Found problems: 85335
Gwen, Eli, and Kat take turns flipping a coin in their respective order. The first one to flip heads wins. What is the probability that Kat will win?
Assume that $ x$ is a positive real number. Which is equivalent to $ \sqrt[3]{x\sqrt{x}}$?
$ \textbf{(A)}\ x^{1/6} \qquad
\textbf{(B)}\ x^{1/4} \qquad
\textbf{(C)}\ x^{3/8} \qquad
\textbf{(D)}\ x^{1/2} \qquad
\textbf{(E)}\ x$
A $2023 \times 2023$ grid of lights begins with every light off. Each light is assigned a coordinate $(x,y).$ For every distinct pair of lights $(x_1, y_1), (x_2, y_2),$ with $x_1<x_2$ and $y_1>y_2,$ all lights strictly between them (i.e. $x_1<x<x_2$ and $y_2<y<y_1$) are toggled. After this procedure is done, how many lights are on?
Ana and Banana are playing a game. Initially, Ana secretly picks a number $1\leq A\leq 10^6$. In each subsequent turn of the game, Banana may pick a positive integer $B$, and Ana will reveal to him the most common digit in the product $A\cdot B$ (written in decimal notation). In the case when at least two digits are tied for being the most common, Ana will reveal all of them to Banana. For example, if $A\cdot B=2022$, Ana will tell Banana that the digit $2$ is the most common, while if $A\cdot B=5783783$, Ana will reveal that $3, 7$ and $8$ are the most common. Banana's goal is to determine with certainty the number $A$ after some number of turns. Does he have a winning strategy?
Given a non negative real $a$ and a sequence $(u_n)$ defined by \[ \begin{cases} u_1=3\\ u_{n+1}=\frac{u_n}{2}+\frac{n^2}{4n^2+a}\sqrt{u_n^2+3} \end{cases} \]
a) Prove that for $a=0$, the sequence is convergent and find its limit.
b) For $a\in [0,1]$, prove that the sequence if convergent.
Incircle of triangle $ABC$ tangent to $AB$ and $AC$ on $E$ and $F$ respectively. If $X$ is the midpoint of $EF$, prove $\angle BXC > 90^{\circ}$
A sequence $(a_n)$ of real numbers is defined by $a_0=1$, $a_1=2015$ and for all $n\geq1$, we have
$$a_{n+1}=\frac{n-1}{n+1}a_n-\frac{n-2}{n^2+n}a_{n-1}.$$
Calculate the value of $\frac{a_1}{a_2}-\frac{a_2}{a_3}+\frac{a_3}{a_4}-\frac{a_4}{a_5}+\ldots+\frac{a_{2013}}{a_{2014}}-\frac{a_{2014}}{a_{2015}}$.
Geoff has an infinite stock of sweets, which come in $n$ flavours. He arbitrarily distributes some of the sweets amongst $n$ children (a child can get sweets of any subset of all flavours, including the empty set). Call a distribution $k-\textit{nice}$ if every group of $k$ children together has sweets in at least $k$ flavours. Find all subsets $S$ of $\{ 1, 2, \dots, n \}$ such that if a distribution of sweets is $s$-nice for all $s \in S$, then it is $s$-nice for all $s \in \{ 1, 2, \dots, n \}$.
[i]Proposed by Kyle Hess, USA[/i]
Four distinct points, $A$, $B$, $C$, and $D$, are to be selected from $1996$ points evenly spaced around a circle. All quadruples are equally likely to be chosen. What is the probability that the chord $AB$ intersects the chord $CD$?
$\text{(A)}\ \frac 14 \qquad \text{(B)}\ \frac 13 \qquad \text{(C)}\ \frac 12 \qquad \text{(D)}\ \frac 23\qquad \text{(E)}\ \frac 34$
Find the sum of all positive integers $n \leq 1000$ with the property that for every prime number $p$ dividing $n,$ we have that $2p-1$ also divides $n.$
Find the number of positive integers that are divisors of at least one of $10^{10},15^7,18^{11}$.
In $\triangle ABC$, shown on the right, let $r$ denote the radius of the inscribed circle, and let $r_A$, $r_B$, and $r_C$ denote the radii of the smaller circles tangent to the inscribed circle and to the sides emanating from $A$, $B$, and $C$, respectively. Prove that
$r \leq r_A + r_B + r_C$
Find all positive integers $n$, such that: $$a+b+c \mid a^{2n}+b^{2n}+c^{2n}-n(a^2b^2+b^2c^2+c^2a^2)$$ for all pairwise different positive integers $a,b$ and $c$
Let $ABCDEF$ be a regular hexagon, and let $G$, $H$, $I$, $J$, $K$, and $L$ be the midpoints of sides $AB$, $BC$, $CD$, $DE$, $EF$, and $FA$, respectively. The intersection of lines $\overline{AH}$, $\overline{BI}$, $\overline{CJ}$, $\overline{DK}$, $\overline{EL}$, and $\overline{FG}$ bound a smaller regular hexagon. Find the ratio of the area of the smaller hexagon to the area of $ABCDEF$.
