Given triangle $ABC, BC$ is extended beyond $B$ to the point $D$ such that $BD = BA$. The bisectors of the exterior angles at vertices $B$ and $C$ intersect at the point $M$. Prove that quadrilateral $ADMC$ is cyclic. (4)

Prove that if the numbers $a, b, c$ are the lengths of the sides of some nondegenerate triangle, then the equation $$b^2x^2 + (b^2 + c^2 - a^2) x + c^2 = 0$$ has imaginary roots.

Let $ABC$ be a triangle with $\angle CAB = 2 \angle ABC$. Assume that a point $D$ is inside the triangle $ABC$ exists such that $AD = BD$ and $CD = AC$. Show that $\angle ACB = 3 \angle DCB$.

Let $ G$ be a simple graph. Suppose that size of largest independent set in $ G$ is $ \alpha$. Prove that: a) Vertices of $ G$ can be partitioned to at most $ \alpha$ paths. b) Suppose that a vertex and an edge are also cycles. Prove that vertices of $ G$ can be partitioned to at most $ \alpha$ cycles.

There are $n ( \ge 3) $ students in a class. Some students are friends each other, and friendship is always mutual. There are $ s ( \ge 1 ) $ couples of two students who are friends, and $ t ( \ge 1 ) $ triples of three students who are each friends. For two students $ x, y $ define $ d(x,y)$ be the number of students who are both friends with $ x $ and $ y $. Prove that there exist three students $ u, v, w $ who are each friends and satisfying \[ d(u,v) + d(v,w) + d(w,u) \ge \frac{9t}{s} . \]

Points $C$, $A$, $D$, $M$, $E$, $B$, $F$ lie on a line in that order such that $CA = AD = EB = BF = 1$ and $M$ is the midpoint of $DB$. Let $X$ be a point such that a quarter circle arc exists with center $D$ and endpoints $C$, $X$. Suppose that line $XM$ is tangent to the unit circle centered at $B$. Compute $AB$. [i]2020 CCA Math Bonanza Individual Round #11[/i]

Let $\triangle ABC$ be a triangle such that $AB=6, BC=8,$ and $AC=10$. Let $M$ be the midpoint of $BC$. Circle $\omega$ passes through $A$ and is tangent to $BC$ at $M$. Suppose $\omega$ intersects segments $AB$ and $AC$ again at points $X$ and $Y$, respectively. If the area of $AXY$ can be expressed as $\frac{p}{q}$ where $p, q$ are relatively prime integers, compute $p+q$.

Solve in the set of rational numbers the equation $$2\sqrt{3(x + 1)^2} -3 \sqrt{2(y - 2)^2}= 4\sqrt2 + 5|\sqrt2 - \sqrt3|$$

Let $ABC$ be a triangle satisfying $BC < CA$. Let $P$ be an arbitrary point on the side $AB$ (different from $A$ and $B$), and let the line $CP$ meet the circumcircle of triangle $ABC$ at a point $S$ (apart from the point $C$). Let the circumcircle of triangle $ASP$ meet the line $CA$ at a point $R$ (apart from $A$), and let the circumcircle of triangle $BPS$ meet the line $CB$ at a point $Q$ (apart from $B$). Prove that the excircle of triangle $APR$ at the side $AP$ is identical with the excircle of triangle $PQB$ at the side $PQ$ if and only if the point $S$ is the midpoint of the arc $AB$ on the circumcircle of triangle $ABC$.

Let $ABC$ be a triangle with $\angle A,\angle B,\angle C <120^{\circ}$, $T$ is its [i]Fermat-Torricelli[/i] point. Let $G$ be the centroid of $\triangle ABC$. Prove that: the distances from $G$ to the perpendicular bisectors of $TA, TB, TC$ are the same.

The paper is written on consecutive integers $1$ through $n$. Then are deleted all numbers ending in $4$ and $9$ and the rest alternating between $-$ and $+$. Finally, an opening parenthesis is added after each character and at the end of the expression the corresponding number of parentheses: $1 - (2 + 3 - (5 + 6 - (7 + 8 - (10 +...))))$. Find all numbers $n$ such that the value of this expression is $13$.

