Found problems: 85335
A [i]Pattano coin[/i] is a coin which has a blue side and a yellow side. A positive integer not exceeding $100$ is written on each side of every coin (the sides may have different integers).
Two Pattano coins are [i]identical [/i] if the number on the blue side of both coins are equal and the number on the yellow side of both coins are equal.
Two Pattano coins are [i]pairable [/i] if the number on the blue side of both coins are equal or the number on the yellow side of both coins are equal.
Given $2559$ Pattano coins such that no two coins are identical. Show that at least one Pattano coin is pairable with at least $50$ other coins
$\sqrt{8}+\sqrt{18}=$
$\textbf{(A)}\ \sqrt{20} \qquad \textbf{(B)}\ 2(\sqrt{2}+\sqrt{3}) \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 5\sqrt{2} \qquad \textbf{(E)}\ 2\sqrt{13}$
Let $k$ be a fixed integer greater than 1, and let ${m=4k^2-5}$. Show that there exist positive integers $a$ and $b$ such that the sequence $(x_n)$ defined by \[x_0=a,\quad x_1=b,\quad x_{n+2}=x_{n+1}+x_n\quad\text{for}\quad n=0,1,2,\dots,\] has all of its terms relatively prime to $m$.
[i]Proposed by Jaroslaw Wroblewski, Poland[/i]
Do there exist 16 three digit numbers, using only three different digits in all, so that the all numbers give different residues when divided by 16?
[i]Bulgaria[/i]
Jay is given $99$ stacks of blocks, such that the $i$th stack has $i^2$ blocks. Jay must choose a positive integer $N$ such that from each stack, he may take either $0$ blocks or exactly $N$ blocks. Compute the value Jay should choose for $N$ in order to maximize the number of blocks he may take from the $99$ stacks.
[i]Proposed by James Lin[/i]
Triangle $ ABC$ has $ AB \equal{} 2 \cdot AC$. Let $ D$ and $ E$ be on $ \overline{AB}$ and $ \overline{BC}$, respectively, such that $ \angle{BAE} \equal{} \angle{ACD}.$ Let $ F$ be the intersection of segments $ AE$ and $ CD$, and suppose that $ \triangle{CFE}$ is equilateral. What is $ \angle{ACB}$?
$ \textbf{(A)}\ 60^{\circ}\qquad \textbf{(B)}\ 75^{\circ}\qquad \textbf{(C)}\ 90^{\circ}\qquad \textbf{(D)}\ 105^{\circ}\qquad \textbf{(E)}\ 120^{\circ}$
Consider the $2\times3$ rectangle below. We fill in the small squares with the numbers $1,2,3,4,5,6$ (one per square). Define a [i]tasty[/i] filling to be one such that each row is [b]not[/b] in numerical order from left to right and each column is [b]not[/b] in numerical order from top to bottom. If the probability that a randomly selected filling is tasty is $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, what is $m+n$?
\begin{tabular}{|c|c|c|c|}
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\end{tabular}
[i]2016 CCA Math Bonanza Lightning #4.2[/i]
Let $I$ be the incenter of triangle $ABC$ with $AB < AC$. Point $X$ is chosen on the external bisector of $\angle ABC$ such that $IC = IX$. Let the tangent to the circumscribed circle of $\triangle BXC$ at point $X$ intersect the line $AB$ at point $Y$. Prove that $AC = AY$.
[i]Proposed by Oleksiy Masalitin[/i]
What is the maximum number of points of intersection of the graphs of two different fourth degree polynomial functions $ y \equal{} p(x)$ and $ y \equal{} q(x)$, each with leading coefficient $ 1$?
$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 8$
Determine all integers $ k\ge 2$ such that for all pairs $ (m$, $ n)$ of different positive integers not greater than $ k$, the number $ n^{n\minus{}1}\minus{}m^{m\minus{}1}$ is not divisible by $ k$.
$p(x_1, x_2, ... , x_n)$ is a polynomial with integer coefficients. For each positive integer $r, k(r)$ is the number of $n$-tuples $(a_1, a_2,... , a_n)$ such that $0 \le a_i \le r-1 $ and $p(a_1, a_2, ... , a_n)$ is prime to $r$. Show that if $u$ and $v$ are coprime then $k(u\cdot v) = k(u)\cdot k(v)$, and if p is prime then $k(p^s) = p^{n(s-1)} k(p)$.
Find all solutions $x,y,z$ in the positive integers of the equation $$3^x -5^y = z^2$$
In the diagram below, square $ABCD$ has side length $4$. Two congruent square $EGIK$ and $FHJL$ are drawn such that $AE=FB=BG=HC=CI=JD=DK=LA=1$ and $EF=GH=IJ=KL=2$. Compute the area of the region that lies in both $EGIK$ and $FHJL$.
