Found problems: 5923
Let $n$ be a positive integer and let $x_1\le x_2\le\cdots\le x_n$ be real numbers.
Prove that
\[
\left(\sum_{i,j=1}^{n}|x_i-x_j|\right)^2\le\frac{2(n^2-1)}{3}\sum_{i,j=1}^{n}(x_i-x_j)^2.
\]
Show that the equality holds if and only if $x_1, \ldots, x_n$ is an arithmetic sequence.
Let $p_1, p_2, ..., p_k$ be $k$ different primes. We consider all positive integers that use only these primes (not necessarily all) in their prime factorization, and arrange those numbers in increasing order, forming an infinite sequence: $a_1 < a_2 < ... < a_n < ...$
Prove that, for every number $c$, there exists $n$ such that $a_{n+1} -a_n > c$.
A 3 × 3 grid of blocks is labeled from 1 through 9. Cindy paints each block orange or
lime with equal probability and gives the grid to her friend Sophia.
Sophia then plays with the grid of blocks. She can take the top row of blocks and move
it to the bottom, as shown.
1 2 3
4 5 6
7 8 9
4 5 6
7 8 9
1 2 3
Grid A Grid A0
She can also take the leftmost column of blocks and move it to the right end, as shown.
1 2 3
4 5 6
7 8 9
2 3 1
5 6 4
8 9 7
Grid B Grid B0
Sophia calls the grid of blocks citrus if it is impossible for her to use a sequence of the
moves described above to obtain another grid with the same coloring but a different numbering
scheme. For example, Grid B is citrus, but Grid A is not citrus because moving the
top row of blocks to the bottom results in a grid with a different numbering but the same
coloring as Grid A.
What is the probability that Sophia receives a citrus grid of blocks?
[u]Round 9[/u]
[b]p25. [/b]Define a sequence $\{a_n\}_{n \ge 1}$ of positive real numbers by $a_1 = 2$ and $a^2_n -2a_n +5 =4a_{n-1}$ for $n \ge 2$. Suppose $k$ is a positive real number such that $a_n <k$ for all positive integers $n$. Find the minimum possible value of $k$.
[b]p26.[/b] Let $\vartriangle ABC$ be a triangle with $AB = 13$, $BC = 14$, and $C A = 15$. Suppose the incenter of $\vartriangle ABC$ is $I$ and the incircle is tangent to $BC$ and $AB$ at $D$ and $E$, respectively. Line $\ell$ passes through the midpoints of $BD$ and $BE$ and point $X$ is on $\ell$ such that $AX \parallel BC$. Find $X I$ .
[b]p27.[/b] Let $x, y, z$ be positive real numbers such that $x y + yz +zx = 20$ and $x^2yz +x y^2z +x yz^2 = 100$. Additionally, let $s = \max (x y, yz,xz)$ and $m = \min(x, y, z)$. If $s$ is maximal, find $m$.
[u]Round 10[/u]
[b]p28.[/b] Let $\omega_1$ be a circle with center $O$ and radius $1$ that is internally tangent to a circle $\omega_2$ with radius $2$ at $T$ . Let $R$ be a point on $\omega_1$ and let $N$ be the projection of $R$ onto line $TO$. Suppose that $O$ lies on segment $NT$ and $\frac{RN}{NO} = \frac4 3$ . Additionally, let $S$ be a point on $\omega_2$ such that $T,R,S$ are collinear. Tangents are drawn from $S$ to $\omega_1$ and touch $\omega_1$ at $P$ and $Q$. The tangent to $\omega_1$ at $R$ intersects $PQ$ at $Z$. Find the area of triangle $\vartriangle ZRS$.
[b]p29.[/b] Let $m$ and $n$ be positive integers such that $k =\frac{ m^2+n^2}{mn-1}$ is also a positive integer. Find the sum of all possible values of $k$.
[b]p30.[/b] Let $f_k (x) = k \cdot \ min (x,1-x)$. Find the maximum value of $k \le 2$ for which the equation $f_k ( f_k ( f_k (x))) = x$ has fewer than $8$ solutions for $x$ with $0 \le x \le 1$.
