On the table, there're $1000$ cards arranged on a circle. On each card, a positive integer was written so that all $1000$ numbers are distinct. First, Vasya selects one of the card, remove it from the circle, and do the following operation: If on the last card taken out was written positive integer $k$, count the $k^{th}$ clockwise card not removed, from that position, then remove it and repeat the operation. This continues until only one card left on the table. Is it possible that, initially, there's a card $A$ such that, no matter what other card Vasya selects as first card, the one that left is always card $A$?

Three spheres are all externally tangent to a plane and to each other. Suppose that the radii of these spheres are $6$, $8$, and, $10$. The tangency points of these spheres with the plane form the vertices of a triangle. Determine the largest integer that is smaller than the perimeter of this triangle.

Suppose that ${f: \Bbb{R} \rightarrow \Bbb{R}}$ is an additive function $($that is ${f(x+y) = f(x)+f(y)}$ for all ${x, y \in \Bbb{R}})$ for which ${x \mapsto f(x)f(\sqrt{1-x^2})}$ is bounded of some nonempty subinterval of ${(0,1)}$. Prove that ${f}$ is continuous. [i]Proposed by Zoltán Boros[/i]

Let $x$- minimal root of equation $x^2-4x+2=0$. Find two first digits of number $ \{x+x^2+....+x^{20} \}$ after $0$, where $\{a\}$- fractional part of $a$.

Regular $2n$-gon is inscribed in the unit circle. Find the sum of the squares of all sides and diagonal lengths in the $2n$-gon.

Let $S$ be the set of integers from $1$ to $13$ inclusive. A permutation of $S$ is a function $f : S \to S$ such that $f(x) \ne f(y)$ if $x \ne y$. For how many distinct permutations $f$ does there exists an $n $ such that $f^n(i) = 13 - i + 1$ for all $i$.

The diagram below shows the first three figures of a sequence of figures. The first figure shows an equilateral triangle $ABC$ with side length $1$. The leading edge of the triangle going in a clockwise direction around $A$ is labeled $\overline{AB}$ and is darkened in on the figure. The second figure shows the same equilateral triangle with a square with side length $1$ attached to the leading clockwise edge of the triangle. The third figure shows the same triangle and square with a regular pentagon with side length $1$ attached to the leading clockwise edge of the square. The fourth figure in the sequence will be formed by attaching a regular hexagon with side length $1$ to the leading clockwise edge of the pentagon. The hexagon will overlap the triangle. Continue this sequence through the eighth figure. After attaching the last regular figure (a regular decagon), its leading clockwise edge will form an angle of less than $180^\circ$ with the side $\overline{AC}$ of the equilateral triangle. The degree measure of that angle can be written in the form $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [asy] size(250); defaultpen(linewidth(0.7)+fontsize(10)); pair x[],y[],z[]; x[0]=origin; x[1]=(5,0); x[2]=rotate(60,x[0])*x[1]; draw(x[0]--x[1]--x[2]--cycle); for(int i=0;i<=2;i=i+1) { y[i]=x[i]+(15,0); } y[3]=rotate(90,y[0])*y[2]; y[4]=rotate(-90,y[2])*y[0]; draw(y[0]--y[1]--y[2]--y[0]--y[3]--y[4]--y[2]); for(int i=0;i<=4;i=i+1) { z[i]=y[i]+(15,0); } z[5]=rotate(108,z[4])*z[2]; z[6]=rotate(108,z[5])*z[4]; z[7]=rotate(108,z[6])*z[5]; draw(z[0]--z[1]--z[2]--z[0]--z[3]--z[4]--z[2]--z[7]--z[6]--z[5]--z[4]); dot(x[2]^^y[2]^^z[2],linewidth(3)); draw(x[2]--x[0]^^y[2]--y[4]^^z[2]--z[7],linewidth(1)); label("A",(x[2].x,x[2].y-.3),S); label("B",origin,S); label("C",x[1],S);[/asy]

A line $l$ is drawn through the intersection point $H$ of altitudes of acute-angle triangles. Prove that symmetric images $l_a, l_b, l_c$ of $l$ with respect to the sides $BC,CA,AB$ have one point in common, which lies on the circumcircle of $ABC.$

Prove that $f(2) \ge 3^n$ where the polynomial $f(x) = x_n + a_1x_{n-1} + ...+ a_{n-1}x + 1$ has non-negative coefficients and $n$ real roots.

