Found problems: 85335
The area of a circle is doubled when its radius $ r$ is increased by $ n$. Then $ r$ equals:
$ \textbf{(A)}\ n(\sqrt{2} \plus{} 1)\qquad
\textbf{(B)}\ n(\sqrt{2} \minus{} 1)\qquad
\textbf{(C)}\ n\qquad
\textbf{(D)}\ n(2 \minus{} \sqrt{2})\qquad
\textbf{(E)}\ \frac{n\pi}{\sqrt{2} \plus{} 1}$
Find all postive integers $n$ for which the number $\frac{4n+1}{n(2n-1)}$ has a terminating decimal expansion.
Let $S$ be a nonempty set of positive integers such that, for any (not necessarily distinct) integers $a$ and $b$ in $S$, the number $ab+1$ is also in $S$. Show that the set of primes that do not divide any element of $S$ is finite.
[i]Proposed by Carl Schildkraut[/i]
A deck consists of $2^n$ cards. The deck is shuffled using the following operation: if the cards are initially in the order
$a_1,a_2,a_3,a_4,...,a_{2^n-1},a_{2^n}$ then after shuffling the order becomes $a_{2^{n-1}+1},a_1,a_{2^{n-1}+2},a_2,...,a_{2^n},a_{2^{n-1}}$ .
Find the smallest number of such operations after which the original order of the cards is restored.
(R. Palm)
Sam and Jonathan play a game where they take turns flipping a weighted coin, and the game ends when one of them wins. The coin has a $\frac89$ chance of landing heads and a $\frac19$ chance of landing tails. Sam wins when he flips heads, and Jonathan wins when he flips tails. Find the probability that Samwins, given that he takes the first turn.
[i]Proposed by Samuel Tsui[/i]
Let $ABCDEFGHIJ$ be a regular $10$-sided polygon that has all its vertices in one circle with center $O$ and radius $5$. The diagonals $AD$ and $BE$ intersect at $P$ and the diagonals $AH$ and $BI$ intersect at $Q$. Calculate the measure of the segment $PQ$.
Let $ ABC$, $ l$ and $ P$ be arbitrary triangle, line and point. $ A',B',C'$ are reflections of $ A,B,C$ in point $ P$. $ A''$ is a point on $ B'C'$ such that $ AA''\parallel l$. $ B'',C''$ are defined similarly. Prove that $ A'',B'',C''$ are collinear.
The order of $\log_{\sin1}\cos1,\log_{\sin1}\tan1,\log_{\cos1}\sin1,\log_{\cos1}\tan1$ is (form small to large)
$\text{(A)}\log_{\sin1}\cos1<\log_{\cos1}\sin1<\log_{\sin1}\tan1<\log_{\cos1}\tan1$
$\text{(B)}\log_{\cos1}\sin1<\log_{\cos1}\tan1<\log_{\sin1}\cos1<\log_{\sin1}\tan1$
$\text{(C)}\log_{\sin1}\tan1<\log_{\cos1}\tan1<\log_{\cos1}\sin1<\log_{\sin1}\cos1$
$\text{(D)}\log_{\cos1}\tan1<\log_{\sin1}\tan1<\log_{\sin1}\cos1<\log_{\cos1}\sin1$
Ten gamblers start playing with the same amount of money. In turn they cast five dice. If the dice show a total of $n$, the player must pay each other player $\frac{1}{n}$ times the sum which that player owns at the moment. They throw and pay one after the other. At the $10^{\text{th}}$ round (i.e. after each player has cast the five die once), the dice shows a total of $12$ and after the payment it turns out that every player has exactly the same sum as he had in the beginning. Is it possible to determine the totals shown by the dice at the nine former rounds?
Prove that the bisectors of the angles of a rectangle, extended to their mutual intersection, form a square.
On the Cartesian coordinate system $Oxy$, consider a sequence of points $A_n(x_n, y_n)$ in which $(x_n)^{\infty}_{n=1}$,$(y_n)^{\infty}_{n=1}$ are two sequences of positive numbers satisfing the following conditions:
$$x_{n+1} =\sqrt{\frac{x_n^2+x_{n+2}^2}{2}}, y_{n+1} =\big( \frac{\sqrt{y_n}+\sqrt{y_{n+2}}}{2} \big)^2 \,\, \forall n \ge 1 $$ Suppose that $O, A_1, A_{2016}$ belong to a line $d$ and $A_1, A_{2016}$ are distinct.
Prove that all the points $A_2, A_3,. .. , A_{2015}$ lie on one side of $d$.
