Found problems: 5923
The sequence $p_1, p_2, p_3, ...$ is defined as follows. $p_1$ and $p_2$ are primes. $p_n$ is the greatest prime divisor of $p_{n-1} + p_{n-2} + 2000$. Show that the sequence is bounded.
Let ${a_1,a_2,\dots,a_n}$ be positive real numbers, ${n>1}$. Denote by $g_n$ their geometric mean, and by $A_1,A_2,\dots,A_n$ the sequence of arithmetic means defined by \[ A_k=\frac{a_1+a_2+\cdots+a_k}{k},\qquad k=1,2,\dots,n. \] Let $G_n$ be the geometric mean of $A_1,A_2,\dots,A_n$. Prove the inequality \[
n \root n\of{\frac{G_n}{A_n}}+ \frac{g_n}{G_n}\le n+1 \] and establish the cases of equality.
[i]Proposed by Finbarr Holland, Ireland[/i]
There are $64$ towns in a country and some pairs of towns are connected by roads but we do not know these pairs. We may choose any pair of towns and find out whether they are connected or not. Our aim is to determine whether it is possible to travel from any town to any other by a sequence of roads. Prove that there is no algorithm which enables us to do so in less than $2016$ questions.
(Proposed by Konstantin Knop)
Determine all sequences $a_0 , a_1 , a_2 , \ldots$ of positive integers with $a_0 \ge 2015$ such that for all integers $n\ge 1$:
(i) $a_{n+2}$ is divisible by $a_n$ ;
(ii) $|s_{n+1} - (n + 1)a_n | = 1$, where $s_{n+1} = a_{n+1} - a_n + a_{n-1} - \cdots + (-1)^{n+1} a_0$ .
[i]Proposed by Pakawut Jiradilok and Warut Suksompong, Thailand[/i]
Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.)
[i]Proposed by Hong Kong[/i]
[url=https://artofproblemsolving.com/community/c677808][b]Cyprus IMO TST 2018[/b][/url]
[url=https://artofproblemsolving.com/community/c6h1666662p10591751][b]Problem 1.[/b][/url] Determine all integers $n \geq 2$ for which the number $11111$ in base $n$ is a perfect square.
[url=https://artofproblemsolving.com/community/c6h1666663p10591753][b]Problem 2.[/b][/url] Consider a trapezium $AB \Gamma \Delta$, where $A\Delta \parallel B\Gamma$ and $\measuredangle A = 120^{\circ}$. Let $E$ be the midpoint of $AB$ and let $O_1$ and $O_2$ be the circumcenters of triangles $AE \Delta$ and $BE\Gamma$, respectively. Prove that the area of the trapezium is equal to six time the area of the triangle $O_1 E O_2$.
[url=https://artofproblemsolving.com/community/c6h1666660p10591747][b]Problem 3.[/b][/url] Find all triples $(\alpha, \beta, \gamma)$ of positive real numbers for which the expression
$$K = \frac{\alpha+3 \gamma}{\alpha + 2\beta + \gamma} + \frac{4\beta}{\alpha+\beta+2\gamma} - \frac{8 \gamma}{\alpha+ \beta + 3\gamma}$$obtains its minimum value.
[url=https://artofproblemsolving.com/community/c6h1666661p10591749][b]Problem 4.[/b][/url] Let $\Lambda= \{1, 2, \ldots, 2v-1,2v\}$ and $P=\{\alpha_1, \alpha_2, \ldots, \alpha_{2v-1}, \alpha_{2v}\}$ be a permutation of the elements of $\Lambda$.
(a) Prove that
$$\sum_{i=1}^v \alpha_{2i-1}\alpha_{2i} \leq \sum_{i=1}^v (2i-1)2i.$$(b) Determine the largest positive integer $m$ such that we can partition the $m\times m$ square into $7$ rectangles for which every pair of them has no common interior points and their lengths and widths form the following sequence:
$$1,2,3,4,5,6,7,8,9,10,11,12,13,14.$$
Suppose that $a_0, a_1, \cdots $ and $b_0, b_1, \cdots$ are two sequences of positive integers such that $a_0, b_0 \ge 2$ and \[ a_{n+1} = \gcd{(a_n, b_n)} + 1, \qquad b_{n+1} = \operatorname{lcm}{(a_n, b_n)} - 1. \] Show that the sequence $a_n$ is eventually periodic; in other words, there exist integers $N \ge 0$ and $t > 0$ such that $a_{n+t} = a_n$ for all $n \ge N$.