Let $x$, $y$, $z$ be positive real numbers satisfying $x+y+z\ge 6$. Find, with proof, the minimum value of \[ x^2+y^2+z^2+\frac{x}{y^2+z+1}+\frac{y}{z^2+x+1}+\frac{z}{x^2+y+1}. \]

On Day $1$, Alice starts with the number $a_1=5$. For all positive integers $n>1$, on Day $n$, Alice randomly selects a positive integer $a_n$ between $a_{n-1}$ and $2a_{n-1}$, inclusive. Given that the probability that all of $a_2,a_3,\ldots,a_7$ are odd can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, compute $m+n$. [i]2021 CCA Math Bonanza Lightning Round #1.4[/i]

Decide whether there exist infinitely many triples $(a,b,c)$ of positive integers such that all prime factors of $a!+b!+c!$ are smaller than $2020$. [i]Pitchayut Saengrungkongka, Thailand[/i]

Compute $\sqrt{2022^2-12^6}.$

Define a common chord between two intersecting circles to be the line segment connecting their two intersection points. Let $\omega_1,\omega_2,\omega_3$ be three circles of radii $3, 5,$ and $7$, respectively. Suppose they are arranged in such a way that the common chord of $\omega_1$ and $\omega_2$ is a diameter of $\omega_1$, the common chord of $\omega_1$ and $\omega_3$ is a diameter of $\omega_1$, and the common chord of $\omega_2$ and $\omega_3$ is a diameter of $\omega_2$. Compute the square of the area of the triangle formed by the centers of the three circles.

To each point of $\mathbb{Z}^3$ we assign one of $p$ colors. Prove that there exists a rectangular parallelepiped with all its vertices in $\mathbb{Z}^3$ and of the same color.

[b]Problem Section #3 c) Let $ABCDE$ be a convex pentagon such that $BC \parallel AE, AB = BC + AE$, and $\angle{ABC} =\angle{CDE}$. Let $M$ be the midpoint of $CE$, and let $O$ be the circumcenter of triangle $BCD$. Given that $\angle{DMO}=90^{o}$, prove that $2\angle{BDA} =\angle{CDE}$.

Let $ABC$ be a triangle such that $|AC|=1$ and $|AB|=\sqrt 2$. Let $M$ be a point such that $|MA|=|AB|$, $m(\widehat{MAB}) = 90^\circ$, and $C$ and $M$ are on the opposite sides of $AB$. Let $N$ be a point such that $|NA|=|AX|$, $m(\widehat{NAC}) = 90^\circ$, and $B$ and $N$ are on the opposite sides of $AC$. If the line passing throung $A$ and the circumcenter of triangle $MAN$ meets $[BC]$ at $F$, what is $\dfrac {|BF|}{|FC|}$? $ \textbf{(A)}\ 2\sqrt 2 \qquad\textbf{(B)}\ 2\sqrt 3 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 3\sqrt 2 $

