Let $ABC$ be a triangle and $K$ be its circumcircle. Let $P$ be the point of intersection of $BC$ with tangent in $A$ to $K$. Let $D$ and $E$ be the symmetrical points of $B$ and $A$, respectively, from $P$. Let $K_1$ be the circumcircle of triangle $DAC$ and let $K_2$ the circumscribed circle of triangle $APB$. We denote with $F$ the second intersection point of the circles $K_1$ and $K_2$ Then denote with $G$ the second intersection point of the circle $K_1$ with $BF$. Show that the lines $BC$ and $EG$ are parallel.

Let $P(x)$ be a polynomial with real coefficients and form the polynomial $$Q(x) = ( x^2 +1) P(x)P'(x) + x(P(x)^2 + P'(x)^2 ).$$ Given that the equation $P(x) = 0$ has $n$ distinct real roots exceeding $1$, prove or disprove that the equation $Q(x)=0$ has at least $2n - 1$ distinct real roots.

Given positive integers $m$ and $n$. Let $a_{i,j} ( 1 \le i \le m, 1 \le j \le n)$ be non-negative real numbers, such that $$ a_{i,1} \ge a_{i,2} \ge \cdots \ge a_{i,n} \text{ and } a_{1,j} \ge a_{2,j} \ge \cdots \ge a_{m,j} $$ holds for all $1 \le i \le m$ and $1 \le j \le n$. Denote $$ X_{i,j}=a_{1,j}+\cdots+a_{i-1,j}+a_{i,j}+a_{i,j-1}+\cdots+a_{i,1},$$ $$ Y_{i,j}=a_{m,j}+\cdots+a_{i+1,j}+a_{i,j}+a_{i,j+1}+\cdots+a_{i,n}.$$ Prove that $$ \prod_{i=1}^{m} \prod_{j=1}^{n} X_{i,j} \ge \prod_{i=1}^{m} \prod_{j=1}^{n} Y_{i,j}.$$

Suppose $a _ { n } > 0$ and $\sum _ { n = 1 } ^ { \infty } a _ { n }$ diverges. Determine whether $\sum _ { n = 1 } ^ { \infty } a _ { n } / S _ { n } ^ { 2 }$ converges, where $S _ { n } = a _ { 1 } + a _ { 2 } + \dots + a _ { n } .$

Let $P$ be a convex polyhedron. Jaroslav writes a non-negative real number to every vertex of $P$ in such a way that the sum of these numbers is $1$. Afterwards, to every edge he writes the product of the numbers at the two endpoints of that edge. Prove that the sum of the numbers at the edges is at most $\frac{3}{8}$.

[b]p1.[/b] What is the value of the sum $2 + 20 + 202 + 2022$? [b]p2.[/b] Find the smallest integer greater than $10000$ that is divisible by $12$. [b]p3.[/b] Valencia chooses a positive integer factor of $6^{10}$ at random. The probability that it is odd can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime integers. Find $m + n$. [b]p4.[/b] How many three digit positive integers are multiples of $4$ but not $8$? [b]p5.[/b] At the Jane Street store, Andy accidentally buys $5$ dollars more worth of shirts than he had planned. Originally, including the tip to the cashier, he planned to spend all of the remaining $90$ dollars on his giftcard. To compensate for his gluttony, Andy instead gives the cashier a smaller, $12.5\%$ tip so that he still spends $90$ dollars total. How much percent tip was Andy originally planning on giving? [b]p6.[/b] Let $A,B,C,D$ be four coplanar points satisfying the conditions $AB = 16$, $AC = BC =10$, and $AD = BD = 17$. What is the minimum possible area of quadrilateral $ADBC$? [b]p7.[/b] How many ways are there to select a set of three distinct points from the vertices of a regular hexagon so that the triangle they form has its smallest angle(s) equal to $30^o$? [b]p8.[/b] Jaeyong rolls five fair $6$-sided die. The probability that the sum of some three rolls is exactly $8$ times the sum of the other two rolls can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [b]p9.[/b] Find the least positive integer n for there exists some positive integer $k > 1$ for which $k$ and $k + 2$ both divide $\underbrace{11...1}_{n\,\,\,1's}$. [b]p10.[/b] For some real constant $k$, line $y = k$ intersects the curve $y = |x^4-1|$ four times: points $A$,$B$,$C$ and $D$, labeled from left to right. If $BC = 2AB = 2CD$, then the value of $k$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [b]p11.[/b] Let a be a positive real number and $P(x) = x^2 -8x+a$ and $Q(x) = x^2 -8x+a+1$ be quadratics with real roots such that the positive difference of the roots of $P(x)$ is exactly one more than the positive difference of the roots of $Q(x)$. The value of a can be written as a common fraction $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [b]p12.[/b] Let $ABCD$ be a trapezoid satisfying $AB \parallel CD$, $AB = 3$, $CD = 4$, with area $35$. Given $AC$ and $BD$ intersect at $E$, and $M$, $N$, $P$, $Q$ are the midpoints of segments $AE$,$BE$,$CE$,$DE$, respectively, the area of the intersection of quadrilaterals $ABPQ$ and $CDMN$ can be expressed as $\frac{m}{n}$ where $m, n$ are relatively prime positive integers. Find $m + n$. [b]p13.[/b] There are $8$ distinct points $P_1, P_2, ... , P_8$ on a circle. How many ways are there to choose a set of three distinct chords such that every chord has to touch at least one other chord, and if any two chosen chords touch, they must touch at a shared endpoint? [b]p14.[/b] For every positive integer $k$, let $f(k) > 1$ be defined as the smallest positive integer for which $f(k)$ and $f(k)^2$ leave the same remainder when divided by $k$. The minimum possible value of $\frac{1}{x}f(x)$ across all positive integers $x \le 1000$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$. Find $m + n$. [b]p15.[/b] In triangle $ABC$, let $I$ be the incenter and $O$ be the circumcenter. If $AO$ bisects $\angle IAC$, $AB + AC = 21$, and $BC = 7$, then the length of segment $AI$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A lattice point in the coordinate plane with origin $O$ is called invisible if the segment $OA$ contains a lattice point other than $O,A$. Let $L$ be a positive integer. Show that there exists a square with side length $L$ and sides parallel to the coordinate axes, such that all points in the square are invisible.

Find the largest positive real number $p$ (if it exists) such that the inequality \[x^2_1+ x_2^2+ \cdots + x^2_n\ge p(x_1x_2 + x_2x_3 + \cdots + x_{n-1}x_n)\] is satisfied for all real numbers $x_i$, and $(a) n = 2; (b) n = 5.$ Find the largest positive real number $p$ (if it exists) such that the inequality holds for all real numbers $x_i$ and all natural numbers $n, n \ge 2.$

[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 $d$.