Found problems: 85335
What is the sum of the infinite series $\frac{20}{3} +\frac{17}{9} + \frac{20}{27} + \frac{17}{81} + \frac{20}{243} + \frac{17}{729} + ...$ ?
The set $S$ of positive real numbers $x$ such that
\[ \left\lfloor\frac{2x}{5}\right\rfloor + \left\lfloor\frac{3x}{5}\right\rfloor + 1 = \left\lfloor x\right\rfloor \]
can be written as $S = \bigcup_{j = 1}^{\infty} I_{j}$, where the $I_{i}$ are disjoint intervals of the form $[a_{i}, b_{i}) = \{x \, | \, a_i \leq x < b_i\}$ and $b_{i} \leq a_{i+1}$ for all $i \geq 1$. Find $\sum_{i=1}^{2017} (b_{i} - a_{i})$.
For every $ n\in\mathbb{N}$ let $ d(n)$ denote the number of (positive) divisors of $ n$. Find all functions $ f: \mathbb{N}\to\mathbb{N}$ with the following properties: [list][*] $ d\left(f(x)\right) \equal{} x$ for all $ x\in\mathbb{N}$.
[*] $ f(xy)$ divides $ (x \minus{} 1)y^{xy \minus{} 1}f(x)$ for all $ x$, $ y\in\mathbb{N}$.[/list]
[i]Proposed by Bruno Le Floch, France[/i]
Find all triples $(a,b,k)$, $k \geq 2$, of positive integers such that $(a^k+b)(b^k+a)$ is a power of two.
In a triangle $ABC$, let $K$ be a point on the median $BM$ such that $CM = CK$. It turned out that $\angle CBM = 2\angle ABM$. Show that $BC = KM$.
A small square $PQRS$ is contained in a big square. Produce $PQ$ to $A$, $QR$ to $B$, $RS$ to $C$ and $SP$ to $D$ so that $A$, $B$, $C$ and $D$ lie on the four sides of the large square in order, produced if necessary. Prove that $AC = BD$ and $AC \perp BD$.
Let $n > 1$ be a positive integer and $S$ be a collection of $\frac{1}{2}\binom{2n}{n}$ distinct $n$-element subsets of $\{1, 2, \dotsc, 2n\}$. Show that there exists $A, B\in S$ such that $|A\cap B|\leq 1$.
[i]Proposed by Michael Ren.[/i]
A single bench section at a school event can hold either $7$ adults or $11$ children. When $N$ bench sections are connected end to end, an equal number of adults and children seated together will occupy all the bench space. What is the least possible positive integer value of $N?$
$\textbf{(A) } 9 \qquad \textbf{(B) } 18 \qquad \textbf{(C) } 27 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 77$
Let $\mathbb{F}_p$ be the set of integers modulo $p$. Call a function $f : \mathbb{F}_p^2 \to \mathbb{F}_p$ [i]quasiperiodic[/i] if there exist $a,b \in \mathbb{F}_p$, not both zero, so that $f(x + a, y + b) = f(x, y)$ for all $x,y \in \mathbb{F}_p$.
Find the number of functions $\mathbb{F}_p^2 \to \mathbb{F}_p$ that can be written as the sum of some number of quasiperiodic functions.
Let $f$ be a real-valued function such that $4f(x)+xf\left(\tfrac1x\right)=x+1$ for every positive real number $x$. Find $f(2014)$.
Let $n$ be a positive integer. Prove that all roots of the equation $$x(x + 2) (x + 4 )... (x + 2n) + (x +1) (x + 3 )... (x + 2n - 1) = 0$$ are real and irrational.
In the isosceles triangle $ABC$, $AB = AC$. Let $E$ be on side $AB$ such that $\angle ACE = \angle ECB = 18^o$, and let $D$ be the midpoint of side $CB$. If we know the length of $AD$ is $3$ units, what is the length of $CE$?
There are finite many coins in David’s purse. The values of these coins are pair wisely distinct positive integers. Is that possible to make such a purse, such that David has exactly $2020$ different ways to select the coins in his purse and the sum of these selected coins is $2020$?
[b]p1.[/b] $2011$ distinct points are arranged along the perimeter of a circle. We choose without replacement four points $P$, $Q$, $R$, $S$. What is the probability that no two of the segments $P Q$, $QR$, $RS$, $SP$ intersect (disregarding the endpoints)?
[b]p2.[/b] In Soviet Russia, all phone numbers are between three and six digits and contain only the digits $1$, $2$, and $3$. No phone number may be the prefix of another phone number, so, for example, we cannot have the phone numbers $123$ and $12332$. If the Soviet bureaucracy has preassigned $10$ phone numbers of length $3$, $20$ numbers of length $4$, and $77$ phone numbers of length $6$, what is the maximum number of phone numbers of length $5$ that the authorities can allocate?
[b]p3.[/b] The sequence $\{a_n\}_{n\ge 1}$ is defined as follows: we have $a_1 = 1$, $a_2 = 0$, and for $n \ge 3$ we have $$a_n = \frac12 \sum\limits_{\substack{1\le i,j\\ i+j+k=n}} a_ia_ja_k.$$
Find $$\sum^{\infty}_{n=1} \frac{a_n}{2^n}$$
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Helena and Luis are going to play a game with two bags with marbles. They play alternately and on each turn they can do one and only one of the following moves:
[list]
Take out a marble from one bag.
