ lies on segment $NP$ so that $AK$ bisects $\angle BKC$. Lines $MN,BK$ intersects at $E$ and lines $MP,CK$ intersects at $F$. Suppose that $H$ be the foot of perpendicular line from $A$ to $BC$ and $L$ the second intersection of circumcircle of triangles $AKH, HEF$. Prove that $MK,EF$ and $HL$ are concurrent. [i]Proposed by Alireza Dadgarnia[/i]

A fair coin is flipped every second and the results are recorded with $1$ meaning heads and $0$ meaning tails. What is the probability that the sequence $10101$ occurs before the first occurance of the sequence $010101$?

Seven two-digit integers form a strictly increasing arithmetic sequence. If the first and last terms of this sequence have the same set of digits, what is the sum of all possible medians of the sequence?

Let $ABC$ be a triangle inscribed in the circle $K_1$ and $I$ be center of the inscribed in $ABC$ circle. The lines $IB$ and $IC$ intersect circle $K_1$ again in $J$ and $L$. Circle $K_2$, circumscribed to $IBC$, intersects again $CA$ and $AB$ in $E$ and $F$. Show that $EL$ and $FJ$ intersects on the circle $K_2$.

A strictly increasing sequence $\{x_i\}_{i=1}^{\infty}$ of positive integers is said to be [i]large[/i] if, for every real number $L$, there exists an integer $n$ such that $\frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n} > L$. Do there exist large sequences $\{a_i\}_{i=1}^\infty$ and $\{b_i\}_{i=1}^{\infty}$ such that the sequence $\{a_i+b_i\}_{i=1}^{\infty}$ is not large? [i]Proposed by Lewis Chen[/i]

Initially the numbers $i^3-i$ for $i=2,3 \ldots 2n+1$ are written on a blackboard, where $n\geq 2$ is a positive integer. On one move we can delete three numbers $a, b, c$ and write the number $\frac{abc} {ab+bc+ca}$. Prove that when two numbers remain on the blackboard, their sum will be greater than $16$.

Sarah is fighting a dragon in DnD. She rolls two fair twenty-sided dice numbered $1, 2, \ldots, 20$. She vanquishes the dragon if the product of her two rolls is a multiple of $4$. What is the probability that the dragon is vanquished?

Consider the number of triangles where the side lengths $a,b,c$ satisfy $(a + b + c) (a + b -c) = 2b^2$. Determine the angles in the triangle for which the angle opposite to the side with the length $a$ is as big as possible.

Let $ABCD$ be a rectangle. Determine the set of all points $P$ from the region between the parallel lines $AB$ and $CD$ such that $\angle APB=\angle CPD$.

Shining tells Prajit a positive integer $n \ge 2025$. Prajit then tries to place n points such that no four points are concyclic and no $3$ points are collinear in Euclidean plane, such that Shining cannot find a group of three points such that their circumcircle contains none of the other remaining points. Is he always able to do so? [i](Prajit Adhikari, Nepal and Shining Sun, USA)[/i]

Given is a natural number $n>4$. There are $n$ points marked on the plane, no three of which lie on the same line. Vasily draws one by one all the segments connecting pairs of marked points. At each step, drawing the next segment $S$, Vasily marks it with the smallest natural number, which hasn't appeared on a drawn segment that has a common end with $S$. Find the maximal value of $k$, for which Vasily can act in such a way that he can mark some segment with the number $k$?

Find all positive integers $p,q,r,s>1$ such that $$p!+q!+r!=2^s.$$

Consider the equilateral triangular lattice in the complex plane defined by the Eisenstein integers; let the ordered pair $(x,y)$ denote the complex number $x+y\omega$ for $\omega=e^{2\pi i/3}$. We define an $\omega$-chessboard polygon to be a (non self-intersecting) polygon whose sides are situated along lines of the form $x=a$ or $y=b$, where $a$ and $b$ are integers. These lines divide the interior into unit triangles, which are shaded alternately black and white so that adjacent triangles have different colors. To tile an $\omega$-chessboard polygon by lozenges is to exactly cover the polygon by non-overlapping rhombuses consisting of two bordering triangles. Finally, a [i]tasteful tiling[/i] is one such that for every unit hexagon tiled by three lozenges, each lozenge has a black triangle on its left (defined by clockwise orientation) and a white triangle on its right (so the lozenges are BW, BW, BW in clockwise order). a) Prove that if an $\omega$-chessboard polygon can be tiled by lozenges, then it can be done so tastefully. b) Prove that such a tasteful tiling is unique. [i]Victor Wang.[/i]