Found problems: 85335
The tourist on an island can play the "getting treasure" game. He has to open a series of doors, each door is colored with one of n colors, according to the following rules:
[i](i)[/i] The tourist has n keys, each key with a different color.
[i](ii)[/i] Once a key is used, it is not permitted to change until it is destroyed.
[i](iii)[/i] Each key can open any door, and keeps intact when it opens the door having different color with it, but is destroyed when it opens the door having the same color with it.
Find the least number of doors to ensure that no tourist, no matter how he choose the order of the keys to use, can get the treasure.
Let $x, y, z$ be positive real numbers such that $xyz(x+y+z) = 1$.
Show that the following equality holds: $$\sqrt{\left( x^2+\frac{1}{y^2}\right)\left( y^2+\frac{1}{z^2}\right)\left( z^2+\frac{1}{x^2}\right)}=(x+y)(y+z)(z+x)$$
Find some numbers $x ,y ,z$ which satisfy the given property.
Consider the following operation. Given a positive integer $n$, if $n$ is a multiple of $3$, then you replace $n$ by $\dfrac{n}3$. If $n$ is not a multiple of $3$, then you replace $n$ by $n + 10$. Then continue this process. For example, beginning with $n = 4$, this procedure gives $4 \to 14 \to 24 \to 8 \to 18 \to 6 \to 2 \to 12 \to \cdots$. Suppose you start with $n = 100$. What value results if you perform this operation exactly $100$ times?
$\textbf{(A) }10\qquad\textbf{(B) }20\qquad\textbf{(C) }30\qquad\textbf{(D) }40\qquad\textbf{(E) }50$
On a plane is given a finite set of points. Prove that the points can be covered by open squares $Q_1,Q_2,...,Q_n$ such that $1 \le\frac{N_j}{S_j} \le 4$ for $j = 1,...,n,$ where $N_j$ is the number of points from the set inside square $Q_j$ and $S_j$ is the area of $Q_j$.
Let $\mathbb{N}=\{1,2,\ldots\}$. Find all functions $f: \mathbb{N}\to\mathbb{N}$ such that for all $m,n\in \mathbb{N}$ the number $f^2(m)+f(n)$ is a divisor of $(m^2+n)^2$.
Let $ I $ be the incenter of a triangle $ ABC $ with $ AB \neq AC $, and let $ M $ be the midpoint of the arc $ BAC $ of the circumcircle of the triangle. The perpendicular line to $ AI $ passing through $ I $ intersects line $ BC $ at point $ D $. The line $ MI $ intersects the circumcircle of triangle $ BIC $ at point $ N $. Prove that line $ DN $ is tangent to the circumcircle of triangle $ BIC $.
A rectangle $ABCD$ is given. On the side $AB$, n different points are chosen strictly between $A$ and $B$. Similarly, $m$ different points are chosen on the side $AD$ between $A$ and $D$. Lines are drawn from the points parallel to the sides. How many rectangles are formed in this way?
An example of a particular rectangle $ABCD$ is shown with a shaded one rectangle that may be formed in this way.
[img]https://cdn.artofproblemsolving.com/attachments/e/4/f7a04300f0c846fb6418d12dc23f5c74b54242.png[/img]
Find all natural numbers $n$ for which each natural number written with $~$ $n-1$ $~$ 'ones' and one 'seven' is prime.
Find all pairs $(x, y)$ of real numbers such that
\[y^2 - [x]^2 = 19.99 \text{ and } x^2 + [y]^2 = 1999\]
where $f(x)=[x]$ is the floor function.
Determine $ Q \in \mathbb{R}$ which is so big that a sequence with non-negative reals elements $ a_1 ,a_2, \ldots$ which satisfies the following two conditions:
[b](i)[/b] $ \forall m,n \geq 1$ we have $ a_{m \plus{} n} \leq 2 \left(a_m \plus{} a_n \right)$
[b](ii)[/b] $ \forall k \geq 0$ we have $ a_{2^k} \leq \frac {1}{(k \plus{} 1)^{2008}}$
such that for each sequence element we have the inequality $ a_n \leq Q.$
Compute the number of real numbers $t$ such that \[t = 50 \sin(t - \lfloor t \rfloor).\] Here $\lfloor \cdot\rfloor$ denotes the greatest integer function.
[i]Proposed by David Altizio[/i]
Let $ABC$ be a triangle with $\angle A = 90^o$ and $\angle B < \angle C$. The tangent at $A$ to the circumcircle $k$ of $\vartriangle ABC$ intersects line $BC$ at $D$. Let $E$ be the reflection of $A$ in $BC$. Also, let $X$ be the feet of the perpendicular from $A$ to $BE$ and let $Y$ be the midpoint of $AX$. Line $BY$ meets $k$ again at $Z$. Prove that line $BD$ is tangent to the circumcircle of $\vartriangle ADZ$.