A point $D$ is chosen in the interior of the side $BC$ of an acute triangle $ABC$, and another point $P$ in the interior of the segment $AD$, but not lying on the median through $C$. This median (through $C$) intersects the circumcircle of a triangle $CPD$ at $K(\ne C)$. Prove that the circumcircle of triangle $AKP$ always passes through a fixed point $M(\ne A)$ independent of the choices of the points $D$ and $P.$
Let $\vartriangle ABC$ be an acute triangle with angles $\angle BAC = 70^o$, $\angle ABC = 60^o$, let $D, E$ be the feet of perpendiculars from $B, C$ to $AC$, $AB$, respectively, and let $H$ be the orthocenter of $ABC$. Let $F$ be a point on the shorter arc $AB$ of circumcircle of $ABC$ satisfying $\angle F AB = 10^o$ and let $G$ be the foot of perpendicular from $H$ to $AF$. If $I = BF \cap EG$ and $J = CF \cap DG$, compute the angle $\angle GIJ$.
A card deck consists of $1024$ cards. On each card, a set of distinct decimal digits is written in such a way that no two of these sets coincide (thus, one of the cards is empty). Two players alternately take cards from the deck, one card per turn. After the deck is empty, each player checks if he can throw out one of his cards so that each of the ten digits occurs on an even number of his remaining cards. If one player can do this but the other one cannot, the one who can is the winner; otherwise a draw is declared.
Determine all possible first moves of the first player after which he has a winning strategy.
[i]Proposed by Ilya Bogdanov & Vladimir Bragin, Russia[/i]
The numbers $1, 2, \dots , 50$ are written on a board. Letícia performs the following actions: she erases two numbers $a$ and $b$ on the board, writes the number $a+b$ on it and notes the number $ab(a+b)$ in her notebook. After performing these operations $49$ times, when there is only one number written on the board, Letícia calculates the sum $S$ of the $49$ numbers in the notebook.
a) Prove that $S$ doesn't depend on the order Letícia chooses the numbers to perform the operations.
b) Find the value of $S$.
Let $f : N \to N$ be a strictly increasing function such that $f(f(n))= 3n$, for all $n \in N$. Find $f(2010)$.
Note: $N = \{0,1,2,...\}$
There are $n{}$ wise men in a hall and everyone sees each other. Each man will wear a black or white hat. The wise men should simultaneously write down on their piece of paper a guess about the color of their hat. If at least one does not guess, they will all be executed.
The wise men can discuss a strategy before the test and they know that the layout of the hats will be chosen randomly from the set of all $2^n$ layouts. They want to choose their strategy so that the number of layouts for which everyone guesses correctly is as high as possible. What is this number equal to?
Kevin has $255$ cookies, each labeled with a unique nonempty subset of $\{1,2,3,4,5,6,7,8\}$. Each day, he chooses one cookie uniformly at random out of the cookies not yet eaten. Then, he eats that cookie, and all remaining cookies that are labeled with a subset of that cookie (for example, if he chooses the cookie labeled with $\{1,2\}$, he eats that cookie as well as the cookies with $\{1\}$ and $\{2\}$). The expected value of the number of days that Kevin eats a cookie before all cookies are gone can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
[i]Proposed by Ray Li[/i]
Find all $(x, y, z, n) \in {\mathbb{N}}^4$ such that $ x^3 +y^3 +z^3 =nx^2 y^2 z^2$.
Find the number of eight-digit positive integers that are multiples of $9$ and have all distinct digits.
A parallelogram $ABCD$ is given. Two circles with centers at the vertices $A$ and $C$ pass through $B$. The straight line $\ell$ that passes through $B$ and crosses the circles at second time at points $X, Y$ respectively. Prove that $DX = DY$.
Each of $1000$ elves has a hat, red on the inside and blue on the outside or vise versa. An elf with a hat that is red outside can only lie, and an elf with a hat that is blue outside can only tell the truth. One day every elf tells every other elf, “Your hat is red on the outside.” During that day, some of the elves turn their hats inside out at any time during the day. (An elf can do that more than once per day.) Find the smallest possible number of times any hat is turned inside out.
Let the inscribed circle $ \omega $ of triangle $ ABC $ touch the side $ BC $ at the point $ A '$. Let $ AA '$ intersect $ \omega $ at $ P \neq A $. Let $ CP $ and $ BP $ intersect $ \omega $, respectively, at points $ N $ and $ M $ other than $ P $. Prove that $ AA ', BN $ and $ CM $ intersect at one point.