Consider the $4 \times 4$ “multiplication table” below. The numbers in the first column multiplied by the numbers in the first row give the remaining numbers in the table. For example, the $3$ in the first column times the $4$ in the first row give the $12 (= 3 \cdot 4)$ in the cell that is in the 3rd row and 4th column. [asy] size(3cm); for (int x=0; x<=4; ++x) draw((x, 0) -- (x, 4), linewidth(.5pt)); for (int y=0; y<=4; ++y) draw((0, y) -- (4, y), linewidth(.5pt)); draw((0,0)--(4,0)--(4,4)--(0,4)--cycle); void foo(int x, int y, string n) { label(n, (x+0.5, y+0.5)); } foo(0, 3, "1"); foo(1, 3, "2"); foo(2, 3, "3"); foo(3, 3, "4"); foo(0, 2, "2"); foo(1, 2, "4"); foo(2, 2, "6"); foo(3, 2, "8"); foo(0, 1, "3"); foo(1, 1, "6"); foo(2, 1, "9"); foo(3, 1, "12"); foo(0, 0, "4"); foo(1, 0, "8"); foo(2, 0, "12"); foo(3, 0, "16"); [/asy] We create a path from the upper-left square to the lower-right square by always moving one cell either to the right or down. For example, here is one such possible path, with all the numbers along the path circled: [asy] import graph; size(3cm); for (int x=0; x<=4; ++x) draw((x, 0) -- (x, 4), linewidth(.5pt)); for (int y=0; y<=4; ++y) draw((0, y) -- (4, y), linewidth(.5pt)); draw((0,0)--(4,0)--(4,4)--(0,4)--cycle); void foo(int x, int y, string n) { label(n, (x+0.5, y+0.5)); } draw(Circle((0.5,3.5),0.5)); draw(Circle((1.5,3.5),0.5)); draw(Circle((2.5,3.5),0.5)); draw(Circle((2.5,2.5),0.5)); draw(Circle((3.5,2.5),0.5)); draw(Circle((3.5,1.5),0.5)); draw(Circle((3.5,0.5),0.5)); foo(0, 3, "1"); foo(1, 3, "2"); foo(2, 3, "3"); foo(3, 3, "4"); foo(0, 2, "2"); foo(1, 2, "4"); foo(2, 2, "6"); foo(3, 2, "8"); foo(0, 1, "3"); foo(1, 1, "6"); foo(2, 1, "9"); foo(3, 1, "12"); foo(0, 0, "4"); foo(1, 0, "8"); foo(2, 0, "12"); foo(3, 0, "16"); [/asy] If we add up the circled numbers in the example above (including the start and end squares), we get $48$. Considering all such possible paths: (a) What is the smallest sum we can possibly get when we add up the numbers along such a path? Prove your answer is correct. (b) What is the largest sum we can possibly get when we add up the numbers along such a path? Prove your answer is correct.

Prove for every integer $n>0$, there exists an integer $k>0$ such that $2^nk$ can be written in decimal notation using only digits 1 and 2.

Light of a blue laser (wavelength $\lambda=475 \, \text{nm}$) goes through a narrow slit which has width $d$. After the light emerges from the slit, it is visible on a screen that is $ \text {2.013 m} $ away from the slit. The distance between the center of the screen and the first minimum band is $ \text {765 mm} $. Find the width of the slit $d$, in nanometers. [i](Proposed by Ahaan Rungta)[/i]

Let $S$ be the set of lattice points $(x,y) \in \mathbb{Z}^2$ such that $-10\leq x,y \leq 10$. Let the point $(0,0)$ be $O$. Let Scotty the Dog's position be point $P$, where initially $P=(0,1)$. At every second, consider all pairs of points $C,D \in S$ such that neither $C$ nor $D$ lies on line $OP$, and the area of quadrilateral $OCPD$ (with the points going clockwise in that order) is $1$. Scotty finds the pair $C,D$ maximizing the sum of the $y$ coordinates of $C$ and $D$, and randomly jumps to one of them, setting that as the new point $P$. After $50$ such moves, Scotty ends up at point $(1, 1)$. Find the probability that he never returned to the point $(0,1)$ during these $50$ moves. [i]Proposed by David Tang[/i]

Let $n \ge 2$ be an integer. Prove that if $k^2 + k + n$ is prime for all integers $k$ such that $0 \leq k \leq \sqrt{\frac{n}{3}}$, then $k^2 +k + n$ is prime for all integers $k$ such that $0 \leq k \leq n - 2$.

Determine the smallest possible value of $| A_{1} \cup A_{2} \cup A_{3} \cup A_{4} \cup A_{5} |$, where $A_{1}, A_{2}, A_{3}, A_{4}, A_{5}$ sets simultaneously satisfying the following conditions: $(i)$ $| A_{i}\cap A_{j} | = 1$ for all $1\leq i < j\leq 5$, i.e. any two distinct sets contain exactly one element in common; $(ii)$ $A_{i}\cap A_{j} \cap A_{k}\cap A_{l} =\varnothing$ for all $1\leq i<j<k<l\leq 5$, i.e. any four different sets contain no common element. Where $| S |$ means the number of elements of $S$.

Find all prime numbers $p, q$ such that$$p(p+1)(p^2+1) = q^2(q^2+q+1) + 2025.$$ [i]Proposed by Md. Fuad Al Alam[/i]

A sequence $a_n$ is defined by $a_0 = 0$, and for all $n \ge 1$, $a_n = a_{n-1} + (-1)^n \cdot n^2$. Compute $a_{100}$