[u]Round 11[/u]
In the following problems, $A$ is the answer to Problem $31$, $B$ is the answer to Problem $32$, and $C$ is the answer to Problem $33$. For this set, you should find the values of $A$,$B$, and $C$ and submit them as answers to problems $31$, $32$, and $33$, respectively. Although these answers depend on each other, each problem will be scored separately.
[b]p31.[/b] Find $$A \cdot B \cdot C + \dfrac{1}{B+ \dfrac{1}{C +\dfrac{1}{B+\dfrac{1}{...}}}}$$
[b]p32.[/b] Let $D = 7 \cdot B \cdot C$. An ant begins at the bottom of a unit circle. Every turn, the ant moves a distance of $r$ units clockwise along the circle, where $r$ is picked uniformly at random from the interval $\left[ \frac{\pi}{2D} , \frac{\pi}{D} \right]$. Then, the entire unit circle is rotated $\frac{\pi}{4}$ radians counterclockwise. The ant wins the game if it doesn’t get crushed between the circle and the $x$-axis for the first two turns. Find the probability that the ant wins the game.
[b]p33.[/b] Let $m$ and $n$ be the two-digit numbers consisting of the products of the digits and the sum of the digits of the integer $2016 \cdot B$, respectively. Find $\frac{n^2}{m^2 - mn}$.
[u]Round 12[/u]
[b]p34.[/b] There are five regular platonic solids: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. For each of these solids, define its adjacency angle to be the dihedral angle formed between two adjacent faces. Estimate the sum of the adjacency angles of all five solids, in degrees. If your estimate is $E$ and the correct answer is $A$, your score for this problem will be $\max \left(0, \lfloor 15 -\frac12 |A-E| \rfloor \right).$
[b]p35.[/b] Estimate the value of $$\log_{10} \left(\prod_{k|2016} k!\right), $$ where the product is taken over all positive divisors $k$ of $2016$. If your estimate is $E$ and the correct answer is $A$, your score for this problem will be $\max \left(0, \lceil 15 \cdot \min \left(\frac{E}{A}, 2- \frac{E}{A}\right) \rceil \right).$
[b]p36.[/b] Estimate the value of $\sqrt{2016}^{\sqrt[4]{2016}}$. If your estimate is $E$ and the correct answer is $A$, your score for this problem will be $\max \left(0, \lceil 15 \cdot \min \left(\frac{\ln E}{\ln A}, 2- \frac{\ln E}{\ln A}\right) \rceil \right).$
PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3158461p28714996]here [/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3158474p28715078]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
There are $n$ cities in a country, where $n \geq 100$ is an integer. Some pairs of cities are connected by direct (two-way) flights. For two cities $A$ and $B$ we define:
$(i)$ A $\emph{path}$ between $A$ and $B$ as a sequence of distinct cities $A = C_0, C_1, \dots, C_k, C_{k+1} = B$, $k \geq 0$, such that there are direct flights between $C_i$ and $C_{i+1}$ for every $0 \leq i \leq k$;
$(ii)$ A $\emph{long path}$ between $A$ and $B$ as a path between $A$ and $B$ such that no other path between $A$ and $B$ has more cities;
$(iii)$ A $\emph{short path}$ between $A$ and $B$ as a path between $A$ and $B$ such that no other path between $A$ and $B$ has fewer cities.
Assume that for any pair of cities $A$ and $B$ in the country, there exist a long path and a short path between them that have no cities in common (except $A$ and $B$). Let $F$ be the total number of pairs of cities in the country that are connected by direct flights. In terms of $n$, find all possible values $F$
Proposed by David-Andrei Anghel, Romania.
