Found problems: 85335
Let $\gamma_1, \gamma_2, \gamma_3$ be three circles with radii $3, 4, 9,$ respectively, such that $\gamma_1$ and $\gamma_2$ are externally tangent at $C,$ and $\gamma_3$ is internally tangent to $\gamma_1$ and $\gamma_2$ at $A$ and $B,$ respectively. Suppose the tangents to $\gamma_3$ at $A$ and $B$ intersect at $X.$ The line through $X$ and $C$ intersect $\gamma_3$ at two points, $P$ and $Q.$ Compute the length of $PQ.$
[i]Proposed by Kyle Lee[/i]
Ben creates an $8\times8$ grid of coins, where each coin faces heads with probability $\tfrac{1}{2}$, and tails with probability $\tfrac{1}{2}$. Ben then makes a series of moves; each move consists of selecting a coin in the grid and flipping over all coins in the same row and column as the selected coin. Suppose that in Ben’s current grid of coins, it is possible to make a series of moves so that all coins in the grid are heads, and that Ben will make the fewest number of moves to do so. What is the expected number of moves that Ben makes?
Eve randomly chooses two $\textbf{distinct}$ points on the coordinate plane from the set of all $11^2$ lattice points $(x, y)$ with $0 \le x \le 10$, $0 \le y \le 10$. Then, Anne the ant walks from the point $(0,0)$ to the point $(10, 10)$ using a sequence of one-unit right steps and one-unit up steps. Let $P$ be the number of paths Anne could take that pass through both of the points that Eve chose. The expected value of $P$ is $\dbinom{20}{10} \cdot \dfrac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100a+b$.
[i]Proposed by Michael Tang[/i]
In the rectangular parallelepiped $ABCDA'B'C'D'$ we denote by $M$ the center of the face $ABB'A'$. We denote by $M_1$ and $M_2$ the projections of $M$ on the lines $B'C$ and $AD'$ respectively. Prove that:
a) $MM_1 = MM_2$
b) if $(MM_1M_2) \cap (ABC) = d$, then $d \parallel AD$;
c) $\angle (MM_1M_2), (A B C)= 45^ o \Leftrightarrow \frac{BC}{AB}=\frac{BB'}{BC}+\frac{BC}{BB'}$.
Before district play, the Unicorns had won $45\%$ of their basketball games. During district play, they won six more games and lost two, to finish the season having won half their games. How many games did the Unicorns play in all?
$\textbf{(A)}\ 48 \qquad
\textbf{(B)}\ 50 \qquad
\textbf{(C)}\ 52 \qquad
\textbf{(D)}\ 54 \qquad
\textbf{(E)}\ 60$
Eric has $2$ boxes of apples, with the first box containing red and yellow apples and the second box containing green apples. Eric observes that the red apples make up $\tfrac{1}{2}$ of the apples in the first box. He then moves all of the red apples to the second box, and observes that the red apples now make up $\tfrac{1}{3}$ of the apples in the second box. Suppose that Eric has $28$ apples in total. How many red apples does Eric have?
We are given tiles in the form of right angled triangles having perpendicular sides of length $1$ cm and $2$ cm. Is it possible to form a square from $20$ such tiles?
( S . Fomin , Leningrad)
At first all $2^n$ rows of a $2^n \times n$ table were filled with all different $n$-tuples of numbers $+1$ and $-1$. Then some of the numbers were replaced by Os. Prove that one can choose a (non-empty) set of rows such that:
(a) the sum of all the numbers in all the chosen rows is $0$;
(b) the sum of all the chosen rows equals the zero row, that is, the sum of numbers in each column of the chosen rows equals $0$.
(G Kondakov, V Chernorutskii)
Find all integers $n$, $n \ge 1$, such that $n \cdot 2^{n+1}+1$ is a perfect square.
Given positive reals $a,b,c;$ show that we have
\[\left(a+\frac 1b\right)\left(b+\frac 1c\right)\left(c+\frac 1a\right)\geq 8.\]
Let $n$ be a positive integer and let $a_1, \ldots, a_{n-1} $ be arbitrary real numbers. Define the sequences $u_0, \ldots, u_n $ and $v_0, \ldots, v_n $ inductively by $u_0 = u_1 = v_0 = v_1 = 1$, and $u_{k+1} = u_k + a_k u_{k-1}$, $v_{k+1} = v_k + a_{n-k} v_{k-1}$ for $k=1, \ldots, n-1.$
Prove that $u_n = v_n.$
In $\vartriangle ABC$, the external bisectors of $\angle A$ and $\angle B$ meet at a point $D$. Prove that the circumcentre of $\vartriangle ABD$ and the points $C, D$ lie on the same straight line.
