Found problems: 85335
Let ${(a_n)_{n\ge1}} $ be a sequence with ${a_1 = 1} $ and ${a_{n+1} = \lfloor a_n +\sqrt{a_n}+\frac{1}{2}\rfloor }$ for all ${n \ge 1}$, where ${\lfloor x \rfloor}$ denotes the greatest integer less than or equal to ${x}$. Find all ${n \le 2013}$ such that ${a_n}$ is a perfect square
In a game, there are three indistinguishable boxes; one box contains two red balls, one contains two blue balls, and the last contains one ball of each color. To play, Raj first predicts whether he will draw two balls of the same color or two of different colors. Then, he picks a box, draws a ball at random,
looks at the color, and replaces the ball in the same box. Finally, he repeats this; however, the boxes are not shuffled between draws, so he can determine whether he wants to draw again from the same box. Raj wins if he predicts correctly; if he plays optimally, what is the probability that he will win?
Integer $n>2$ is given. Find the biggest integer $d$, for which holds, that from any set $S$ consisting of $n$ integers, we can find three different (but not necesarilly disjoint) nonempty subsets, such that sum of elements of each of them is divisible by $d$.
Let $(m,n)$ be pair of positive integers. Julia has carefully planted $m$ rows of $n$ dandelions in an $m \times n$ array in her back garden. Now, Jana un Viviane decides to play a game with a lawnmower they just found. Taking alternating turns and starting with Jana, they can now mow down all the dandelions in a straight horizontal or vertical line (and they must mow down at least one dandelion ). The winner is the player who mows down the final dandelion. Determine all pairs of $(m,n)$ for which Jana has a winning strategy.
Find, with proof, the least integer $N$ such that if any $2016$ elements are removed from the set ${1, 2,...,N}$, one can still find $2016$ distinct numbers among the remaining elements with sum $N$.
We have a regular polygon $P$ with 2019 vertices, and in each vertex there is a coin. Two players [i]Azul[/i] and [i]Rojo[/i] take turns alternately, beginning with Azul, in the following way: first, Azul chooses a triangle with vertices in $P$ and colors its interior with blue, then Rojo selects a triangle with vertices in $P$ and colors its interior with red, so that the triangles formed in each move don't intersect internally the previous colored triangles. They continue playing until it's not possible to choose another triangle to be colored. Then, a player wins the coin of a vertex if he colored the greater quantity of triangles incident to that vertex (if the quantities of triangles colored with blue or red incident to the vertex are the same, then no one wins that coin and the coin is deleted). The player with the greater quantity of coins wins the game. Find a winning strategy for one of the players.
[i]Note.[/i] Two triangles can share vertices or sides.
In a national park there is a group of sequoia trees, all of which have a positive integer age. Their average age is $41$ years. After a $2010$ year old building was destroyed by lightning, the average age drops to $40$ years. How many trees were originally in the group? At most, how many of them were exactly $2010$ years old?
(W. Janous, WRG Ursulinen, Innsbruck)
Show that $n!=a^{n-1}+b^{n-1}+c^{n-1}$ has only finitely many solutions in positive integers.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
Floyd looked at a standard $12$ hour analogue clock at $2\!:\!36$. When Floyd next looked at the clock, the angles through which the hour hand and minute hand of the clock had moved added to $247$ degrees. How many minutes after $3\!:\!00$ was that?
Three unit circles are inscribed inside an equilateral triangle such that each circle is tangent to each of the other $2$ circles and to $2$ sides of the triangle. Compute the area of the triangle.
Cat and Claire are having a conversation about Cat's favorite number.
Cat says, "My favorite number is a two-digit positive integer that is the product of three distinct prime numbers!"
Claire says, "I don't know your favorite number yet, but I do know that among four of the numbers that might be your favorite number, you could start with any one of them, add a second, subtract a third, and get the fourth!"
Cat says, "That's cool! My favorite number is not among those four numbers, though."
Claire says, "Now I know your favorite number!"
What is Cat's favorite number?
[i]Proposed by Andrew Wu and Andrew Milas[/i]
Let $F$ be the boundary and $M,N$ be any interior points of a triangle $ABC$. Consider the function $f_{M,N}: F \to R$ defined by $f_{M,N}(P) = MP^2 +NP^2$ and let $\eta_{M,N}$ be the number of points $P$ for which $f{M,N}$ attains its minimum.
(a) Prove that $1 \le \eta_{M,N} \le 3$.
(b) If $M$ is fixed, find the locus of $N$ for which $\eta_{M,N} > 1$.
(c) Prove that the locus of $M$ for which there are points $N$ such that $\eta_{M,N} = 3$ is the interior of a tangent hexagon.
Let $C_1, ..., C_d$ be compact and connected sets in $R^d$, and suppose that each convex hull of $C_i$ contains the origin. Prove that for every i there is a $c_i \in C_i$ for which the origin is contained in the convex hull of the points $c_1, ..., c_d$.