Let $S=\{(a,b)|a=1,2,\dots,n,b=1,2,3\}$. A [i]rook tour[/i] of $S$ is a polygonal path made up of line segments connecting points $p_1,p_2,\dots,p_{3n}$ is sequence such that
(i) $p_i\in S,$
(ii) $p_i$ and $p_{i+1}$ are a unit distance apart, for $1\le i<3n,$
(iii) for each $p\in S$ there is a unique $i$ such that $p_i=p.$
How many rook tours are there that begin at $(1,1)$ and end at $(n,1)?$
(The official statement includes a picture depicting an example of a rook tour for $n=5.$ This example consists of line segments with vertices at which there is a change of direction at the following points, in order: $(1,1),(2,1),(2,2),(1,2), (1,3),(3,3),(3,1),(4,1), (4,3),(5,3),(5,1).$)
Let $n$ be a positive integer. We start with $n$ piles of pebbles, each initially containing a single pebble. One can perform moves of the following form: choose two piles, take an equal number of pebbles from each pile and form a new pile out of these pebbles. Find (in terms of $n$) the smallest number of nonempty piles that one can obtain by performing a finite sequence of moves of this form.
Let $ a_{0} \equal{} 1994$ and $ a_{n \plus{} 1} \equal{} \frac {a_{n}^{2}}{a_{n} \plus{} 1}$ for each nonnegative integer $ n$. Prove that $ 1994 \minus{} n$ is the greatest integer less than or equal to $ a_{n}$, $ 0 \leq n \leq 998$
Let $a,b,$ and $c$ be pairwise distinct positive integers such that $\tfrac{1}{a}, \tfrac{1}{b}, \tfrac{1}{c}$ is an increasing arithmetic sequence in that order. Prove that $\gcd(a,b)>1.$
Prove that there are infinitely many positive integers $n$ with the following property: For any $n$ integers $a_{1},a_{2},...,a_{n}$ which form in arithmetic progression, both the mean and the standard deviation of the set $\{a_{1},a_{2},...,a_{n}\}$ are integers.
[i]Remark[/i]. The mean and standard deviation of the set $\{x_{1},x_{2},...,x_{n}\}$ are defined by $\overline{x}=\frac{x_{1}+x_{2}+...+x_{n}}{n}$ and $\sqrt{\frac{\sum (x_{i}-\overline{x})^{2}}{n}}$, respectively.
[b]p1.[/b] Find the greatest integer $n$ such that $n \log_{10} 4$ does not exceed $\log_{10} 1998$.
[b]p2.[/b] Rectangle $ABCD$ has sides $AB = CD = 12/5$, $BC = DA = 5$. Point $P$ is on $AD$ with $\angle BPC = 90^o$. Compute $BP + PC$.
[b]p3.[/b] Compute the number of sequences of four decimal digits $(a, b, c, d)$ (each between $0$ and $9$ inclusive) containing no adjacent repeated digits. (That is, each digit is distinct from the digits directly before and directly after it.)
[b]p4.[/b] Solve for $t$, $-\pi/4 \le t \le \pi/4 $:
$$\sin^3 t + \sin^2 t \cos t + \sin t \cos^2 t + \cos^3 t =\frac{\sqrt6}{2}$$
[b]p5.[/b] Find all integers $n$ such that $n - 3$ divides $n^2 + 2$.
[b]p6.[/b] Find the maximum number of bishops that can occupy an $8 \times 8$ chessboard so that no two of the bishops attack each other. (Bishops can attack an arbitrary number of squares in any diagonal direction.)
[b]p7.[/b] Points $A, B, C$, and $D$ are on a Cartesian coordinate system with $A = (0, 1)$, $B = (1, 1)$, $C = (1,-1)$, and $D = (-1, 0)$. Compute the minimum possible value of $PA + PB + PC + PD$ over all points $P$.
[b]p8.[/b] Find the number of distinct real values of $x$ which satisfy
$$(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)(x-8)(x-9)(x-10)+(1^2 \cdot 3^2\cdot 5^2\cdot 7^2\cdot 9^2)/2^{10} = 0.$$
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
p1. Find all real numbers that satisfy the equation $$(1 + x^2 + x^4 + .... + x^{2014})(x^{2016} + 1) = 2016x^{2015}$$
p2. Let $A$ be an integer and $A = 2 + 20 + 201 + 2016 + 20162 + ... + \underbrace{20162016...2016}_{40\,\, digits}$
Find the last seven digits of $A$, in order from millions to units.
p3. In triangle $ABC$, points $P$ and $Q$ are on sides of $BC$ so that the length of $BP$ is equal to $CQ$, $\angle BAP = \angle CAQ$ and $\angle APB$ is acute. Is triangle $ABC$ isosceles? Write down your reasons.
p4. Ayu is about to open the suitcase but she forgets the key. The suitcase code consists of nine digits, namely four $0$s (zero) and five $1$s. Ayu remembers that no four consecutive numbers are the same. How many codes might have to try to make sure the suitcase is open?
p5. Fulan keeps $100$ turkeys with the weight of the $i$-th turkey, being $x_i$ for $i\in\{1, 2, 3, ... , 100\}$. The weight of the $i$-th turkey in grams is assumed to follow the function $x_i(t) = S_it + 200 - i$ where $t$ represents the time in days and $S_i$ is the $i$-th term of an arithmetic sequence where the first term is a positive number $a$ with a difference of $b =\frac15$. It is known that the average data on the weight of the hundred turkeys at $t = a$ is $150.5$ grams. Calculate the median weight of the turkey at time $t = 20$ days.