The sequence $(a_k)$, $k> 0$ is Fibonacci, with $a_0 = a_1 = 1$. Calculate the value of $$\sum_{j = 0}^{\infty} \frac{a_j}{2^j}$$
[i]George the grasshopper[/i] lives of the real line, starting at $0$ . He is given the following sequence of numbers: $2, 3, 4, 8, 9, ... ,$ which are all the numbers of the form $2^k$ or $3^l$, $k, l \in \mathbb{N}$, arranged in increasing order. Starting from $2$, for each number $x$ in the sequence in order, he (currently at $a$) must choose to jump to either $a+x$ or $a-x$. Show that [i]George the grasshopper[/i] can jump in a way that he reaches every integer on the real line.
Natural numbers have been divided in groups as follow: $(1), (2, 4), (3, 5, 7), (6, 8, 10, 12), (9, 11, 13, 15, 17), \ldots$. Let $S_n$ be the sum of the elements of the $n$th group. Prove that $\frac{S_{2n+1}}{2n+1}-\frac{S_{2n}}{2n}$ is even.
On a table there is a pile with $ T$ tokens which incrementally shall be converted into piles with three tokens each. Each step is constituted of selecting one pile removing one of its tokens. And then the remaining pile is separated into two piles. Is there a sequence of steps that can accomplish this process?
a.) $ T \equal{} 1000$ (Cono Sur)
b.) $ T \equal{} 2001$ (BWM)
Fix a sequence $a_1,a_2,a_3\ldots$ of integers satisfying the following condition:for all prime numbers $p$ and all positive integers $k$,we have $a_{pk+1}=pa_k-3a_p+13$.Determine all possible values of $a_{2013}$.
Consider a sequence of positive integers with total sum $2019$ such that no number and no sum of a set of consecutive num bers is equal to $40$. What is the greatest possible length of such a sequence?
(Alexandr Shapovalov)
A country has $100$ cities and $n$ airplane companies which take care of a total of $2018$ two-way direct flights between pairs of cities. There is a pair of cities such that one cannot reach one from the other with just one or two flights. What is the largest possible value of $n$ for which between any two cities there is a route (a sequence of flights) using only one of the airplane companies?
Let $n$ be a positive integer. Show that there is a prime $p$ and a sequence $\left(a_k\right)_{k\ge1}$ of positive integers such that the sequence $\left(p+na_k\right)_{k\ge1}$ consists of distinct primes.
Consider a sequence of circles $K_1,K_2,K_3,K_4, \ldots$ of radii $r_1, r_2, r_3, r_4, \ldots$ , respectively, situated inside a triangle $ABC$. The circle $K_1$ is tangent to $AB$ and $AC$; $K_2$ is tangent to $K_1$, $BA$, and $BC$; $K_3$ is tangent to $K_2$, $CA$, and $CB$; $K_4$ is tangent to $K_3$, $AB$, and $AC$; etc.
(a) Prove the relation
\[r_1 \cot \frac 12 A+ 2 \sqrt{r_1r_2} + r_2 \cot \frac 12 B = r \left(\cot \frac 12 A + \cot \frac 12 B \right) \]
where $r$ is the radius of the incircle of the triangle $ABC$. Deduce the existence of a $t_1$ such that
\[r_1=r \cot \frac 12 B \cot \frac 12 C \sin^2 t_1\]
(b) Prove that the sequence of circles $K_1,K_2, \ldots $ is periodic.
A [i]pucelana[/i] sequence is an increasing sequence of $16$ consecutive odd numbers whose sum is a perfect cube. How many pucelana sequences are there with $3$-digit numbers only?
We consider the real sequence ($x_n$) defined by $x_0=0, x_1=1$ and $x_{n+2}=3x_{n+1}-2 x_{n}$ for $n=0,1,2,...$
We define the sequence ($y_n$) by $y_n=x^2_n+2^{n+2}$ for every nonnegative integer $n$.
Prove that for every $n>0, y_n$ is the square of an odd integer.