Two circle $\Gamma_1$ and $\Gamma_2$ intersect at points $A$ and $B$, point $P$ is not on the line $AB$. Line $AP$ intersects again $\Gamma_1$ and $\Gamma_2$ at points $K$ and $L$ respectively, line $BP$ intersects again $\Gamma_1$ and $\Gamma_2$ at points $M$ and $N$ respectively and all the points mentioned so far are different. The centers of the circles circumscribed around the triangles $KMP$ and $LNP$ are $O_1$ and $O_2$ respectively. Prove that $O_1O_2$ is perpendicular to $AB$.
Let $ ABC$ be a triangle, line $ l$ cuts its sides $ BC,CA,AB$ at $ D,E,F$, respectively. Denote by $ O_{1},O_{2},O_{3}$ the circumcenters of triangle $ AEF,BFD,CDE$, respectively. Prove that the orthocenter of triangle $ O_{1}O_{2}O_{3}$ lies on line $ l$.
For every $0 < \alpha < 1$, let $R(\alpha)$ be the region in $\mathbb{R}^2$ whose boundary is the convex pentagon of vertices $(0,1-\alpha), (\alpha, 0), (1, 0), (1,1)$ and $(0, 1)$. Let $R$ be the set of points that belong simultaneously to each of the regions $R(\alpha)$ with $0 < \alpha < 1$, that is, $R =\bigcap_{0<\alpha<1} R(\alpha)$.
Determine the area of $R$.
There are $n \geq 2$ coins numbered from $1$ to $n$. These coins are placed around a circle, not necesarily in order.
In each turn, if we are on the coin numbered $i$, we will jump to the one $i$ places from it, always in a clockwise order, beginning with coin number 1. For an example, see the figure below.
Find all values of $n$ for which there exists an arrangement of the coins in which every coin will be visited.
Let $n\ge 3$ be a fixed integer. There are $m\ge n+1$ beads on a circular necklace. You wish to paint the beads using $n$ colors, such that among any $n+1$ consecutive beads every color appears at least once. Find the largest value of $m$ for which this task is $\emph{not}$ possible.
[i]Carl Schildkraut, USA[/i]
Let $\ell$ be a real number satisfying the equation $\tfrac{(1+\ell)^2}{1+\ell^2}=\tfrac{13}{37}$. Then
\[\frac{(1+\ell)^3}{1+\ell^3}=\frac mn,\] where $m$ and $n$ are positive coprime integers. Find $m+n$.
From the vertices of a regular 27-gon, seven are chosen arbitrarily. Prove that among these seven points there are three points that form an isosceles triangle or four points that form an isosceles trapezoid.
For a positive integer $k$ and real numbers $x$ and $y$, let
\[f_k(x,y)=(x+y)-\left(x^{2k+1}+y^{2k+1}\right)\mbox{.}\]
If $x^2+y^2=1$, then determine the maximal possible value $c_k$ of $f_k(x,y)$.
Let $ S$ be the set of all real numbers strictly greater than −1. Find all functions $ f: S \to S$ satisfying the two conditions:
(a) $ f(x \plus{} f(y) \plus{} xf(y)) \equal{} y \plus{} f(x) \plus{} yf(x)$ for all $ x, y$ in $ S$;
(b) $ \frac {f(x)}{x}$ is strictly increasing on each of the two intervals $ \minus{} 1 < x < 0$ and $ 0 < x$.
How many $5 \times 5$ grids are possible such that each element is either 1 or 0 and each row sum and column sum is $4 ?$
[list=1]
[*] 64
[*] 32
[*] 120
[*] 96
[/list]
Let $p = 491$ be prime. Let $S$ be the set of ordered $k$-tuples of nonnegative integers that are less than $p$. We say that a function $f\colon S \to S$ is \emph{$k$-murine} if, for all $u,v\in S$, $\langle f(u), f(v)\rangle \equiv \langle u,v\rangle \pmod p$, where $\langle(a_1,\dots ,a_k) , (b_1, \dots , b_k)\rangle = a_1b_1+ \dots +a_kb_k$ for any $(a_1, \dots a_k), (b_1, \dots b_k) \in S$.
Let $m(k)$ be the number of $k$-murine functions. Compute the remainder when $m(1) + m(2) + m(3) + \cdots + m(p)$ is divided by $488$.
[i]Proposed by Brandon Wang[/i]
Let $\{a_{n}\}$ be a sequence which satisfy
$a_{1}=5$ and $a_{n=}\sqrt[n]{a_{n-1}^{n-1}+2^{n-1}+2.3^{n-1}} \qquad \forall n\geq2$
[b](a)[/b] Find the general fomular for $a_{n}$
[b](b)[/b] Prove that $\{a_{n}\}$ is decreasing sequences