Let $<a_n>$ be a sequence of non-negative real numbers such that $a_{m+n} \le a_m +a_n$ for all $m,n \in \mathbb{N}$.
Prove that
\[\sum_{k=1}^{N} \frac{a_k}{k^2}\ge \frac{a_N}{4N}\ln N\]
for any $N \in \mathbb{N}$, where $\ln$ denotes the natural logarithm.
Let $(F_n)$ be the sequence defined recursively by $F_1=F_2=1$ and $F_{n+1}=F_n+F_{n-1}$ for $n\geq 2$. Find all pairs of positive integers $(x,y)$ such that
$$5F_x-3F_y=1.$$
Suppose that a sequence $x_1,x_2,\ldots,x_{2001}$ of positive real numbers satisfies
$$3x^2_{n+1}=7x_nx_{n+1}-3x_{n+1}-2x^2_n+x_n\enspace\text{ and }\enspace x_{37}=x_{2001}.$$Find the maximum possible value of $x_1$.
Let $(a_n)_{n\ge0}$ be a sequence of positive real numbers such that
\[\sum_{k=0}^nC_n^ka_ka_{n-k}=a_n^2,\ \text{for any }n\ge 0.\]
Prove that $(a_n)_{n\ge0}$ is a geometric sequence.
[i]Lucian Dragomir[/i]
Determine whether there exists an infinite sequence of nonzero digits $a_1 , a_2 , a_3 , \cdots $ and a positive integer $N$ such that for every integer $k > N$, the number $\overline{a_k a_{k-1}\cdots a_1 }$ is a perfect square.
Find all strictly increasing sequences $\{a_n\}_{n=0}^\infty$ of positive integers such that for all positive integers $k,m,n$
$$\frac{a_{n+1} +a_{n+2} +\dots +a_{n+k}}{k+m}$$ is not an integer larger than $2020$.
Let $X_0, \xi_{i, j}, \epsilon_k$ (i, j, k ∈ N) be independent, non-negative integer random variables. Suppose that $\xi_{i, j}$ (i, j ∈ N) have the same distribution, $\epsilon_k$ (k ∈ N) also have the same distribution.
$\mathbb{E}(\xi_{1,1})=1$ , $\mathbb{E}(X_0^l)<\infty$ , $\mathbb{E}(\xi_{1,1}^l)<\infty$ , $\mathbb{E}(\epsilon_1^l)<\infty$ for some $l\in\mathbb{N}$
Consider the random variable $X_n := \epsilon_n + \sum_{j=1}^{X_{n-1}} \xi_{n,j}$ (n ∈ N) , where $\sum_{j=1}^0 \xi_{n,j} :=0$
Introduce the sequence $M_n := X_n-X_{n-1}-\mathbb{E}(\epsilon_n)$ (n ∈ N)
Prove that there is a polynomial P of degree $\leq l/2$ such that $\mathbb{E}(M_n^l) = P_l(n)$ (n ∈ N).
Consider the non-decreasing sequence of positive integers
\[ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5,... \] in which the $n^{\text{th}}$ positive integer appears $n$ times. The remainder when the $1993^{\text{rd}}$ term is divided by $5$ is
$ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4 $
Dedalo buys a finite number of binary strings, each of finite length and made up of the binary digits 0 and 1. For each string, he pays $(\frac{1}{2})^L$ drachmas, where $L$ is the length of the string. The Minotaur is able to escape the labyrinth if he can find an infinite sequence of binary digits that does not contain any of the strings Dedalo bought. Dedalo’s aim is to trap the Minotaur.
For instance, if Dedalo buys the strings $00$ and $11$ for a total of half a drachma, the Minotaur is able to escape using the infinite string $01010101 \ldots$.
On the other hand, Dedalo can trap the Minotaur by spending $75$ cents of a drachma: he could for example buy the strings $0$ and $11$, or the strings $00, 11, 01$.
Determine all positive integers $c$ such that Dedalo can trap the Minotaur with an expense of at most $c$ cents of a drachma.
Find the number of ordered pairs of integers $(a, b)$ such that the sequence $$3, 4, 5, a, b, 30, 40, 50$$ is strictly increasing and no set of four (not necessarily consecutive) terms forms an arithmetic progression.