A sequence $(a_n)$ is defined by $a_0=-1,a_1=0$, and $a_{n+1}=a_n^2-(n+1)^2a_{n-1}-1$ for all positive integers $n$. Find $a_{100}$.
An infinite sequence $(x_{n})$ is defined by its first term $x_{1}>1$, which is a rational number, and the relation $x_{n+1}=x_{n}+\frac{1}{\lfloor x_{n}\rfloor}$ for all positive integers $n$. Prove that this sequence contains an integer.
[i]A. Golovanov[/i]
Let $ABC$ be an equilateral triangle with side length $8.$ Let $X$ be on side $AB$ so that $AX=5$ and $Y$ be on side $AC$ so that $AY=3.$ Let $Z$ be on side $BC$ so that $AZ,BY,CX$ are concurrent. Let $ZX,ZY$ intersect the circumcircle of $AXY$ again at $P,Q$ respectively. Let $XQ$ and $YP$ intersect at $K.$ Compute $KX\cdot KQ.$
How many ordered pairs of integers $(x, y)$ satisfy $2 |y| \le x \le 40\,$?
The bisector of $\angle BAD$ of a parallelogram $ABCD$ meets $BC$ at $K$. The point $L$ lies on $AB$ such that $AL=CK$. The lines $AK$ and $CL$ meet at $M$. Let $(ALM)$ meet $AD$ after $D$ at $N$. Prove that $\angle CNL=90^{o}$
Suppose $ABC$ is a triangle with inradius $r$. The incircle touches the sides $BC, CA,$ and $AB$ at $D,E$ and $F$ respectively. If $BD=x, CE=y$ and $AF=z$, then show that
\[r^2=\frac{xyz}{x+y+z}\]
Does there exist a natural number $n$ in whose decimal representation each digit occurs at least $2006$ times and which has the property that you can find two different digits in its decimal representation such that the number obtained from $n$ by interchanging these two digits is different from $n$ and has the same set of prime divisors as $n$ ?
In a certain language words are formed using an alphabet of three letters. Some words of two or more letters are not allowed, and any two such distinct words are of different lengths. Prove that one can form a word of arbitrary length that does not contain any non-allowed word.
Let $ABC$ be an equilateral triangle with unit side length and circumcircle $\Gamma$. Let $D_1, D_2$ be the points on $\Gamma$ such that $BD_i = 3CD_i$. Let $E_1, E_2$ be the points on $\Gamma$ such that $CE_i = 3AE_i$. Let $F_1, F_2$ be the points on $\Gamma$ such that $AF_i = 3BF_i$. Then points $D_1, D_2, E_1, E_2, F_1, F_2$ are the vertices of a convex hexagon. What is the area of this hexagon?
Let $r$ and $k$ be positive integers such that all prime divisors of $r$ are greater than $50$.
A positive integer, whose decimal representation (without leading zeroes) has at least $k$ digits, will be called [i]nice[/i] if every sequence of $k$ consecutive digits of this decimal representation forms a number (possibly with leading zeroes) which is a multiple of $r$.
Prove that if there exist infinitely many nice numbers, then the number $10^k-1$ is nice as well.
For each $n \ge 2$ define the polynomial $$f_n(x)=x^n-x^{n-1}-\dots-x-1.$$ Prove that
(a) For each $n \ge 2$, $f_n(x)=0$ has a unique positive real root $\alpha_n$;
(b) $(\alpha_n)_n$ is a strictly increasing sequence;
(c) $\lim_{n \rightarrow \infty} \alpha_n=2.$
Let $BD$ and $CE$ be the bisectors of the interior angles $\angle B$ and $\angle C$, respectively ($D\in AC$, $E\in AB$). Consider the circumcircle of $ABC$ with center $O$ and the excircle corresponding to the side $BC$ with center $I_a$. These two circles intersect at points $P$ and $Q$.
(a) Prove that $PQ$ is parallel to $DE$.
(b) Prove that $I_aO$ is perpendicular to $DE$.
SEEMOUS 2016 COMPETITION PROBLEMS
Points $E$ and $F$ lie on the sides $AB$ and $AC$ of a triangle $ABC$. Lines $EF$ and $BC$ meet at point $S$. Let $M$ and $N$ be the midpoints of $BC$ and $EF$, respectively. The line passing through $A$ and parallel to $MN$ meets $BC$ at point $K$. Prove that $\frac{BK}{CK}=\frac{FS}{ES}$ .
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