Let $ABCD$ be a convex quadrilateral. We assume that there exists a point $P$ inside the quadrilateral such that the areas of the triangles $ABP, BCP, CDP$, and $DAP$ are equal. Show that at least one of the diagonals of the quadrilateral bisects the other diagonal.

In the plane let $\,C\,$ be a circle, $\,L\,$ a line tangent to the circle $\,C,\,$ and $\,M\,$ a point on $\,L$. Find the locus of all points $\,P\,$ with the following property: there exists two points $\,Q,R\,$ on $\,L\,$ such that $\,M\,$ is the midpoint of $\,QR\,$ and $\,C\,$ is the inscribed circle of triangle $\,PQR$.

Decide if there are primes $p, q, r$ such that $(p^2 + p) (q^2 + q) (r^2 + r)$ is a square of an integer.

Can an asymptotically stable equilibrium position become unstable in the Lyapunov sense under linearization?

Given any set $S$ of positive integers, show that at least one of the following two assertions holds: (1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x\in F}1/x=\sum_{x\in G}1/x$; (2) There exists a positive rational number $r<1$ such that $\sum_{x\in F}1/x\neq r$ for all finite subsets $F$ of $S$.

Let $ a_1,a_2,...,a_{2003}\geq 0$, such that $ a_1\plus{}a_2\plus{}...\plus{}a_{2003}\equal{}2$ and $ a_1a_2\plus{}a_2a_3\plus{}...\plus{}a_{2003}a_1\equal{}1$. Determine the minimum and maximum value of $ a_1^2\plus{}a_2^2\plus{}...\plus{}a_{2003}^2$.

Find the number of permutations of the letters $AAAABBBCC$ where no letter is next to another letter of the same type. For example, count $ABCABCABA$ and $ABABCABCA$ but not $ABCCBABAA$.

Let $ABCDEF$ be a convex hexagon such that $\angle ACE = \angle BDF$ and $\angle BCA = \angle EDF$. Let $A_1=AC\cap FB$, $B_1=BD\cap AC$, $C_1=CE\cap BD$, $D_1=DF\cap CE$, $E_1=EA\cap DF$, and $F_1=FB\cap EA$. Suppose $B_1, C_1, D_1, F_1$ lie on the same circle $\Gamma$. The circumcircles of $\triangle BB_1F_1$ and $ED_1F_1$ meet at $F_1$ and $P$. The line $F_1P$ meets $\Gamma$ again at $Q$. Prove that $B_1D_1$ and $QC_1$ are parrallel. (Here, we use $l_1\cap l_2$ to denote the intersection point of lines $l_1$ and $l_2$)

Solve the system of equations $$\begin{cases} \sin \frac{\pi}{2}xy =z \\ \sin \frac{\pi}{2}yz =x \\ \sin \frac{\pi}{2}zx =y \end{cases} \,\,\, ?$$

Find the distance $\overline{CF}$ in the diagram below where $ABDE$ is a square and angles and lengths are as given: [asy] markscalefactor=0.15; size(8cm); pair A = (0,0); pair B = (17,0); pair E = (0,17); pair D = (17,17); pair F = (-120/17,225/17); pair C = (17+120/17, 64/17); draw(A--B--D--E--cycle^^E--F--A--cycle^^D--C--B--cycle); label("$A$", A, S); label("$B$", B, S); label("$C$", C, dir(0)); label("$D$", D, N); label("$E$", E, N); label("$F$", F, W); label("$8$", (F+E)/2, NW); label("$15$", (F+A)/2, SW); label("$8$", (C+B)/2, SE); label("$15$", (D+C)/2, NE); draw(rightanglemark(E,F,A)); draw(rightanglemark(D,C,B)); [/asy] The length $\overline{CF}$ is of the form $a\sqrt{b}$ for integers $a, b$ such that no integer square greater than $1$ divides $b$. What is $a + b$?

Suppose that a bounded subset $ S$ of the plane is a union of congruent, homothetic, closed triangles. Show that the boundary of $ S$ can be covered by a finite number of rectifiable arcs. [i]L. Geher[/i]

Find all angles a for which the set of numbers $\sin a$, $\sin 2a$, $\sin 3a$ coincides with the set $cos a$, $cos 2a$, $cos 3a$.

A teacher writes the positive integers from $1$ to $12$ on a blackboard. Every minute, they choose a number $k$ uniformly at random from the written numbers, subtract $k$ from each number $n \geq k$ on the blackboard (without touching the numbers $n<k$), and erase every $0$ on the board. Estimate the expected number of minutes that pass before the board is empty. An estimate of $E$ earns $2^{1-0.5|E-A|}$ points, where $A$ is the actual answer. [i]2020 CCA Math Bonanza Lightning Round #5.2[/i]

Is $\frac{19^{92} - 91^{29}}{90}$ an integer?