[b]1.[/b] Let $a_{v} $ and $b_{v} $, ${v= 1,2,\dots,n} $, be real numbers such that
$a_{1}\geq a_{2} \geq a_{3}\geq\dots\geq a_{n}> 0 $ and
$b_{1}\geq a_{1}, b_{1}b_{2}\geq a_{1}a_{2},\dots,b_{1}b_{2}\dots b_{n}\geq a_{1}a_{2}\dots a_{n} $
Show that $b_{1}+b_{2}+\dots+b_{n}\geq a_{1}+a_{2}+\dots+a_{n} $ [b](S. 4)[/b]
The sequence $a_1, a_2, \dots, a_n$ consists of the numbers $1, 2, \dots, n$ in some order. For which positive integers $n$ is it possible that the $n+1$ numbers $0, a_1, a_1+a_2, a_1+a_2+a_3,\dots, a_1 + a_2 +\cdots + a_n$ all have di fferent remainders when divided by $n + 1$?
A sequence $a_n$ is determined by the rules $a_0 = 9$ and for any nonnegative $k$,
$$a_{k+1}=3a_k^4+4a_k^3.$$
Prove that $a_{10}$ contains more than $1000$ nines in decimal notation.
(Yao)
Consider the sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1,...$ Find $n$ such that the fi rst $n$ terms sum up
to $2010$.
Define the sequence $a_1 = 2$ and $a_n = 2^{a_{n-1}} + 2$ for all integers $n \ge 2$. Prove that $a_{n-1}$ divides $a_n$ for all integers $n \ge 2$.
[i]Proposed by Sam Korsky[/i]
Let $p>2$ be a prime number, and $(a_k)_{k\geqslant 1}$ be a sequence defined by $a_1=p$ and $a_{k+1}=2a_k+1$, $k\geqslant 1$. Show that one of the first $p$ terms of the sequence is not prime.
[i]Marcel Țena[/i]
We write one of the numbers $0$ and $1$ into each unit square of a chessboard with $40$ rows and $7$ columns. If any two rows have different sequences, at most how many $1$s can be written into the unit squares?
$
\textbf{(A)}\ 198
\qquad\textbf{(B)}\ 128
\qquad\textbf{(C)}\ 82
\qquad\textbf{(D)}\ 40
\qquad\textbf{(E)}\ \text{None of above}
$
In a rectangular plot of land, a man walks in a very peculiar fashion. Labeling the corners $ABCD$, he starts at $A$ and walks to $C$. Then, he walks to the midpoint of side $AD$, say $A_1$. Then, he walks to the midpoint of side $CD$ say $C_1$, and then the midpoint of $A_1D$ which is $A_2$. He continues in this fashion, indefinitely. The total length of his path if $AB=5$ and $BC=12$ is of the form $a + b\sqrt{c}$. Find $\displaystyle\frac{abc}{4}$.
[b]p1.[/b] Eric is playing Brawl Stars. If he starts playing at $11:10$ AM, and plays for $2$ hours total, then how many minutes past noon does he stop playing?
[b]p2.[/b] James is making a mosaic. He takes an equilateral triangle and connects the midpoints of its sides. He then takes the center triangle formed by the midsegments and connects the midpoints of its sides. In total, how many equilateral triangles are in James’ mosaic?
[b]p3.[/b] What is the greatest amount of intersections that $3$ circles and $3$ lines can have, given that they all lie on the same plane?
[b]p4.[/b] In the faraway land of Arkesia, there are two types of currencies: Silvers and Gold. Each Silver is worth $7$ dollars while each Gold is worth $17$ dollars. In Daniel’s wallet, the total dollar value of the Silvers is $1$ more than that of the Golds. What is the smallest total dollar value of all of the Silvers and Golds in his wallet?
[b]p5.[/b] A bishop is placed on a random square of a $8$-by-$8$ chessboard. On average, the bishop is able to move to $s$ other squares on the chessboard. Find $4s$.
Note: A bishop is a chess piece that can move diagonally in any direction, as far as it wants.
[b]p6.[/b] Andrew has a certain amount of coins. If he distributes them equally across his $9$ friends, he will have $7$ coins left. If he apportions his coins for each of his $15$ classmates, he will have $13$ coins to spare. If he splits the coins into $4$ boxes for safekeeping, he will have $2$ coins left over. What is the minimum number of coins Andrew could have?
