Found problems: 85335
The phrase "COLORFUL TARTAN'' is spelled out with wooden blocks, where blocks of the same letter are indistinguishable. How many ways are there to distribute the blocks among two bags of different color such that neither bag contains more than one of the same letter?
Let $a$, $b$, $c$ and $d$ be real numbers with $a^2 + b^2 + c^2 + d^2 = 4$. Prove that the inequality
$$(a+2)(b+2) \ge cd$$
holds and give four numbers $a$, $b$, $c$ and $d$ such that equality holds.
(Walther Janous)
There are $2009$ cities in country, and every two are connected by road. Businessman and Road Ministry play next game. Every morning Businessman buys one road and every evening Minisrty destroys 10 free roads. Can Business create cyclic route without self-intersections through exactly $75$ different cities?
In a certain language there are only two letters, $A$ and $B$. We know that
(i) There are no words of length $1$, and the only words of length $2$ are $AB$ and $BB$.
(ii) A segment of length $n > 2$ is a word if and only if it can be obtained from a word of length less than $n$ by replacing each letter $B$ by some (not necessarily the same) word.
Prove that the number of words of length $n$ is equal to $\frac{2^n +2\cdot (-1)^n}{3}$
All the cells of a $10\times10$ board are colored white initially. Two players are playing a game with alternating moves. A move consists of coloring any un-colored cell black. A player is considered to loose, if after his move no white domino is left. Which of the players has a winning strategy?
[I]Proposed by A. Khrabrov[/i]
Let $P$ be an interior point of triangle $ABC$ and extend lines from the vertices through $P$ to the opposite sides. Let $a$, $b$, $c$, and $d$ denote the lengths of the segments indicated in the figure. Find the product $abc$ if $a + b + c = 43$ and $d = 3$.
[asy]
size(200);
defaultpen(fontsize(10));
pair A=origin, B=(14,0), C=(9,12), D=midpoint(B--C), E=midpoint(A--C), F=midpoint(A--B), P=centroid(A,B,C);
draw(D--A--B--C--A^^B--E^^C--F);
dot(A^^B^^C^^P);
label("$a$", P--A, dir(-90)*dir(P--A));
label("$b$", P--B, dir(90)*dir(P--B));
label("$c$", P--C, dir(90)*dir(P--C));
label("$d$", P--D, dir(90)*dir(P--D));
label("$d$", P--E, dir(-90)*dir(P--E));
label("$d$", P--F, dir(-90)*dir(P--F));
label("$A$", A, SW);
label("$B$", B, SE);
label("$C$", C, N);
label("$P$", P, 1.8*dir(285));[/asy]
How many ordered pairs of integers $(x,y)$ are there such that $2011y^2=2010x+3$?
$ \textbf{(A)}\ 3
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 1
\qquad\textbf{(D)}\ 0
\qquad\textbf{(E)}\ \text{Infinitely many}
$
Let $n > 1$ and $p(x)=x^n+a_{n-1}x^{n-1} +...+a_0$ be a polynomial with $n$ real roots (counted
with multiplicity). Let the polynomial $q$ be defined by
$$q(x) = \prod_{j=1}^{2015} p(x + j)$$.
We know that $p(2015) = 2015$. Prove that $q$ has at least $1970$ different roots $r_1, ..., r_{1970}$
such that $|r_j| < 2015$ for all $ j = 1, ..., 1970$.
Let $a, b, c$ be positive real numbers. Prove that
$$\frac {3(ab + bc + ca)}{2(a^2b^2+b^2c^2+c^2a^2)}\leq \frac1{a^2 + bc} + \frac1{b^2 + ca} + \frac1{c^2 + ab}\leq\frac{a+b+c}{2abc}.$$
Let $ n \geq 3$ and consider a set $ E$ of $ 2n \minus{} 1$ distinct points on a circle. Suppose that exactly $ k$ of these points are to be colored black. Such a coloring is [b]good[/b] if there is at least one pair of black points such that the interior of one of the arcs between them contains exactly $ n$ points from $ E$. Find the smallest value of $ k$ so that every such coloring of $ k$ points of $ E$ is good.
A single elimination tournament is held with $2016$ participants. In each round, players pair up to play games with each other. There are no ties, and if there are an odd number of players remaining before a round then one person will get a bye for the round. Find the minimum number of rounds needed to determine a winner.
