We define a sequence of positive integers $a_1,a_2,a_3,\dots$ as follows: Let $a_1=1$ and iteratively, for $k =2,3,\dots$ let $a_k$ be the largest prime factor of $1+a_1a_2\cdots a_{k-1}$. Show that the number $11$ is not an element of this sequence.

Let $n\ge 2$ be an integer. Elwyn is given an $n\times n$ table filled with real numbers (each cell of the table contains exactly one number). We define a [i]rook set[/i] as a set of $n$ cells of the table situated in $n$ distinct rows as well as in n distinct columns. Assume that, for every rook set, the sum of $n$ numbers in the cells forming the set is nonnegative.\\ \\ By a move, Elwyn chooses a row, a column, and a real number $a,$ and then he adds $a$ to each number in the chosen row, and subtracts $a$ from each number in the chosen column (thus, the number at the intersection of the chosen row and column does not change). Prove that Elwyn can perform a sequence of moves so that all numbers in the table become nonnegative.

How many ways are there to paint each of the integers $2, 3, \dots, 9$ either red, green, or blue so that each number has a different color from each of its proper divisors? $\textbf{(A)}\ 144\qquad\textbf{(B)}\ 216\qquad\textbf{(C)}\ 256\qquad\textbf{(D)}\ 384\qquad\textbf{(E)}\ 432$

An experiment consists of performing $n$ independent tests. The $i$-th test is successful with the probability equal to $p_i$. Let $r_k$ be the probability that exactly $k$ tests succeed. Prove that $$\sum_{i=1}^n p_i =\sum_{k=0}^n kr_k.$$

In a tetrahedron $ABCD$, the medians of the faces $ABD$, $ACD$, $BCD$ from $D$ make equal angles with the corresponding edges $AB$, $AC$, $BC$. Prove that each of these faces has area less than or equal to the sum of the areas of the other two faces. [hide="Comment"][i]Equivalent version of the problem:[/i] $ABCD$ is a tetrahedron. $DE$, $DF$, $DG$ are medians of triangles $DBC$, $DCA$, $DAB$. The angles between $DE$ and $BC$, between $DF$ and $CA$, and between $DG$ and $AB$ are equal. Show that: area $DBC$ $\leq$ area $DCA$ + area $DAB$. [/hide]

Rectangular sheet of paper $ABCD$ is folded as shown in the figure. Find the rato $DK: AB$, given that $C_1$ is the midpoint of $AD$. [img]https://3.bp.blogspot.com/-9EkSdxpGnPU/W6dWD82CxwI/AAAAAAAAJHw/iTkEOejlm9U6Dbu427vUJwKMfEOOVn0WwCK4BGAYYCw/s400/Yasinsky%2B2017%2BVIII-IX%2Bp1.png[/img]

A [i]lattice point[/i] in the Cartesian plane is a point whose coordinates are both integers. A [i]lattice polygon[/i] is a polygon all of whose vertices are lattice points. Let $\Gamma$ be a convex lattice polygon. Prove that $\Gamma$ is contained in a convex lattice polygon $\Omega$ such that the vertices of $\Gamma$ all lie on the boundary of $\Omega$, and exactly one vertex of $\Omega$ is not a vertex of $\Gamma$.

Let $f$ be a function on nonnegative integers such that $f(0)=0$ and $$f(3n+2)=f(3n+1)=f(3n)+1=3f(n)+1$$ for all integers $n \ge 0.$ Compute the sum of all nonnegative integers $m$ such that $f(m)=13.$

For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.

a) $r_1, r_2, ... , r_{100}, c_1, c_2, ... , c_{100}$ are distinct reals. The number $r_i + c_j$ is written in position $i, j$ of a $100 \times 100$ array. The product of the numbers in each column is $1$. Show that the product of the numbers in each row is $-1$. b) $r_1, r_2, ... , r_{2n}, c_1, c_2, ... , c_{2n}$ are distinct reals. The number $r_i + c_j$ is written in position $i, j$ of a $2n \times 2n$ array. The product of the numbers in each column is the same. Show that the product of the numbers in each row is also the same.

Answer either (i) or (ii). (i) In a right prism with triangular base, given the sum of the areas of three mutually adjacent faces (that is, of two lateral faces and one base), show that these faces are of equal area and perpendicular to each other when the volume attains its maximum. (ii) Show that \[ \frac{\frac x1 + \frac {x^3} {1 \cdot 3} + \frac {x^5} {1 \cdot 3 \cdot 5} + \frac {x^7} {1 \cdot 3 \cdot 5 \cdot 7} + \cdots }{1 + \frac {x^2} 2 + \frac {x^4}{2 \cdot 4} + \frac{x^6}{2 \cdot 4 \cdot 6} + \cdots} = \int_0^x e^{-t^2} \mathrm dt.\]

How many natural numbers $n$ satisfy the following conditions: i) $219<=n<=2019$, ii) there exist integers $x, y$, so that $1<=x<n<y$, and $y$ is divisible by all natural numbers from $1$ to $n$ with the exception of the numbers $x$ and $x + 1$ with which $y$ is not divisible by.