A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won $10$ games and lost $10$ games; there were no ties. How many sets of three teams $\{A, B, C\}$ were there in which $A$ beat $B$, $B$ beat $C$, and $C$ beat $A?$ $\textbf{(A)}\ 385 \qquad \textbf{(B)}\ 665 \qquad \textbf{(C)}\ 945 \qquad \textbf{(D)}\ 1140 \qquad \textbf{(E)}\ 1330$

Let $ x$, $ y$, $ z$ be real numbers such that $ 0 < x,y,z < 1$ and $ xyz \equal{} (1 \minus{} x)(1 \minus{} y)(1 \minus{} z)$. Show that at least one of the numbers $ (1 \minus{} x)y,(1 \minus{} y)z,(1 \minus{} z)x$ is greater than or equal to $ \frac {1}{4}$

Consider a 4-point configuration in the plane such that every 3 points can be covered by a strip of a unit width. Prove that: 1) the four points can be covered by a strip of length at most $\sqrt2$ and 2)if no strip of length less that $\sqrt2$ covers all the four points, then the points are vertices of a square of length $\sqrt2$

For $x=7$, which of the following is the smallest? $ \text{(A)}\ \frac{6}{x}\qquad\text{(B)}\ \frac{6}{x+1}\qquad\text{(C)}\ \frac{6}{x-1}\qquad\text{(D)}\ \frac{x}{6}\qquad\text{(E)}\ \frac{x+1}{6} $

A list of \(n\) positive integers \(a_1, a_2,a_3,\ldots,a_n\) is said to be [i]good[/i] if it checks simultaneously: \(\bullet a_1<a_2<a_3<\cdots<a_n,\) \(\bullet a_1+a_2^2+a_3^3+\cdots+a_n^n\le 2023.\) For each \(n\ge 1\), determine how many [i]good[/i] lists of \(n\) numbers exist.

In a building whose floors are numbered $1$ to $8$, the builder wants to place elevators so that, for every choice of two floors, there are always at least three elevators that stop on those floors. Furthermore, each elevator can only stop at a maximum of $5$ floors. What is the minimum number of elevators that need to be placed?

Determine all positive integers $d$ such that whenever $d$ divides a positive integer $n$, $d$ will also divide any integer obtained by rearranging the digits of $n$.

In the triangle$ABC$ ($AC > AB$), point $N$ is the midpoint of $BC$, and $I$ is the intersection point of the angle bisectors. Ray $AI$ intersects the circumscribed circle of triangle $ABC$ at point $W$, a perpendicular $WF$ is drawn from it on side $AC$. Find the length of the segment $CF$ , if the radius of the circle inscribed in the triangle $ABC$ is equal to $r$ and $\angle INB = 45^o$. (Gryhoriy Filippovskyi)

Two points are selected independently and uniformly at random inside a regular hexagon. Compute the probability that a line passing through both of the points intersects a pair of opposite edges of the hexagon.

Each of six fruit baskets contains pears, plums and apples. The number of plums in each basket equals the total number of apples in all other baskets combined while the number of apples in each basket equals the total number of pears in all other baskets combined. Prove that the total number of fruits is a multiple of $31$.

$x\geq y\geq z\geq \frac{\pi}{12},x+y+z=\frac{\pi}{2}$, find the maximum and minumum value of $\cos x\sin y\cos z$.

A regular polyhedron is a polyhedron that is convex and all of its faces are regular polygons. We call a regular polhedron a "[i]Choombam[/i]" iff none of its faces are triangles. a) prove that each choombam can be inscribed in a sphere. b) Prove that faces of each choombam are polygons of at most 3 kinds. (i.e. there is a set $\{m,n,q\}$ that each face of a choombam is $n$-gon or $m$-gon or $q$-gon.) c) Prove that there is only one choombam that its faces are pentagon and hexagon. (Soccer ball) [img]http://aycu08.webshots.com/image/5367/2001362702285797426_rs.jpg[/img] d) For $n>3$, a prism that its faces are 2 regular $n$-gons and $n$ squares, is a choombam. Prove that except these choombams there are finitely many choombams.

Suppose $a$, $b$, $c$ are three positive real numbers with $a + b + c = 3$. Prove that $$\frac{a}{b^2 + c}+\frac{b}{c^2 + a}+\frac{c}{a^2 + b}\geq \frac{3}{2}$$

[b]p1.[/b] Is it possible to put $9$ numbers $1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9$ in a circle in a way such that the sum of any three circularly consecutive numbers is divisible by $3$ and is, moreover: a) greater than $9$ ? b) greater than $15$? [b]p2.[/b] You can cut the figure below along the sides of the small squares into several (at least two) identical pieces. What is the minimal number of such equal pieces? [img]https://cdn.artofproblemsolving.com/attachments/8/e/9cd09a04209774dab34bc7f989b79573453f35.png[/img] [b]p3.[/b] There are $100$ colored marbles in a box. It is known that among any set of ten marbles there are at least two marbles of the same color. Show that the box contains $12$ marbles of the same color. [b]p4.[/b] Is it possible to color squares of a $ 8\times 8$ board in white and black color in such a way that every square has exactly one black neighbor square separated by a side? [b]p5.[/b] In a basket, there are more than $80$ but no more than $200$ white, yellow, black, and red balls. Exactly $12\%$ are yellow, $20\%$ are black. Is it possible that exactly $2/3$ of the balls are white? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].