Let $a$, $b$, and $c$ be positive integers such that the product $ab$ divides the product $c(c^2-c+1)$ and the sum $a+b$ is divisible the number $c^2+1$. Prove that the sets ${a,b}$ and ${c,c^2-c+1}$ coincide.

A point $D$ is chosen on the side $AC$ of a triangle $ABC$ with $\angle C < \angle A < 90^\circ$ in such a way that $BD=BA$. The incircle of $ABC$ is tangent to $AB$ and $AC$ at points $K$ and $L$, respectively. Let $J$ be the incenter of triangle $BCD$. Prove that the line $KL$ intersects the line segment $AJ$ at its midpoint.

Let $n \ge 2$ be an integer. Alex writes the numbers $1, 2, ..., n$ in some order on a circle such that any two neighbours are coprime. Then, for any two numbers that are not comprime, Alex draws a line segment between them. For each such segment $s$ we denote by $d_s$ the difference of the numbers written in its extremities and by $p_s$ the number of all other drawn segments which intersect $s$ in its interior. Find the greatest $n$ for which Alex can write the numbers on the circle such that $p_s \le |d_s|$, for each drawn segment $s$.

Alistar wants to wreak havoc on Jhin's yard, which is a 2D plane of grass. First, he selects a number $n$, randomly and uniformly from $[0,1]$, and then he eats all grass within $n$ meters from where he's standing. He then moves 2 meters in a random direction, and repeats his process. He stops if any of the grass that he wants to eat (or, in other words, in his intended eating territory) is already eaten. Estimate the amount of grass Alistar is expected to eat. An estimate $E$ earns $\frac{2}{1+|A-E|}$ points, where $A$ is the actual answer. [i]2022 CCA Math Bonanza Lightning Round 5.1[/i]

How many ways are there to arrange the five letters P,U,M,A,C, such that the two vowels are not adjacent?

Find all functions $ f:\mathbb{R^{\ast }}\rightarrow \mathbb{ R^{\ast }}$ satisfying $f(\frac{f(x)}{f(y)})=\frac{1}{y}f(f(x))$ for all $x,y\in \mathbb{R^{\ast }}$ and are strictly monotone in $(0,+\infty )$

Starting from a pyramid $T_0$ whose edges are all of length $2019$, we construct the Figure $T_1$ when considering the triangles formed by the midpoints of the edges of each face of $T_0$, building in each of these new pyramid triangles with faces identical to base. Then the bases of these new pyramids are removed. Figure $T_2$ is constructed by applying the same process from $T_1$ on each triangular face resulting from $T_1$, and so on for $T_3, T_4, ...$ Let $D_0= \max \{d(x,y)\}$, where $x$ and $y$ are vertices of $T_0$ and $d(x,y)$ is the distance between $x$ and $y$. Then we define $D_{n + 1} = \max \{d (x, y) |d (x, y) \notin \{D_0, D_1,...,D_n\}$, where $x, y$ are vertices of $T_{n+1}$. Find the value of $D_n$ for all $n$.

Let $n \ge 3$ be an odd integer. In a $2n \times 2n$ board, we colour $2(n-1)^2$ cells. What is the largest number of three-square corners that can surely be cut out of the uncoloured figure? [i]Proposed by G. Sharafetdinova[/i]

Given a triangle $A_1 A_2 A_3$ and a point $P$ inside. Let $B_i$ be a point on the side opposite to $A_i$ for $i=1,2,3$, and let $C_i$ and $D_i$ be the midpoints of $A_i B_i$ and $P B_i$, respectively. Prove that the triangles $C_1 C_2 C_3$ and $D_1 D_2 D_3$ have equal area.

A polygon (not necessarily convex) on the coordinate plane is called [i]plump[/i] if it satisfies the following conditions: $\bullet$ coordinates of vertices are integers; $\bullet$ each side forms an angle of $0^\circ$, $90^\circ$, or $45^\circ$ with the abscissa axis; $\bullet$ internal angles belong to the interval $[90^\circ, 270^\circ]$. Prove that if a square of each side length of a plump polygon is even, then such a polygon can be cut into several convex plump polygons. [i](A. Yuran)[/i]

Let $P$ be the midpoint of the arc $AC$ of a circle, and $B$ be a point on the arc $AP$. Let $M$ and $N$ be the projections of $P$ onto the segments $AC$ and $BC$ respectively. Prove that if $D$ is the intersection of the bisector of $\angle ABC$ and the segment $AC$, then every diagonal of the quadrilateral $BDMN$ bisects the area of the triangle $ABC$.