Determine the smallest positive prime $p$ which satisfies the congruence \[p+p^{-1}\equiv 25\pmod{143}.\] Here, $p^{-1}$ as usual denotes multiplicative inverse.
Find all pairs of integers $(x,y)$ such that
\[ y^3=8x^6+2x^3y-y^2.\]
Ankit wants to create a pseudo-random number generator using modular arithmetic. To do so he starts with a seed $x_0$ and a function $f(x) = 2x + 25$ (mod $31$). To compute the $k$-th pseudo random number, he calls $g(k)$ de ned as follows:
$$g(k) = \begin{cases} x_0 \,\,\, \text{if} \,\,\, k = 0 \\
f(g(k- 1)) \,\,\, \text{if} \,\,\, k > 0 \end{cases}$$
If $x_0$ is $2017$, compute $\sum^{2017}_{j=0} g(j)$ (mod $31$).
Given an integer $ k > 1.$ We call a $ k \minus{}$digits decimal integer $ a_{1}a_{2}\cdots a_{k}$ is $ p \minus{}$monotonic, if for each of integers $ i$ satisfying $ 1\le i\le k \minus{} 1,$ when $ a_{i}$ is an odd number, $ a_{i} > a_{i \plus{} 1};$ when $ a_{i}$ is an even number, $ a_{i}<a_{i \plus{} 1}.$ Find the number of $ p \minus{}$monotonic $ k \minus{}$digits integers.
Let $S$ be the subset of $N$($N$ is the set of all natural numbers) satisfying:
i)Among each $2003$ consecutive natural numbers there exist at least one contained in $S$;
ii)If $n \in S$ and $n>1$ then $[\frac{n}{2}] \in S$
Prove that:$S=N$
I hope it hasn't posted before. :lol: :lol:
Let $n$ be an even positive integer, and let $G$ be an $n$-vertex graph with exactly $\tfrac{n^2}{4}$ edges, where there are no loops or multiple edges (each unordered pair of distinct vertices is joined by either 0 or 1 edge). An unordered pair of distinct vertices $\{x,y\}$ is said to be [i]amicable[/i] if they have a common neighbor (there is a vertex $z$ such that $xz$ and $yz$ are both edges). Prove that $G$ has at least $2\textstyle\binom{n/2}{2}$ pairs of vertices which are amicable.
[i]Zoltán Füredi (suggested by Po-Shen Loh)[/i]
In the triangle $ABC$, the inscribed circle touches the sides $CA{}$ and $AB{}$ at the points $B_1{}$ and $C_1{}$, respectively. An arbitrary point $D{}$ is selected on the side $AB{}$. The point $L{}$ is the center of the inscribed circle of the triangle $BCD$. The bisector of the angle $ACD$ intersects the line $B_1C_1$ at the point $M{}$. Prove that $\angle CML=90^\circ$.
[i]Proposed by Chan Quang Heung (Vietnam)[/i]
Prove that the intersection of a plane and a regular tetrahedron can be an obtuse-angled triangle and that the obtuse angle in any such triangle is always smaller than $ 120^{\circ}.$
A square matrix is sum-magic if the sum of all elements in each row, column and major diagonal is constant. Similarly, a square matrix is product-magic if the product of all elements in each row, column and major diagonal is constant.
Determine if there exist $3\times 3$ matrices of real numbers which are both sum-magic and product-magic.
Compute $\sin^4\left(7.5^\circ\right)+\sin^4\left(82.5^\circ\right)$.
[i]2019 CCA Math Bonanza Lightning Round #2.3[/i]
Prove that the arithmetic sequence $5, 11, 17, 23, 29, \ldots$ contains infinitely many primes.
There are three flies of negligible size that start at the same position on a circular track with circumference 1000 meters. They fly clockwise at speeds of 2, 6, and $k$ meters per second, respectively, where $k$ is some positive integer with $7\le k \le 2013$. Suppose that at some point in time, all three flies meet at a location different from their starting point. How many possible values of $k$ are there?
[i]Ray Li[/i]
Let $m$ be a positive integer. Define the sequence $a_0, a_1, a_2, \cdots$ by $a_0 = 0,\; a_1 = m,$ and $a_{n+1} = m^2a_n - a_{n-1}$ for $n = 1,2,3,\cdots$.
Prove that an ordered pair $(a,b)$ of non-negative integers, with $a \leq b$, gives a solution to the equation
\[ {\displaystyle \frac{a^2 + b^2}{ab + 1} = m^2} \]
if and only if $(a,b)$ is of the form $(a_n,a_{n+1})$ for some $n \geq 0$.