A quarry wants to sell a large pile of gravel. At full price, the gravel would sell for $3200$ dollars. But during the first week the quarry only sells $60\%$ of the gravel at full price. The following week the quarry drops the price by $10\%$, and, again, it sells $60\%$ of the remaining gravel. Each week, thereafter, the quarry reduces the price by another $10\%$ and sells $60\%$ of the remaining gravel. This continues until there is only a handful of gravel left. How many dollars does the quarry collect for the sale of all its gravel?

Twenty-one girls and twenty-one boys took part in a mathematical competition. It turned out that each contestant solved at most six problems, and for each pair of a girl and a boy, there was at least one problem that was solved by both the girl and the boy. Show that there is a problem that was solved by at least three girls and at least three boys.

The circles $\omega_1$ and $\omega_2$ intersect at $K{}$ and $L{}$. The line $\ell$ touches the circles $\omega_1$ and $\omega_2$ at the points $X{}$ and $Y{}$, respectively. The point $K{}$ lies inside the triangle $XYL$. The line $XK$ intersects $\omega_2$ a second time at the point $Z{}$. Prove that $LY$ is the bisector of the angle $XLZ$.

Prove that for every positive integer $n$ there exists a (not necessarily convex) polygon with no three collinear vertices, which admits exactly $n$ diffferent triangulations. (A [i]triangulation[/i] is a dissection of the polygon into triangles by interior diagonals which have no common interior points with each other nor with the sides of the polygon)

Let $1,2,3,4,5,6,7,8,9,11,12,\cdots$ be the sequence of all positive integers which do not contain the digit zero. Write $\{a_n\}$ for this sequence. By comparing with a geometric series, show that $\sum_{k=1}^n \frac{1}{a_k} < 90$.

In the scalene triangle $ABC$,$\angle ACB=60$ and $\Omega$ is its cirumcirle.On the bisectors of the angles $BAC$ and $CBA$ points $A^\prime$,$B^\prime$ are chosen respectively such that $AB^\prime \parallel BC$ and $BA^\prime \parallel AC$.$A^\prime B^\prime$ intersects with $\Omega$ at $D,E$.Prove that triangle $CDE$ is isosceles.(A. Kuznetsov)

In triangle $ABC$, interior point $D$ is chosen such that triangle $BCD$ is equilateral. Let $E$ be the isogonal conjugate of point $D$ with respect to triangle $ABC$. Define point $P$ on the ray $AB$ such that $AP=BE$. Similarly, define point $Q$ on the ray $AC$ such that $AQ=CE$. Prove that line $AD$ bisects segment $PQ$. [i]Proposed by Áron Bán-Szabó, Budapest[/i]

Given that the height of a greater sage grouse flying through the air is defined by the function $64x-x^2$ for $0<x<64$, what is the first time at which the bird reaches a height of 903? [i]2022 CCA Math Bonanza Lightning Round 2.3[/i]

Find all prime numbers $ p,q,r$, such that $ \frac{p}{q}\minus{}\frac{4}{r\plus{}1}\equal{}1$

For any real number $a$, prove the inequality: $$\left(a^3+a^2+3\right)^2>4a^3(a-1)^2.$$

Let $r,s$ and $t$ be integers with $0 \leq r$, $0 \leq s$ and $r+s \leq t$. Prove that \[ \frac{\binom s0}{\binom tr} + \frac{\binom s1}{\binom{t}{r+1}} + \cdots + \frac{\binom ss}{\binom{t}{r+s}} = \frac{t+1}{(t+1-s)\binom{t-s}{r}}. \]

Katelyn is building an integer (in base $10$). She begins with $9$. Each step, she appends a randomly chosen digit from $0$ to $9$ inclusive to the right end of her current integer. She stops immediately when the current integer is $0$ or $1$ (mod $11$). The probability that the final integer ends up being $0$ (mod $11$) is $\tfrac ab$ for coprime positive integers $a$, $b$. Find $a + b$. [i]Proposed by Evan Chang[/i]

$ABCD$ is parallelogram. Line $l$ is perpendicular to $BC$ at $B$. Two circles passes through $D,C$, such that $l$ is tangent in points $P$ and $Q$. $M$ - midpoint $AB$. Prove that $\angle DMP=\angle DMQ$

$n\in \mathbb{N}$ is called [i]“good”[/i], if $n$ can be presented as a sum of the fourth powers of five of its divisors (different). a) Prove that each [i]good[/i] number is divisible by 5; b) Find a [i]good[/i] number; c) Does there exist infinitely many [i]good[/i] numbers?

