Points $ A’,B,C’ $ are arbitrarily taken on edges $ SA,SB, $ respectively, $ SC $ of a tetrahedron $ SABC. $ Plane forrmed by $ ABC $ intersects the plane $ \rho , $ formed by $ A’B’C’, $ in a line $ d. $ Prove that, meanwhile the plane $ \rho $ rotates around $ d, $ the lines $ AA’,BB’ $ and $ CC’ $ are, and remain concurrent. Find de locus of the respective intersections.

$9.3$ A natural number is called square-free, if it is not divisible by the square of any prime number. For a natural number $a$, we consider the number $f(a) = a^{a+1} + 1$. Prove that: a) if $a$ is even, then $f(a)$ is not square-free b) there exist infinitely many odd $a$ for which $f(a)$ is not square-free $9.4$ We will call a generalized $2n$-parallelogram a convex polygon with $2n$ sides, so that, traversed consecutively, the $k$th side is parallel and equal to the $(n+k)$th side for $k=1, 2, ... , n$. In a rectangular coordinate system, a generalized parallelogram is given with $50$ vertices, each with integer coordinates. Prove that its area is at least $300$.

Given a right triangular prism $A_1B_1C_1 - ABC$ with $\angle BAC = \frac{\pi}{2}$ and $AB = AC = AA_1$, let $G$, $E$ be the midpoints of $A_1B_1$, $CC_1$ respectively, and $D$, $F$ be variable points lying on segments $AC$, $AB$ (not including endpoints) respectively. If $GD \bot EF$, the range of the length of $DF$ is ${ \textbf{(A)}\ [\frac{1}{\sqrt{5}}, 1)\qquad\textbf{(B)}\ [\frac{1}{5}, 2)\qquad\textbf{(C)}\ [1, \sqrt{2})\qquad\textbf{(D)}} [\frac{1}{\sqrt{2}}, \sqrt{2})\qquad $

Lukas Nepomuceno draws 5 congruent circles equally spaced around a 6th, and colors all of them 1 of 3 colors. Assume that rotations and reflections of colorings are indistinguishable. How many distinct colorings are there? [i]2022 CCA Math Bonanza Lightning Round 4.4[/i]

A regular icosahedron is a regular solid of $20$ faces, each in the form of an equilateral triangle, with $12$ vertices, so that each vertex is in $5$ edges. Twelve indistinguishable candies are glued to the vertices of a regular icosahedron (one at each vertex), and four of these twelve candies are special. André and Lucas want to together create a strategy for the following game: • First, André is told which are the four special sweets and he must remove exactly four sweets that are not special from the icosahedron and leave the solid on a table, leaving afterwards without communicating with Lucas. • Later, Sponchi, who wants to prevent Lucas from discovering the special sweets, can pick up the icosahedron from the table and rotate it however he wants. • After Sponchi makes his move, he leaves the room, Lucas enters and he must determine the four special candies out of the eight that remain in the icosahedron. Determine if there is a strategy for which Lucas can always properly discover the four special sweets.

Show that the number $1989$ as well as each of its powers $1989^n$ ($n \in N$), can be expressed as a sum of two positive squares in at least two ways.

Let $x$ be a number such that $10000x+2=4$. What is the value of $5000x+1$? $\textbf{(A) }{-}1\qquad\textbf{(B) }0\qquad\textbf{(C) }1\qquad\textbf{(D) }2\qquad\textbf{(E) }3$

What percent of the numbers $1, 2, 3, ... 1000$ are divisible by exactly one of the numbers $4$ and $5?$

Let $x,y,z$ be positive real numbers satisfying the equations $$xyz=1\text{ and }\frac yz(y-x^2)+\frac zx(z-y^2)+\frac xy(x-z^2)=0$$ What is the minimum value of the ratio of the sum of the largest and smallest numbers among $x,y,z$ to the median of them.

Consider the equations $x^2+y^2=16$ and $xy=\tfrac{9}{2}.$ Find the sum, over all ordered pairs $(x,y)$ satisfying these equations, of $|x+y|.$

Evaluate \[\int_{0}^{1}\frac{x\ dx}{(x^{2}+x+1)^{\frac{3}{2}}}\]

Let $n \geq 2$ be a positive integer, and suppose buildings of height $1, 2, \ldots, n$ are built in a row on a street. Two distinct buildings are said to be $\emph{roof-friendly}$ if every building between the two is shorter than both buildings in the pair. For example, if the buildings are arranged $5, 3, 6, 2, 1, 4,$ there are $8$ roof-friendly pairs: $(5, 3), (5, 6), (3, 6), (6, 2), (6, 4), (2, 1),$ $(2, 4), (1, 4).$ Find, with proof, the minimum and maximum possible number of roof-friendly pairs of buildings, in terms of $n.$

Find all points on the plane such that the sum of the distances of each of the four lines defined by the unit square of that plane is $4$.