Take out a marble from each bag.
Take out a marble from one bag and then put it into the other bag.
[/list]
The player who leaves both bags empty wins the game.
Before starting the game, Helena counted out the marbles of each bag and said to Luis: "You may start!", while she thought "I will certainly win...". What are the possible distributions of the marbles in the bags?
If $a, b, c$ are real numbers such that $$a^2 + b^2 + c^2 = 1$$ prove the inequalities $$- \frac12 \le ab + bc + ca \le 1$$
The expression $\sin2^\circ\sin4^\circ\sin6^\circ\cdots\sin90^\circ$ is equal to $p\sqrt{5}/2^{50}$, where $p$ is an integer. Find $p$.
An acute-angled triangle $ABC$ has two altitudes $BE$ and $CF$. The circle with diameter $AC$ intersects the segment $BE$ at point $P$. A circle with diameter $AB$ intersects the segment $CF$ at point $Q$ and the extension of this altitude at point $Q'$. Prove that $\angle PQ'Q = \angle PQB$.
Let $B$ be a set of $k$ sequences each having $n$ terms equal to $1$ or $-1$. The product of two such sequences $(a_1, a_2, \ldots , a_n)$ and $(b_1, b_2, \ldots , b_n)$ is defined as $(a_1b_1, a_2b_2, \ldots , a_nb_n)$. Prove that there exists a sequence $(c_1, c_2, \ldots , c_n)$ such that the intersection of $B$ and the set containing all sequences from $B$ multiplied by $(c_1, c_2, \ldots , c_n)$ contains at most $\frac{k^2}{2^n}$ sequences.
In Skinien there 2009 towns where each of them is connected with exactly 1004 other town by a highway. Prove that starting in an arbitrary town one can make a round trip along the highways such that each town is passed exactly once and finally one returns to its starting point.
Find all real values of $x$ such that $(\frac{1}{5}(x^2-10x+26))^{x^2-6x+5}=1$
Mr. Ramos gave a test to his class of $20$ students. The dot plot below shows the distribution of test scores.
[asy]
//diagram by pog . give me 1,000,000,000 dollars for this diagram
size(5cm);
defaultpen(0.7);
dot((0.5,1));
dot((0.5,1.5));
dot((1.5,1));
dot((1.5,1.5));
dot((2.5,1));
dot((2.5,1.5));
dot((2.5,2));
dot((2.5,2.5));
dot((3.5,1));
dot((3.5,1.5));
dot((3.5,2));
dot((3.5,2.5));
dot((3.5,3));
dot((4.5,1));
dot((4.5,1.5));
dot((5.5,1));
dot((5.5,1.5));
dot((5.5,2));
dot((6.5,1));
dot((7.5,1));
draw((0,0.5)--(8,0.5),linewidth(0.7));
defaultpen(fontsize(10.5pt));
label("$65$", (0.5,-0.1));
label("$70$", (1.5,-0.1));
label("$75$", (2.5,-0.1));
label("$80$", (3.5,-0.1));
label("$85$", (4.5,-0.1));
label("$90$", (5.5,-0.1));
label("$95$", (6.5,-0.1));
label("$100$", (7.5,-0.1));
[/asy]
Later Mr. Ramos discovered that there was a scoring error on one of the questions. He regraded the tests, awarding some of the students $5$ extra points, which increased the median test score to $85$. What is the minimum number of students who received extra points?
(Note that the [i]median[/i] test score equals the average of the $2$ scores in the middle if the $20$ test scores are arranged in increasing order.)
$\textbf{(A)} ~2\qquad\textbf{(B)} ~3\qquad\textbf{(C)} ~4\qquad\textbf{(D)} ~5\qquad\textbf{(E)} ~6\qquad$
An eccentric mathematician has a ladder with $ n$ rungs that he always ascends and descends in the following way: When he ascends, each step he takes covers $ a$ rungs of the ladder, and when he descends, each step he takes covers $ b$ rungs of the ladder, where $ a$ and $ b$ are fixed positive integers. By a sequence of ascending and descending steps he can climb from ground level to the top rung of the ladder and come back down to ground level again. Find, with proof, the minimum value of $ n,$ expressed in terms of $ a$ and $ b.$
Given $a,b,c,d>0$, prove that:
\[\sum_{cyc}\frac{c}{a+2b}+\sum_{cyc}\frac{a+2b}{c}\geq 8(\frac{(a+b+c+d)^2}{ab+ac+ad+bc+bd+cd}-1),\]
where $\sum_{cyc}f(a,b,c,d)=f(a,b,c,d)+f(d,a,b,c)+f(c,d,a,b)+f(b,c,d,a)$.
Let $\vartriangle ABC$ be a right triangle such that $AB = 3$, $BC = 4$, $AC = 5$. Let point $D$ be on $AC$ such that the incircles of $\vartriangle ABD$ and $\vartriangle BCD$ are mutually tangent. Find the length of $BD$.