[b]R[/b]thea, a distant planet, is home to creatures whose DNA consists of two (distinguishable) strands of bases with a fixed orientation. Each base is one of the letters H, M, N, T, and each strand consists of a sequence of five bases, thus forming five pairs. Due to the chemical properties of the bases, each pair must consist of distinct bases. Also, the bases H and M cannot appear next to each other on the same strand; the same is true for N and T. How many possible DNA sequences are there on Rthea?

Prove that $ \left( \left.\left\{\begin{pmatrix} a & b & c \\ 3c & a & b \\ 3b & 3c & a\end{pmatrix} \right| a,b,c\in\mathbb{Q}\right\} ,+,\cdot\right) $ is a field.

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]