In the convex quadrilateral $ABCD$, $AD = BD$ and $\angle ACD = 3 \angle BAC$. Let $M$ be the midpoint of side $AD$. If the lines $CM$ and $AB$ are parallel, prove that the angle $\angle ACB$ is right.
Given the sequence $ 10^{\frac {1}{11}},10^{\frac {2}{11}},10^{\frac {3}{11}},\ldots,10^{\frac {n}{11}}$, the smallest value of $ n$ such that the product of the first $ n$ members of this sequence exceeds $ 100000$ is:
$ \textbf{(A)}\ 7 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 11$
Given $8$ coins, at most one of them is counterfeit. A counterfeit coin is lighter than a real coin. You have a free weight balance. What is the minimum number of weighings necessary to determine the identity of the counterfeit coin if it exists
Prove that the following inequality holds with the exception of finitely many positive integers $n$:
$\sum^{n}_{i=1}\sum^{n}_{j=1}gcd(i,j)>4n^2$.
Let the incircle of the triangle $ABC$ touch $[BC]$ at $D$ and $I$ be the incenter of the triangle. Let $T$ be midpoint of $[ID]$. Let the perpendicular from $I$ to $AD$ meet $AB$ and $AC$ at $K$ and $L$, respectively. Let the perpendicular from $T$ to $AD$ meet $AB$ and $AC$ at $M$ and $N$, respectively. Show that $|KM|\cdot |LN|=|BM|\cdot|CN|$.
Xenia and Yagve take turns in playing the following game: A coin is placed on the first box in a row of nine cells. At each turn the player may choose to move the coin forward one step, move the coin forward four steps, or move coin back two steps. For a move to be allowed, the coin must land on one of them of nine cells. The winner is one who gets to move the coin to the last ninth cell. Who wins, given that Xenia makes the first move, and both players play optimally?
Adithya and Bill are playing a game on a connected graph with $n > 2$ vertices, two of which are labeled $A$ and $B$, so that $A$ and $B$ are distinct and non-adjacent and known to both players. Adithya starts on vertex $A$ and Bill starts on $B$. Each turn, both players move simultaneously: Bill moves to an adjacent vertex, while Adithya may either move to an adjacent vertex or stay at his current vertex. Adithya loses if he is on the same vertex as Bill, and wins if he reaches $B$ alone. Adithya cannot see where Bill is, but Bill can see where Adithya is. Given that Adithya has a winning strategy, what is the maximum possible number of edges the graph may have? (Your answer may be in terms of $n$.)
[i]Proposed by Steven Liu[/i]
Consider equilateral triangle $ABC$ and $M,N\in (BC)$, $P,Q\in (CA)$, $R,S\in (AB)$ such that $MN=PQ=RS$ and $M\in (BN)$, $P\in(CQ)$, $R\in(AS)$. Prove that there exist three noncollinear points inside hexagon $MNPQRS$ with the same sum of distances to the sides of the hexagon if and only if triangles $ARQ$, $BMS$ and $CPN$ are congruent.
[i]Vasile Pop[/i]
Let $ABCD$ be a rectangle with area $2012$. There exist points $E$ on $AB$ and $F$ on $CD$ such that $DE = EF = F B$. Diagonal $AC$ intersects $DE$ at $X$ and $EF$ at $Y$ . Compute the area of triangle $EXY$ .
Let $ABCD$ a trapezium with $AB$ parallel to $CD, \Omega$ the circumcircle of $ABCD$ and $A_1,B_1$ points on segments $AC$ and $BC$ respectively, such that $DA_1B_1C$ is a cyclic cuadrilateral. Let $A_2$ and $B_2$ the symmetric points of $A_1$ and $B_1$ with respect of the midpoint of $AC$ and $BC$, respectively. Prove that points $A, B, A_2, B_2$ are concyclic.
Last year Mr. John Q. Public received an inheritance. He paid $ 20\%$ in federal taxes on the inheritance, and paid $ 10\%$ of what he had left in state taxes. He paid a total of $ \$10,500$ for both taxes. How many dollars was the inheritance?
$ \textbf{(A)}\ 30,000 \qquad \textbf{(B)}\ 32,500 \qquad \textbf{(C)}\ 35,000 \qquad \textbf{(D)}\ 37,500 \qquad \textbf{(E)}\ 40,000$
If $a$ and $b$ are the roots of the equation $x^2-2cx-5d = 0$, $c$ and $d$ are the roots of the equation $x^2-2ax-5b=0$, where $a,b,c,d$ are distinct real numbers, what is $a+b+c+d$?
$
\textbf{(A)}\ 10
\qquad\textbf{(B)}\ 15
\qquad\textbf{(C)}\ 20
\qquad\textbf{(D)}\ 25
\qquad\textbf{(E)}\ 30
$
Given a triangle $ABC$ satisfying $AC+BC=3\cdot AB$. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Let $K$ and $L$ be the reflections of the points $D$ and $E$ with respect to $I$. Prove that the points $A$, $B$, $K$, $L$ lie on one circle.
[i]Proposed by Dimitris Kontogiannis, Greece[/i]