To each number with $n^2$ digits, we associate the $n\times n$ determinant of the matrix obtained by writing the digits of the number in order along the rows. For example : $8617\mapsto \det \left(\begin{matrix}{\;8}& 6\;\\ \;1 &{ 7\;}\end{matrix}\right)=50$. Find, as a function of $n$, the sum of all the determinants associated with $n^2$-digit integers. (Leading digits are assumed to be nonzero; for example, for $n = 2$, there are $9000$ determinants.)

The two brothers, without waiting for the bus, decided to walk to the next stop. After passing $1/3$ of the way, they looked back and saw a bus approaching the stop. One of the brothers ran backwards, and the other ran forward at the same speed. It turned out that everyone ran to their stop exactly at the moment when the bus approached it. Find the speed of the brothers, if the bus speed is $30$ km / h, neglect the bus stop time.

On a plane are given $6$ red, $6$ blue, and $6$ green points, such that no three of the given points lie on a line. Prove that the sum of the areas of the triangles whose vertices are of the same color does not exceed quarter the sum of the areas of all triangles with vertices in the given points.

For some positive integer $n,$ Elmo writes down the equation \[x_1+x_2+\dots+x_n=x_1+x_2+\dots+x_n.\] Elmo inserts at least one $f$ to the left side of the equation and adds parentheses to create a valid functional equation. For example, if $n=3,$ Elmo could have created the equation \[f(x_1+f(f(x_2)+x_3))=x_1+x_2+x_3.\] Cookie Monster comes up with a function $f: \mathbb{Q}\to\mathbb{Q}$ which is a solution to Elmo's functional equation. (In other words, Elmo's equation is satisfied for all choices of $x_1,\dots,x_n\in\mathbb{Q})$. Is it possible that there is no integer $k$ (possibly depending on $f$) such that $f^k(x)=x$ for all $x$? [i]Srinivas Arun[/i]

The points $P(a,b)$ and $Q(0,c)$ are on the curve $\dfrac{y}{c} = \cosh{(\dfrac{x}{c})}.$ The line through $Q$ parallel to the normal at $P$ cuts the $x-$axis at $R.$ Prove that $QR = b.$

Let $ABC$ be a triangle with ${AB\ne BC}$; and let ${BD}$ be the internal bisector of $\angle ABC,\ $, $\left( D\in AC \right)$. Denote by ${M}$ the midpoint of the arc ${AC}$ which contains point ${B}$. The circumscribed circle of the triangle ${\vartriangle BDM}$ intersects the segment ${AB}$ at point ${K\neq B}$. Let ${J}$ be the reflection of ${A}$ with respect to ${K}$. If ${DJ\cap AM=\left\{O\right\}}$, prove that the points ${J, B, M, O}$ belong to the same circle.

The Frontier Lands have $50$ towns, some pairs of which are directly connected by Morton’s railroad tracks (which are bidirectional and may pass over each other), and it is possible to travel from any town to any other town via these tracks, possibly stopping at other towns on the way. Morton decides that he wants some tracks destroyed so that each town is directly connected to an odd number of other towns. (After Morton destroys the tracks, it might no longer be possible to travel from any town to any other town.) Prove that this is possible.

The reals $b>0$ and $a$ are such that the quadratic $x^2+ax+b$ has two distinct real roots, exactly one of which lies in the interval $[-1;1]$. Prove that one of the roots lies in the interval $(-b;b)$.

Show that there exist $2009$ consecutive positive integers such that for each of them the ratio between the largest and the smallest prime divisor is more than $20.$

[color=darkred]Let $n$ and $k$ be two natural numbers such that $n\ge 2$ and $1\le k\le n-1$ . Prove that if the matrix $A\in\mathcal{M}_n(\mathbb{C})$ has exactly $k$ minors of order $n-1$ equal to $0$ , then $\det (A)\ne 0$ .[/color]

The managing director of AS Mull, a brokerage company for soap bubbles, air castles and cheese holes, kissed the sales manager lazily, claiming that the company's sales volume in December had decreased by more than $10\%$ compared to October. Muugijuht, on the other hand, wrote in his quarterly report that although each, in the first half of the month, sales decreased compared to the second half of the previous month $30\%$ of the time, it increased in the second half of each month compared to the first half of the same month by $35\%$. Was the CEO wrong when the sales manager's report is true?