[i]Proposed by Nathan Ramesh
Let $ABCD$ be a cyclic quadrilateral with $\angle BAD < \angle ADC$. Let $M$ be the midpoint of the arc $CD$ not containing $A$. Suppose there is a point $P$ inside $ABCD$ such that $\angle ADB = \angle CPD$ and $\angle ADP = \angle PCB$.
Prove that lines $AD, PM$, and $BC$ are concurrent.
Welcome to the [b]USAYNO[/b], a twelve-part question where each part has a yes/no answer. If you provide $C$ correct answers, your score on this problem will be $\frac{C}{6}$.
Your answer should be a twelve-character string containing `Y' (for yes) and `N' (for no). For instance if you think a, c, and f are `yes' and the rest are `no', you should answer YNYNNYNNNNNN.
(a) Is there a positive integer $n$ such that the sum of the digits of $2018n+1337$ in base $10$ is $2018$ more than the sum of the digits of $2018n+1337$ in base $4$?
(b) Is there a fixed constant $\theta$ such that for all triangles $ABC$ with $$2018AB^2=2018CA^2+2017CA\cdot CB+2018CB^2,$$ one of the angles of $ABC$ is $\theta$?
(c) Adam lists out every possible way to arrange the letters of ``CCACCACCA'' (including the given arrangement) at $1$ arrangement every $5$ seconds. Madam lists out every possible way to arrange the letters of ``CCACCAA'' (including the given arrangement) at $1$ arrangement every $12$ seconds. Does Adam finish first?
(d) Do there exist real numbers $a,b,c$, none of which is the average of the other two, such that \[\frac{1}{b+c-2a}+\frac{1}{c+a-2b}+\frac{1}{a+b-2c}=0?\]
(e) Let $f\left(x\right)=\frac{2^x-2}{x}-1$. Is there an integer $n$ such that $$f\left(n\right),f\left(f\left(n\right)\right),f\left(f\left(f\left(n\right)\right)\right),\ldots$$ are all integers?
(f) In an elementary school with $2585$ students and $159$ classes (every student is in exactly one class), each student reports the size of their class. The principal of the school takes the average of all of these numbers and calls it $X$. The principal then computes the average size of each class and calls it $Y$. Is it necessarily true that $X>Y$?
(g) Six sticks of lengths $3$, $5$, $7$, $11$, $13$, and $17$ are put together to form a hexagon. From a point inside the hexagon, a circular water balloon begins to expand and will stop expanding once it hits any stick. Is it possible that once the balloon stops expanding, it is touching each of the six sticks?
(h) A coin is biased so that it flips heads and tails (and only heads or tails) each with a positive rational probability (not necessarily $\frac{1}{2}$). Is it possible that on average, it takes exactly twice as long to flip two heads in a row as it is to flip two tails in a row?
(i) Does there exist a base $b$ such that $2018_b$ is prime?
(j) Does there exist a sequence of $2018$ distinct real numbers such that no $45$ terms (not necessarily consecutive) can be examined, in order, and be in strictly increasing or strictly decreasing order?
(k) Does there exist a scalene triangle $ABC$ such that there exist two distinct rectangles $PQRS$ inscribed in $\triangle{ABC}$ with $P\in AB$, $Q,R\in BC$, $S\in AC$ such that the angle bisectors of $\angle{PAS}$, $\angle{PQR}$, and $\angle{SRQ}$ concur?
(l) For three vectors $\mathbf{u}_1,\mathbf{u}_2,\mathbf{u}_3$ with $\mathbf{u}_i=\left(x_{i,1},x_{i,2},x_{i,3},x_{i,4}\right)$, define \[f\left(\mathbf{u}_1,\mathbf{u}_2,\mathbf{u}_3\right)=1-\displaystyle\prod_{j=1}^4\left(1+\left(x_{2,j}-x_{3,j}\right)^2+\left(x_{3,j}-x_{1,j}\right)^2+\left(x_{1,j}-x_{2,j}\right)^2\right).\] Are there any sequences $\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_{18}$ of distinct vectors with four components, with all components in $\left\{1,2,3\right\}$, such that \[\displaystyle\prod_{1\leq i<j<k\leq18}f\left(\mathbf{v}_i,\mathbf{v}_j,\mathbf{v}_k\right)\equiv1\pmod3?\]
[i]2018 CCA Math Bonanza Lightning Round #5.4[/i]
We are given the following sequence: $a_1=8,a_2=20,a_{n+2}=a_{n+1}^2+12a_n a_{n+1}+11a_n$. Prove that none of the members of the sequence can be presented as a sum of three seventh powers of natural numbers.
