Found problems: 85335
Alan, Jason, and Shervin are playing a game with MafsCounts questions. They each start with $2$ tokens. In each round, they are given the same MafsCounts question. The first person to solve the MafsCounts question wins the round and steals one token from each of the other players in the game. They all have the same probability of winning any given round. If a player runs out of tokens, they are removed from the game. The last player remaining wins the game.
If Alan wins the first round but does not win the second round, what is the probability that he wins the game?
[i]2020 CCA Math Bonanza Individual Round #4[/i]
$(HUN 3)$ In the plane $4000$ points are given such that each line passes through at most $2$ of these points. Prove that there exist $1000$ disjoint quadrilaterals in the plane with vertices at these points.
All cells of $2015 \times 2015$ table colored in one of $4$ colors. We count number of ways to place one tetris T-block in table. Prove that T-block has cell of all $4$ colors in less than $51\%$ of total number of ways.
A triangle $ABC$ is given. Let $C\ensuremath{'}$ be the vertex of an isosceles triangle $ABC\ensuremath{'}$ with $\angle C\ensuremath{'} = 120^{\circ}$ constructed on the other side of $AB$ than $C$, and $B\ensuremath{'}$ be the vertex of an equilateral triangle $ACB\ensuremath{'}$ constructed on the same side of $AC$ as $ABC$. Let $K$ be the midpoint of $BB\ensuremath{'}$
Find the angles of triangle $KCC\ensuremath{'}$.
[i]Proposed by A.Zaslavsky[/i]
What is the smallest value of $k$ for which the inequality
\begin{align*}
ad-bc+yz&-xt+(a+c)(y+t)-(b+d)(x+z)\leq \\
&\leq k\left(\sqrt{a^2+b^2}+\sqrt{c^2+d^2}+\sqrt{x^2+y^2}+\sqrt{z^2+t^2}\right)^2
\end{align*}
holds for any $8$ real numbers $a,b,c,d,x,y,z,t$?
Edit: Fixed a mistake! Thanks @below.
Let $1=d_1<d_2<\ldots<d_k=n$ be all divisors of $n$. It turned out that numbers $d_2-d_1,\ldots,d_k-d_{k-1}$ are $1,3,\ldots,2k-3$ in some order. Find all possible values of $n$
[i]M. Zorka[/i]
Find the least positive integer of the form [u]a[/u] [u]b[/u] [u]a[/u] [u]a[/u] [u]b[/u] [u]a[/u], where a and b are distinct digits, such that the integer can be written as a product of six distinct primes
A subset $ B$ of the set of integers from $ 1$ to $ 100$, inclusive, has the property that no two elements of $ B$ sum to $ 125$. What is the maximum possible number of elements in $ B$?
$ \textbf{(A)}\ 50\qquad
\textbf{(B)}\ 51\qquad
\textbf{(C)}\ 62\qquad
\textbf{(D)}\ 65\qquad
\textbf{(E)}\ 68$
Determine an infinite family of quadruples $(a, b, c, d)$ of positive integers, each of which is a solution to $a^4+b^5+c^6=d^7$.
What is the greatest number of integers that can be selected from a set of $2015$ consecutive numbers so that no sum of any two selected numbers is divisible by their difference?
What is the difference between the median and the mean of the following data set: $12,41, 44, 48, 47, 53, 60, 62, 56, 32, 23, 25, 31$?
In each cell of a table with $m$ rows and $n$ columns, where $m<n$, we put a non-negative real number such that each column contains at least one positive number.
Show that there is a cell with a positive number such that the sum of the numbers in its row is larger than the sum of the numbers in its column.
Let $a,b$ and $c$ be positive numbers with $a^2+b^2+c^2=3$. Prove that $a+b+c\ge 3\sqrt[5]{abc}$.
The numbers $1, 2, 3$, $\ldots$, $n^2$ are arranged in an $n\times n$ array, so that the numbers in each row increase from left to right, and the numbers in each column increase from top to bottom. Let $a_{ij}$ be the number in position $i, j$. Let $b_j$ be the number of possible values for $a_{jj}$. Show that \[ b_1 + b_2 + \cdots + b_n = \frac{ n(n^2-3n+5) }{3} . \]
Let $S$ be an infinite set of real numbers such that $|x_1+x_2+\cdots + x_n | \leq 1 $ for all finite subsets $\{x_1,x_2,\ldots,x_n\} \subset S$. Show that $S$ is countable.
One hundred tennis players took part in a tournament where they played with each other
exactly one game, with no draws. At the end of the tournament a table (ranking) is formed depending on the number of victories. It is known that one tennis player finished the tournament on
$k$-th place and is the only one with that number of victories, and he has beaten every tennis player who is placed above him in the table and lost to anyone ranked weaker than him on the table. Find the smallest value of $k$.
In an acute scalene triangle $ABC$, the incircle $\omega$ touches the sides $BC$, $CA$, and $AB$ at points $D$, $E$, and $F$, respectively. Let $P$ be the foot of the perpendicular from $F$ to $DE$. The line $BP$ intersects segment $AC$ at $K$, and the line $AP$ intersects segment $BC$ at $L$. The altitude through vertex $C$ in $\triangle ABC$ intersects the circumcircle of $\triangle CKL$ at a point $Q$. Prove that line $PQ$ passes through the center of $\omega$.
Given that
\begin{align*}
x_{1}&=211,\\
x_{2}&=375,\\
x_{3}&=420,\\
x_{4}&=523, \text{ and}\\
x_{n}&=x_{n-1}-x_{n-2}+x_{n-3}-x_{n-4} \text{ when } n \geq 5,
\end{align*}
find the value of $x_{531}+x_{753}+x_{975}$.
Find the digits left and right of the decimal point in the decimal form of the number \[ (\sqrt{2} + \sqrt{3})^{1980}. \]
All possible vertical lines $x = k$ and horizontal lines $y = m$ are drawn on the coordinate plane, where $k$ and $m$ are integers. Let's imagine that all these straight lines are black. A red straight line is also drawn, the equation of which is $19x+96y= c$. Let us denote by $M$ the number of segments of different lengths formed on the red line when intersecting with the black ones.(The ends of each segment are the intersection points of the red and black lines. There are no such intersection points inside the segment.) What values can $M$ take when $c$ changes?
A sled loaded with children starts from rest and slides down a snowy $25^\circ$ (with respect to the horizontal) incline traveling $85$ meters in $17$ seconds. Ignore air resistance. What is the coefficient of kinetic friction between the sled and the slope?
$ \textbf {(A) } 0.36 \qquad \textbf {(B) } 0.40 \qquad \textbf {(C) } 0.43 \qquad \textbf {(D) } 1.00 \qquad \textbf {(E) } 2.01 $
Let $ABCD$ be a cyclic quadrilateral. Points $E$ and $F$ lie on the sides $AD$ and $CD$ in such a way that $AE = BC$ and $AB = CF$. Let $M$ be the midpoint of $EF$. Prove that $\angle AMC = 90^{\circ}$.
Determine the radius of a sphere $S$ that passes through the centroids of each face of a given tetrahedron $T$ inscribed in a unit sphere with center $O$. Also, determine the distance from $O$ to the center of $S$ as a function of the edges of $T.$
Find all pairs of real numbers $(x,y)$ satisfying both equations
\[
3x^2+3xy+2y^2 =2
\]
\[
x^2+2xy+2y^2 =1.
\]
[i]2020 CCA Math Bonanza Team Round #5[/i]
Find out whether it is possible to determine the volume of a tetrahedron knowing the areas of its faces and its circumradius.