[b]p7.[/b] A regular polygon $P$ has three times as many sides as another regular polygon $Q$. The interior angle of $P$ is $16^o$ greater than the interior angle of $Q$. Compute how many more diagonals $P$ has compared to $Q$.
[b]p8.[/b] In an certain airport, there are three ways to switch between the ground floor and second floor that are 30 meters apart: either stand on an escalator, run on an escalator, or climb the stairs. A family on vacation takes 65 seconds to climb up the stairs. A solo traveller late for their flight takes $25$ seconds to run upwards on the escalator. The amount of time (in seconds) it takes for someone to switch floors by standing on the escalator can be expressed as $\frac{u}{v}$ , where $u$ and $v$ are relatively prime. Find $u + v$.
(Assume everyone has the same running speed, and the speed of running on an escalator is the sum of the speeds of riding the escalator and running on the stairs.)
[b]p9.[/b] Avanish, being the studious child he is, is taking practice tests to improve his score. Avanish has a $60\%$ chance of passing a practice test. However, whenever Avanish passes a test, he becomes more confident and instead has a $70\%$ chance of passing his next immediate test. If Avanish takes $3$ practice tests in a row, the expected number of practice tests Avanish will pass can be expressed as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime. Find $a + b$.
[b]p10.[/b] Triangle $\vartriangle ABC$ has sides $AB = 51$, $BC = 119$, and $AC = 136$. Point $C$ is reflected over line $\overline{AB}$ to create point $C'$. Next, point $B$ is reflected over line $\overline{AC'}$ to create point $B'$. If $[B'C'C]$ can be expressed in the form of $a\sqrt{b}$, where $b$ is not divisible by any perfect square besides $1$, find $a + b$.
[b]p11[/b]. Define the following infinite sequence $s$: $$s = \left\{\frac{1}{1},\frac{1}{1 + 3},\frac{1}{1 + 3 + 6}, ... ,\frac{1}{1 + 3 + 6 + ...+ t_k},...\right\},$$
where $t_k$ denotes the $k$th triangular number. The sum of the first $2024$ terms of $s$, denoted $S$, can be
expressed as $$S = 3 \left(\frac{1}{2}+\frac{1}{a}-\frac{1}{b}\right),$$ where $a$ and $b$ are positive integers. Find the minimal possible value of $a + b$.
[b]p12.[/b] Omar writes the numbers from $1$ to $1296$ on a whiteboard and then converts each of them into base $6$. Find the sum of all of the digits written on the whiteboard (in base $10$), including both the base $10$ and base $6$ numbers.
[b]p13.[/b] A mountain number is a number in a list that is greater than the number to its left and right. Let $N$ be the amount of lists created from the integers $1$ - $100$ such that each list only has one mountain number. $N$ can be expressed as
$$N = 2^a(2^b - c^2),$$
where $a$, $b$ and $c$ are positive integers and $c$ is not divisible by $2$. Find $a + b+c$.
(The numbers at the beginning or end of a list are not considered mountain numbers.)[hide]Original problem was voided because the original format of the answer didn't match the result's format. So I changed it in the wording, in order the problem to be correct[/hide]
[b]p14.[/b] A circle $\omega$ with center $O$ has a radius of $25$. Chords $\overline{AB}$ and $\overline{CD}$ are drawn in $\omega$ , intersecting at $X$ such that $\angle BXC = 60^o$ and $AX > BX$. Given that the shortest distance of $O$ with $\overline{AB}$ and $\overline{CD}$ is $7$ and $15$ respectively, the length of $BX$ can be expressed as $x - \frac{y}{\sqrt{z}}$ , where $x$, $y$, and $z$ are positive integers such that $z$ is not divisible by any perfect square. Find $x + y + z.$ [hide]two answers were considered correct according to configuration[/hide]
[b]p15.[/b] How many ways are there to split the first $10$ natural numbers into $n$ sets (with $n \ge 1$) such that all the numbers are used and each set has the same average?
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].