Found problems: 85335
$f(0)=0, f(1) = 1$, and for every $n \geq 1$, $f(3n-1)= f(n)-1, f(3n)= f(n), f(3n+1)=f(n)+1$. So $f(2011)$ is
$\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ 0$
The extensions of two opposite sides of the convex quadrilateral intersect and form an angle of $20^o$ , the extensions of the other two sides also intersect and form an angle of $20^o$. It is known that exactly one angle of the quadrilateral is $80^o$. Find all of its other angles.
Convex quadrilateral $ABCD$ is inscribed in circle $w.$Rays $AB$ and $DC$ intersect at $K.\ L$ is chosen on the diagonal $BD$ so that $\angle BAC= \angle DAL.\ M$ is chosen on the segment $KL$ so that $CM \mid\mid BD.$ Prove that line $BM$ touches $w.$
[i](Kungozhin M.)[/i]
Find
$$ \lim_{t\to \infty} e^{-t} \int_{0}^{t} \int_{0}^{t} \frac{e^x -e^y }{x-y} \,dx\,dy,$$
or show that the limit does not exist.
Let $ABC$ be a triangle with incenter $I$. Suppose the reflection of $AB$ across $CI$ and the reflection of $AC$ across $BI$ intersect at a point $X$. Prove that $XI$ is perpendicular to $BC$.
The cells of some rectangular $M \times n$ array are colored, each by one of two colors, so that for any two columns the number of pairs of cells situated on a same row and bearing the same color is less than the number of pairs of cells situated on a same row and bearing different colors.
i) Prove that if $M=2011$ then $n \leq 2012$ (a model for the extremal case $n=2012$ does indeed exist, but you are not asked to exhibit one).
ii) Prove that if $M=2011=n$, each of the colors appears at most $1006\cdot 2011$ times, and at least $1005\cdot 2011$ times.
iii) Prove that if however $M=2012$ then $n \leq 1007$.
([i]Dan Schwarz[/i])
For which maximal $N$ there exists an $N$-digit number with the following property: among any sequence of its consecutive decimal digits some digit is present once only?
Alexey Glebov
A circle is circumscribed around an isosceles triangle whose two base angles are equal to $x^{\circ}$. Two points are chosen independently and randomly on the circle, and a chord is drawn between them. The probability that the chord intersects the triangle is $\frac{14}{25}.$ Find the sum of the largest and smallest possible value of $x$.
Six circles form a ring with with each circle externally tangent to two circles adjacent to it. All circles are internally tangent to a circle $C$ with radius $30$. Let $K$ be the area of the region inside circle $C$ and outside of the six circles in the ring. Find $\lfloor K \rfloor$.
How many nonzero coefficients can a polynomial $ P(x)$ have if its coefficients are integers and $ |P(z)| \le 2$ for any complex number $ z$ of unit length?
The game of circulate is played with a deck of $kn$ cards each with a number in $1,2,\ldots,n$ such that there are $k$ cards with each number. First, $n$ piles numbered $1,2,\ldots,n$ of $k$ cards each are dealt out face down. The player then flips over a card from pile $1$, places that card face up at the bottom of the pile, then next flips over a card from the pile whose number matches the number on the card just flipped. The player repeats this until he reaches a pile in which every card has already been flipped and wins if at that point every card has been flipped. Hamster has grown tired of losing every time, so he decides to cheat. He looks at the piles beforehand and rearranges the $k$ cards in each pile as he pleases. When can Hamster perform this procedure such that he will win the game?
[i]Brian Hamrick.[/i]
We visit all squares exactly once on a $n\times n$ chessboard (colored in the usual way) with a king. Find the smallest number of times we had to switch colors during our walk.
[i]Proposed by Dömötör Pálvölgyi, Budapest[/i]
$2024$ otters live in the river. Some are friends with each other. Is it possible that, for any collection of $1012$ otters, there is exactly one additional otter that is friends with all $1012$ otters?
Find the minimal $N$ such that any $N$-element subset of $\{1, 2, 3, 4,...,7\}$ has a subset $S$ such that the sum of elements of $S$ is divisible by $7$.
Suppose $n$ lines in plane are such that no two are parallel and no three are concurrent. For each two lines their angle is a real number in $[0,\frac{\pi}2]$. Find the largest value of the sum of the $\binom n2$ angles between line.
[i]By Aliakbar Daemi[/i]
Let $ABCD$ be a regular tetrahedron. To an arbitrary point $M$ on one edge, say $CD$, corresponds the point $P = P(M)$ which is the intersection of two lines $AH$ and $BK$, drawn from $A$ orthogonally to $BM$ and from $B$ orthogonally to $AM$. What is the locus of $P$ when $M$ varies ?
Suppose the polynomial $P(x) \equiv x^3 + ax^2 + bx +c$ has only real zeroes and let $Q(x) \equiv 5x^2 - 16x + 2004$. Assume that $P(Q(x)) = 0$ has no real roots. Prove that $P(2004) > 2004$
Determine the number of all numbers which are represented as $x^2+y^2$ with $x, y \in \{1, 2, 3, \ldots, 1000\}$ and which are divisible by 121.
Euler's formula states that for a convex polyhedron with $V$ vertices, $E$ edges, and $F$ faces, $V - E + F = 2$. A particular convex polyhedron has 32 faces, each of which is either a triangle or a pentagon. At each of its $V$ vertices, $T$ triangular faces and $P$ pentagonal faces meet. What is the value of $100P + 10T + V$?
An aquarium on a level table has rectangular faces and is $10$ inches wide and $8$ inches high. When it was tilted, the water in it covered an $8"\times 10"$ end but only three-fourths of the rectangular room. The depth of the water when the bottom was again made level, was
$\textbf{(A) }2\textstyle{\frac{1}{2}}"\qquad\textbf{(B) }3"\qquad\textbf{(C) }3\textstyle{\frac{1}{4}}"\qquad\textbf{(D) }3\textstyle{\frac{1}{2}}"\qquad \textbf{(E) }4"$
Four complex numbers lie at the vertices of a square in the complex plane. Three of the numbers are $1+2i,-2+i$ and $-1-2i$. The fourth number is
A. $2+i$
B. $2-i$
C. $1-2i$
D. $-1+2i$
E. $-2-i$
Two sequences of nonzero reals $a_1, a_2, a_3, \dots$ and $b_2, b_3, \dots$ are such that $b_n=\prod_{i=1}^{n} a_i$ and $a_n=\frac{b_n^2}{3b_n-3}$ for all integers $n > 1$. Given that $a_1=\frac{1}{2}$, find $\lvert b_{60}\rvert$.
[i]Proposed by Andrew Zhao[/i]
Let $ACE$ be a triangle with $\angle ECA=60^{\circ}, \angle AEC=90^{\circ}$. Let $B$ and $D$ be points on the sides $AC$ and $CE$ respectively such that the $\triangle BCD$ is equilateral. Now suppose $BD \cap AE=F$. Find $\angle EAC+\angle EFD$.
Given a positive integer $n$, an allowed move is to form $2n+1$ or $3n+2$. The set $S_{n}$ is the set of all numbers that can be obtained by a sequence of allowed moves starting with $n$. For example, we can form $5 \rightarrow 11 \rightarrow 35$ so $5, 11$ and $35$ belong to $S_{5}$. We call $m$ and $n$ compatible if $S_{m}$ and $S_{n}$ has a common element. Which members of $\{1, 2, 3, ... , 2002\}$ are compatible with $2003$?
For all nonnegative integer $i$, there are seven cards with $2^i$ written on it.
How many ways are there to select the cards so that the numbers add up to $n$?