A regular unit $7$-simplex is a polytope in $7$-dimensional space with $8$ vertices that are all exactly a distance of $ 1$ apart. (It is the $7$-dimensional analogue to the triangle and the tetrahedron.) In this $7$-dimensional space, there exists a point that is equidistant from all $8$ vertices, at a distance $d$. Determine $d$.

Find the arithmetic sequences of $ 5 $ integers $ n_1,n_2,n_3,n_4,n_5 $ that verify $ 5|n_1,2|n_2,11|n_3,7|n_4,17|n_5. $

The real numbers $a_1 \leq a_2 \leq \cdots \leq a_{672}$ are given such that $$a_1 + a_2 + \cdots + a_{672} = 2018.$$ For any $n \leq 672$, there are $n$ of these numbers with an integer sum. What is the smallest possible value of $a_{672}$?

There are 3 lines $r, s$ and $t$ on a plane. The lines $r$ and $s$ intersect perpendicularly at point $A$. the line $t$ intersects the line $r$ at point $B$ and the line $s$ at point $C$. There exist exactly 4 circumferences on the plane that are simultaneously tangent to all those 3 lines. Prove that the radius of one of those circumferences is equal to the sum of the radius of the other three circumferences.

For which $ n\in \mathbb{N}$ do there exist rational numbers $ a,b$ which are not integers such that both $ a \plus{} b$ and $ a^n \plus{} b^n$ are integers?

Let $m$ and $n$ be positive integers such that $m^4 - n^4 = 3439$. What is the value of $mn$?

Each vertex of an $m\times n$ grid is colored blue, green or red in such a way that all the boundary vertices are red. We say that a unit square of the grid is properly colored if: $(i)$ all the three colors occur at the vertices of the square, and $(ii)$ one side of the square has the endpoints of the same color. Show that the number of properly colored squares is even.

Points $A, B, C, D, E$ and F are given on a circle in such a way that the three chords $AB, CD$ and $EF$ intersect in one point. Express angle $\angle EFA$ in terms of angles $\angle ABC$ and $\angle CDE$ (find all possibilities).

Let $N$ be a positive integer. In each of the $N^2$ unit squares of an $N\times N$ board, one of the two diagonals is drawn. The drawn diagonals divide the $N\times N$ board into $K$ regions. For each $N$, determine the smallest and the largest possible values of $K$. [asy] draw((0,0)--(3,0)--(3,3)--(0,3)--cycle); draw((1,0)--(1,3), dotted); draw((2,0)--(2,3), dotted); draw((0,1)--(3,1), dotted); draw((0,2)--(3,2), dotted); draw((1,0)--(0,1)--(2,3)--(3,2)--(2,1)--(0,3)); draw((1,1)--(2,0)--(3,1)); label("$1$",(0.35,2)); label("$2$",(1,2.65)); label("$3$",(2,2)); label("$4$",(2.65,2.65)); label("$5$",(0.35,0.35)); label("$6$",(1.3,1.3)); label("$7$",(2.65,0.35)); label("Example with $N=3$, $K=7$",(0,-0.3)--(3,-0.3),S); [/asy]

Let $ABC$ be an acute-angled triangle. The line through $C$ perpendicular to $AC$ meets the external angle bisector of $\angle ABC$ at $D$. Let $H$ be the foot of the perpendicular from $D$ onto $BC$. The point $K$ is chosen on $AB$ so that $KH \parallel AC$. Let $M$ be the midpoint of $AK$. Prove that $MC = MB + BH$. Giorgi Arabidze, Georgia,

In a dancing hall, there are $n$ girls standing in one row and $n$ boys in the other row across them (so that all $2n$ dancers form a $2 \times n$ board). Each dancer gives her / his left hand to a neighboring person standing to the left, across, or diagonally to the left. The analogous rule applies for right hands. No dancer gives both hands to the same person. In how many ways can the dancers do this?