There are 99 seagulls labeled 2-100 and 100 bagels labeled 1-100. Starting from Seagull 2, each Seagull $N$ eats $\frac{1}{N}$ of whatever remains of each Bagel $I$ where $N$ divides $I$. How many bagels still have more than $\frac{2}{3}$ of their original size after Seagull 100 finishes eating? [i]2022 CCA Math Bonanza Lightning Round 4.1[/i]

Consider a function $f: R \to R$ that fulfills the following two properties: $f$ is periodic of period $5$ (that is, for all $x\in R$, $f (x + 5) = f (x)$), and by restricting $f$ to the interval $[-2,3]$, $f$ coincides to $x^2$. Determine the value of $f(2018).$

Consider the sequence $a_{1}, a_{2}, a_{3},\ldots$ defined by $a_{1}=2024^{2024}$ and for each positive integer $n$, $$a_{n+1}=\left|a_{n}-\sqrt{2}\right|.$$ Prove that there exists an integer $k$ such that $a_{k+2}=a_k$. [i]Here [/i]$\left|x\right|$[i] denotes the absolute value of [/i]$x$.

For $a>2$, let $f(t)=\frac{\sin ^ 2 at+t^2}{at\sin at},\ g(t)=\frac{\sin ^ 2 at-t^2}{at\sin at}\ \left(0<|t|<\frac{\pi}{2a}\right)$ and let $C: x^2-y^2=\frac{4}{a^2}\ \left(x\geq \frac{2}{a}\right).$ Answer the questions as follows. (1) Show that the point $(f(t),\ g(t))$ lies on the curve $C$. (2) Find the normal line of the curve $C$ at the point $\left(\lim_{t\rightarrow 0} f(t),\ \lim_{t\rightarrow 0} g(t)\right).$ (3) Let $V(a)$ be the volume of the solid generated by a rotation of the part enclosed by the curve $C$, the nornal line found in (2) and the $x$-axis. Express $V(a)$ in terms of $a$, then find $\lim_{a\to\infty} V(a)$.

Let $k$ be the circle inscribed in convex quadrilateral $ABCD$. Lines $AD$ and $BC$ meet at $P$ ,and circumcircles of $\triangle PAB$ and $\triangle PCD$ meet in $X$ . Prove that tangents from $X$ to $k$ form equal angles with lines $AX$ and $CX$ .

Find all second degree polynomials $P(x)$ such that for all $a \in\mathbb{R} , a \geq 1$, then $P(a^2+a) \geq a.P(a+1)$

Four points are chosen at random on the surface of a sphere. What is the probability that the center of the sphere lies inside the tetrahedron whose vertices are at the four points?

The number of real solutions of the equation $\left(\frac{9}{10}\right)^x=-3+x-x^2$ is [list=1] [*] 2 [*] 0 [*] 1 [*] None of these [/list]

Angle bisectors of angles by vertices $A$, $B$ and $C$ in triangle $ABC$ intersect opposing sides in points $A_1$, $B_1$ and $C_1$, respectively. Let $M$ be an arbitrary point on one of the lines $A_1B_1$, $B_1C_1$ and $C_1A_1$. Let $M_1$, $M_2$ and $M_3$ be orthogonal projections of point $M$ on lines $BC$, $CA$ and $AB$, respectively. Prove that one of the lines $MM_1$, $MM_2$ and $MM_3$ is equal to sum of other two

The number $1000$ can be written as the sum of $16$ consecutive natural numbers: $$1000 = 55 + 56 + ... + 70.$$ Determines all natural numbers that cannot be written as the sum of two or more consecutive natural numbers .

Given $50$ segments on the line. Prove that one of the following statements is valid: 1. Some $8$ segments have the common point. 2. Some $8$ segments do not intersect each other.

Let $ n \geq 2$ be an integer, let $ S$ be a set of $ n$ elements, and let $ A_i , \; 1\leq i \leq m$, be distinct subsets of $ S$ of size at least $ 2$ such that \[ A_i \cap A_j \not\equal{} \emptyset, A_i \cap A_k \not\equal{} \emptyset, A_j \cap A_k \not\equal{} \emptyset, \;\textrm{imply}\ \;A_i \cap A_j \cap A_k \not\equal{} \emptyset \ .\] Show that $ m \leq 2^{n\minus{}1}\minus{}1$. [i]P. Erdos[/i]

Let $a,b,c,d$ be positive real numbers satisfying $a+b+c+d=4$. Prove that $$\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{d}+\frac{d^2}{a}\geq 4+(a-d)^2$$