Find all polynomials $p(x)$ with integer coefficients such that for each positive integer $n$, the number $2^n - 1$ is divisible by $p(n)$.
a) A bus network is organized so that:
1) one can reach any stop from any other stop without changing buses;
2) every pair of routes has a single stop at which one can change buses;
3) each route has exactly three stops?
How many bus routes are there? It is assumed that there are at least two routes.
b) A town has $57$ bus routes. How many stops does each route have if it is known that
1) one can reach any stop from any other stop without changing buses;
2) for every pair of routes there is a single stop where one can change buses;
3) each route has three or more stops?
Together, Abe and Bob have less than or equal to \$ $100$. When Corey asks them how much money they have, Abe says that the reciprocal of his money added to Bob’s money is thirteen times as much as the sum of Abe’s money and the reciprocal of Bob’s money. If Abe and Bob both have integer amounts of money, how many possible values are there for Abe’s money?
Let $P(x)=x^3+ax^2+b$ and $Q(x)=x^3+bx+a$, where $a$ and $b$ are nonzero real numbers. Suppose that the roots of the equation $P(x)=0$ are the reciprocals of the roots of the equation $Q(x)=0$. Prove that $a$ and $b$ are integers. Find the greatest common divisor of $P(2013!+1)$ and $Q(2013!+1)$.
Kostya and Sergey play a game on a white strip of length 2016 cells. Kostya (he plays first) in one move should paint black over two neighboring white cells. Sergey should paint either one white cell either three neighboring white cells. It is forbidden to make a move, after which a white cell is formed the doesn't having any white neighbors. Loses the one that can make no other move. However, if all cells are painted, then Kostya wins. Who will win if he plays the right game (has a winning strategy)?
On each vertex of a regular $ n\minus{}$gon there was a crow. Call this as initial configuration. At a signal, they all flew by and after a while, those $ n$ crows came back to the $ n\minus{}$gon, one crow for each vertex. Call this as final configuration. Determine all $ n$ such that: there are always three crows such that the triangle they formed in the initial configuration and the triangle they formed in the final configuration are both right-angled triangle.
Find the number of pairs of sets $(A, B)$ satisfying $A \subseteq B \subseteq \{1, 2, ...,10\}$
Circles $\Omega_1$ and $\Omega_2$ with different radii intersect at two points, denote one of them by $P$. A variable line $l$ passing through $P$ intersects the arc of $\Omega_1$ which is outside of $\Omega_2$ at $X_1$, and the arc of $\Omega_2$ which is outside of $\Omega_1$ at $X_2$. Let $R$ be the point on segment $X_1X_2$ such that $X_1P = RX_2$. The tangent to $\Omega_1$ through $X_1$ meets the tangent to $\Omega_2$ through $X_2$ at $T$. Prove that line $RT$/is tangent to a fixed circle, independent of the choice of $l$.
Let $ \Omega$ is circle with radius $ R$ and center $ O$. Let $ \omega$ is a circle inside of the $ \Omega$ with center $ I$ radius $ r$. $ X$ is variable point of $ \omega$ and tangent line of $ \omega$ pass through $ X$ intersect the circle $ \Omega$ at points $ A,B$. A line pass through $ X$ perpendicular with $ AI$ intersect $ \omega$ at $ Y$ distinct with $ X$.Let point $ C$ is symmetric to the point $ I$ with respect to the line $ XY$.Find the locus of circumcenter of triangle $ ABC$ when $ X$ varies on $ \omega$
For which integers $k$ does there exist a function $f : N \to Z$ such that
$f(1995) =1996$ and $f(xy) = f(x)+ f(y)+k f(gcd(x,y))$ for all $x,y \in N$?
Let $ABC$ be an acute angled triangle with incenter $I$. Line perpendicular to $BI$ at $I$ meets $BA$ and $BC$ at points $P$ and $Q$ respectively. Let $D, E$ be the incenters of $\triangle BIA$ and $\triangle BIC$ respectively. Suppose $D,P,Q,E$ lie on a circle. Prove that $AB=BC$.