[b]p1.[/b] In the following $3$ by $3$ grid, $a, b, c$ are numbers such that the sum of each row is listed at the right and the sum of each column is written below it:
[center][img]https://cdn.artofproblemsolving.com/attachments/d/9/4f6fd2bc959c25e49add58e6e09a7b7eed9346.png[/img][/center]
What is $n$?
[b]p2.[/b] Suppose in your sock drawer of $14$ socks there are 5 different colors and $3$ different lengths present. One day, you decide you want to wear two socks that have both different colors and different lengths. Given only this information, what is the maximum number of choices you might have?
[b]p3.[/b] The population of Arveymuddica is $2014$, which is divided into some number of equal groups. During an election, each person votes for one of two candidates, and the person who was voted for by $2/3$ or more of the group wins. When neither candidate gets $2/3$ of the vote, no one wins the group. The person who wins the most groups wins the election. What should the size of the groups be if we want to minimize the minimum total number of votes required to win an election?
[b]p4.[/b] A farmer learns that he will die at the end of the year (day $365$, where today is day $0$) and that he has a number of sheep. He decides that his utility is given by ab where a is the money he makes by selling his sheep (which always have a fixed price) and $b$ is the number of days he has left to enjoy the profit; i.e., $365-k$ where $k$ is the day. If every day his sheep breed and multiply their numbers by $103/101$ (yes, there are small, fractional sheep), on which day should he sell them all?
[b]p5.[/b] Line segments $\overline{AB}$ and $\overline{AC}$ are tangent to a convex arc $BC$ and $\angle BAC = \frac{\pi}{3}$ . If $\overline{AB} = \overline{AC} = 3\sqrt3$, find the length of arc $BC$.
[b]p6.[/b] Suppose that you start with the number $8$ and always have two legal moves:
$\bullet$ Square the number
$\bullet$ Add one if the number is divisible by $8$ or multiply by $4$ otherwise
How many sequences of $4$ moves are there that return to a multiple of $8$?
[b]p7.[/b] A robot is shuffling a $9$ card deck. Being very well machined, it does every shuffle in exactly the same way: it splits the deck into two piles, one containing the $5$ cards from the bottom of the deck and the other with the $4$ cards from the top. It then interleaves the cards from the two piles, starting with a card from the bottom of the larger pile at the bottom of the new deck, and then alternating cards from the two piles while maintaining the relative order of each pile. The top card of the new deck will be the top card of the bottom pile. The robot repeats this shuffling procedure a total of n times, and notices that the cards are in the same order as they were when it started shuffling. What is the smallest possible value of $n$?
[b]p8.[/b] A secant line incident to a circle at points $A$ and $C$ intersects the circle's diameter at point $B$ with a $45^o$ angle. If the length of $AB$ is $1$ and the length of $BC$ is $7$, then what is the circle's radius?
[b]p9.[/b] If a complex number $z$ satisfies $z + 1/z = 1$, then what is $z^{96} + 1/z^{96}$?
[b]p10.[/b] Let $a, b$ be two acute angles where $\tan a = 5 \tan b$. Find the maximum possible value of $\sin (a - b)$.
[b]p11.[/b] A pyramid, represented by $SABCD$ has parallelogram $ABCD$ as base ($A$ is across from $C$) and vertex $S$. Let the midpoint of edge $SC$ be $P$. Consider plane $AMPN$ where$ M$ is on edge $SB$ and $N$ is on edge $SD$. Find the minimum value $r_1$ and maximum value $r_2$ of $\frac{V_1}{V_2}$ where $V_1$ is the volume of pyramid $SAMPN$ and $V_2$ is the volume of pyramid $SABCD$. Express your answer as an ordered pair $(r_1, r_2)$.
[b]p12.[/b] A $5 \times 5$ grid is missing one of its main diagonals. In how many ways can we place $5$ pieces on the grid such that no two pieces share a row or column?
[b]p13.[/b] There are $20$ cities in a country, some of which have highways connecting them. Each highway goes from one city to another, both ways. There is no way to start in a city, drive along the highways of the country such that you travel through each city exactly once, and return to the same city you started in. What is the maximum number of roads this country could have?
[b]p14.[/b] Find the area of the cyclic quadrilateral with side lengths given by the solutions to $$x^4-10x^3+34x^2- 45x + 19 = 0.$$
[b]p15.[/b] Suppose that we know $u_{0,m} = m^2 + m$ and $u_{1,m} = m^2 + 3m$ for all integers $m$, and that $$u_{n-1,m} + u_{n+1,m} = u_{n,m-1} + u_{n,m+1}$$
Find $u_{30,-5}$.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].