For all positive real numbers $a, b,c.$ Prove the folllowing inequality$$\frac{a}{c+5b}+\frac{b}{a+5c}+\frac{c}{b+5a}\geq\frac{1}{2}.$$

There are $2014$ balls with $106$ different colors, $19$ of each color. Determine the least possible value of $n$ so that no matter how these balls are arranged around a circle, one can choose $n$ consecutive balls so that amongst them, there are $53$ balls with different colors.

Let $a, b, c$ be real numbers such that $0 < a, b, c < 1/2$ and $a + b + c= 1$. Prove that for all real numbers $x,y,z$, $$abc(x + y + z)^2 \ge ayz( 1- 2a) + bxz( 1 - 2b) + cxy( 1 - 2c)$$. When does equality hold?

How many $1 < n \le 2018$ such that the set $$\{0, 1, 1+2,...,1+2+3+...+i,..., 1+2+...+n-1\}$$ is a permutation of $\{0, 1, 2, 3, 4,...,; n -1\}$ when reduced modulo $n$?

What is the smallest positive integer $n$ such that $n=x^3+y^3$ for two different positive integer tuples $(x,y)$?

Let $ABC$ be a triangle. Let $ABC_1, BCA_1, CAB_1$ be three equilateral triangles that do not overlap with $ABC$. Let $P$ be the intersection of the circumcircles of triangle $ABC_1$ and $CAB_1$. Let $Q$ be the point on the circumcircle of triangle $CAB_1$ so that $PQ$ is parallel to $BA_1$. Let $R$ be the point on the circumcircle of triangle $ABC_1$ so that $PR$ is parallel to $CA_1$. Show that the line connecting the centroid of triangle $ABC$ and the centroid of triangle $PQR$ is parallel to $BC$. [i]Proposed by usjl[/i]

Determine all positive integers $n$ for which the square $n \times n$ can be cut into squares $2\times 2$ and $3\times3$ (with the sides parallel to the sides of the big square).

Problem. The sum of two consecutive squares can be a square; for instance $3^2 + 4^2 = 5^2$. (a) Prove that the sum of $m$ consecutive squares cannot be a square for $m \in \{3, 4, 5, 6\}$. (b) Find an example of eleven consecutive squares whose sum is a square. Can anyone help me with this? Thanks.

Let $ABC$ be a triangle, $\Gamma$ its circumcircle and $l$ the tangent to $\Gamma$ through $A$. The altitudes from $B$ and $C$ are extended and meet $l$ at $D$ and $E$, respectively. The lines $DC$ and $EB$ meet $\Gamma$ again at $P$ and $Q$, respectively. Show that the triangle $APQ$ is isosceles.

Sixteen wooden Cs are placed in a $4$-by-$4$ grid, all with the same orientation, and each is to be colored either red or blue. A quadrant operation on the grid consists of choosing one of the four two-by-two subgrids of Cs found at the corners of the grid and moving each C in the subgrid to the adjacent square in the subgrid that is $90$ degrees away in the clockwise direction, without changing the orientation of the C. Given that two colorings are the considered same if and only if one can be obtained from the other by a series of quadrant operations, determine the number of distinct colorings of the Cs. [img]https://cdn.artofproblemsolving.com/attachments/a/9/1e59dce4d33374960953c0c99343eef807a5d2.png[/img]

Find the area of the region bounded by $C: y=-x^4+8x^3-18x^2+11$ and the tangent line which touches $C$ at distinct two points.

Let $ ABCD$ be a trapezoid with parallel sides $ AB > CD$. Points $ K$ and $ L$ lie on the line segments $ AB$ and $ CD$, respectively, so that $AK/KB=DL/LC$. Suppose that there are points $ P$ and $ Q$ on the line segment $ KL$ satisfying \[\angle{APB} \equal{} \angle{BCD}\qquad\text{and}\qquad \angle{CQD} \equal{} \angle{ABC}.\] Prove that the points $ P$, $ Q$, $ B$ and $ C$ are concyclic. [i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]

The segment $AB$ divides the square into two parts, in each of which a circle can be inscribed. The radii of these circles are equal to $r_1$ and $r_2$ respectively, where $r_1> r_2$. Find the length of $AB$.

Two different points $X$ and $Y$ are chosen in the plane. Find all the points $Z$ in this plane for which the triangle $XYZ$ is isosceles.