Let $ n\geq 3 $ be an integer and let $ x_1,x_2,\ldots,x_{n-1} $ be nonnegative integers such that \begin{eqnarray*} \ x_1 + x_2 + \cdots + x_{n-1} &=& n \\ x_1 + 2x_2 + \cdots + (n-1)x_{n-1} &=& 2n-2. \end{eqnarray*} Find the minimal value of $ F(x_1,x_2,\ldots,x_n) = \sum_{k=1}^{n-1} k(2n-k)x_k $.

Consider the function \[ f(x,n) = \frac{\binom n0 + \binom n2 x + \binom n4x^2 + \cdots}{\binom n1 + \binom n3 x + \binom n5 x^2 + \cdots}, \] where $n$ is a positive integer. Express $f(x,n+1)$ rationally in terms of $f(x,n)$ and $x$. Hence, or otherwise, evaluate $\textstyle\lim_{n\to\infty}f(x,n)$ for suitable fixed values of $x$. (The symbols $\textstyle\binom nr$ represent the binomial coefficients.)

Let a space figure consist of $n$ vertices and $l$ lines connecting these vertices, with $n=q^2+q+1$, $l\ge q^2(q+1)^2+1$, $q\ge2$, $q\in\mathbb{N}$. Suppose the figure satisfies the following conditions: every four vertices are non-coplaner, every vertex is connected by at least one line, and there is a vertex connected by at least $p+2$ lines. Prove that there exists a space quadrilateral in the figure, i.e. a quadrilateral with four vertices $A, B, C, D$ and four lines $ AB, BC, CD, DA$ in the figure.

A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has diameter $ 10$ and altitude $ 12$, and the axes of the cylinder and cone coincide. Find the radius of the cylinder. $ \textbf{(A)}\ \frac83 \qquad \textbf{(B)}\ \frac{30}{11} \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ \frac{25}{8} \qquad \textbf{(E)}\ \frac{7}{2}$

How many positive even multiples of $3$ less than $2020$ are perfect squares? $\textbf{(A) }7 \qquad \textbf{(B) }8 \qquad \textbf{(C) }9 \qquad \textbf{(D) }10 \qquad\textbf{(E) }12$

For positive integers $t,a$, and $b$, Lucy and Windy play the $(t,a,b)$- [i]game [/i] defined by the following rules. Initially, the number $t$ is written on a blackboard. On her turn, a player erases the number on the board and writes either the number $t - a$ or $t - b$ on the board. Lucy goes first and then the players alternate. The player who first reaches a negative losses the game. Prove that there exist infinitely many values of $t$ in which Lucy has a winning strategy for all pairs $(a, b)$ with $a + b = 2005$.

Prove that a complex number $ z $ is real and positive if for any nonnegative integers $ n, $ the number $$ z^{2^n} +\bar{z}^{2^n} $$ is real and positive. [i]Sorin Rădulescu[/i]

Given four straight lines, $m_1, m_2, m_3, m_4$, intersecting at $O$ and numbered clockwise with $O$ as the center of the clock, we draw a line through an arbitrary point $A_1$ on $m_1$ parallel to $m_4$ until the line meets $m_2$ at $A_2$. We draw a line through $A_2$ parallel to $m_1$ until it meets $m_3$ at $A_3$. We also draw a line through $A_3$ parallel to $m_2$ until it meets $m_4$ at $A_4$. Now, we draw a line through$ A_4$ parallel to $m_3$ until it meets $m_1$ at $B$. Prove that a) $OB< \frac{OA_1}{2}$ . b) $OB \le \frac{OA_1}{4}$ . [img]https://cdn.artofproblemsolving.com/attachments/5/f/5ea08453605e02e7e1253fd7c74065a9ffbd8e.png[/img]

Let $f(n)$ be the sum of the first $n$ terms of the sequence \[ 0, 1,1, 2,2, 3,3, 4,4, 5,5, 6,6, \ldots\, . \] a) Give a formula for $f(n)$. b) Prove that $f(s+t)-f(s-t)=st$ where $s$ and $t$ are positive integers and $s>t$.

Let $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f(a)-f(b)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f(c)$ for some positive integer $c$. [i]Proposed by Juhan Aru, Estonia[/i]

Let $k$ be the circumference of the triangle $ABC.$ The point $D$ is an arbitrary point on the segment $AB.$ Let $I$ and $J$ be the centers of the circles which are tangent to the side $AB,$ the segment $CD$ and the circle $k.$ We know that the points $A, B, I$ and $J$ are concyclic. The excircle of the triangle $ABC$ is tangent to the side $AB$ in the point $M.$ Prove that $M \equiv D.$

Show that there are infinitely many positive integers $n$ such that the number of distinct odd prime factors of $n(n + 3)$ is a multiple of $3$. (For instance, $180 = 2^2 \cdot 3^2 \cdot 5$ has two distinct odd prime factors and $840 = 2^3 \cdot 3 \cdot 5 \cdot 7$ has three.)