On sides $BC$, $CA$ and $AB$ of triangle $ABC$, points $A_1$, $B_1$, $C_1$ are taken, respectively, so that the radii of the circles inscribed in triangles $A_1BC_1$, $AB_1C_1$ and $A_1B_1C$ are equal to each other and equal to $r$. The radius of the circle inscribed in triangle $A_1B_1C_1$ is equal to $r_1$. Find the radius of the circle inscribed in triangle $ABC$.
Prove that if $a,b,c \ge 1$ and $a+b+c=9$, then $\sqrt{ab+bc+ca} \le \sqrt{a} +\sqrt{b} + \sqrt{c}$
Let $A,B \in \mathbb{R}^{n\times n}$ with $A^2+B^2=AB$. Prove that if $BA-AB$ is invertible then $3|n$.
The bisector of angle $\angle BAC$ of triangle $ABC$ intersects arc $BC$ (not containing point $A$) of the circle circumscribed around this triangle at point $P$. Segment $AP$ is divided by side $BC$ in ratio $k$ (counting from vertex $A$). Find the perimeter of triangle $ABC$ if $BC = a$.
Let $ABC$ a right isosceles triangle with hypotenuse $AB=\sqrt2$ . Determine the positions of the points $X,Y,Z$ on the sides $BC,CA,AB$ respectively so that the triangle $XYZ$ is isosceles, right, and with minimum area.
Compute $e^{\pi}+\pi^e$. If your answer is $A$ and the correct answer is $C$, then your score on this problem will be $\frac{4}{\pi}\arctan\left(\frac{1}{\left|C-A\right|}\right)$ (note that the closer you are to the right answer, the higher your score is).
[i]2017 CCA Math Bonanza Lightning Round #5.2[/i]
Given two positives integers $m$ and $n$, prove that there exists a positive integer $k$ and a set $S$ of at least $m$ multiples of $n$ such that the numbers $\frac {2^k{\sigma({s})}} {s}$ are odd for every $s \in S$. $\sigma({s})$ is the sum of all positive integers of $s$ (1 and $s$ included).
Let $(a_1, a_2,\ldots , a_n)$ be a permutation of $1, 2, \ldots , n,$ where $n \geq 2.$ For each $k = 1, \ldots , n$, we know that $a_k$ apples are placed at the point $k$ on the real axis. Children named $A,B,C$ are assigned respective points $x_A, x_B, x_C \in \{1, \ldots , n\}.$ For each $k,$ the children whose points are closest to $ k$ divide $a_k$ apples equally among themselves. We call $(x_A, x_B, x_C)$ a [i]stable configuration[/i] if no child’s total share can be increased by assigning a new point to this child and not changing the points of the other two. Determine the values of $n$ for which a stable configuration exists for some distribution $(a_1, \ldots, a_n)$ of the apples.
Three sidelines of on acute-angled triangle are drawn on the plane. Fyodor wants to draw the altitudes of this triangle using a ruler and a compass. Ivan obstructs him using an eraser. For each move Fyodor may draw one line through two markeed points or one circle centered at a marked point and passing through another marked point. After this Fyodor may mark an arbitrary number of points (the common points of drawn lines, arbitrary points on the drawn lines or arbitrary points on the plane). For each move Ivan erases at most three of marked point. (Fyodor may not use the erased points in his constructions but he may mark them for the second time). They move by turns, Fydors begins. Initially no points are marked. Can Fyodor draw the altitudes?
Prove that if $m$ is a natural number and $P,Q,R$ polynomials of degrees less than $m$ satisfying
$$x^{2m}P(x,y)+y^{2m}Q(x,y) = (x+y)^{2m}R(x,y),$$
then each of the polynomials is zero.
Let $m<n$ be two positive integers, let $I$ and $J$ be two index sets such that $|I|=|J|=n$ and $|I\cap J|=m$, and let $u_k$, $k\in I\cup J$ be a collection of vectors in the Euclidean plane such that \[|\sum_{i\in I}u_i|=1=|\sum_{j\in J}u_j|.\] Prove that \[\sum_{k\in I\cup J}|u_k|^2\geq \frac{2}{m+n}\] and find the cases of equality.