Found problems: 5923
[u]Round 1[/u]
[b]p1.[/b] At a fortune-telling exam, $13$ witches are sitting in a circle. To pass the exam, a witch must correctly predict, for everybody except herself and her two neighbors, whether they will pass or fail. Each witch predicts that each of the $10$ witches she is asked about will fail. How many witches could pass?
[b]p2.[/b] Out of $152$ coins, $7$ are counterfeit. All counterfeit coins have the same weight, and all real coins have the same weight, but counterfeit coins are lighter than real coins. How can you find $19$ real coins if you are allowed to use a balance scale three times?
[b]p3.[/b] The digits of a number $N$ increase from left to right. What could the sum of the digits of $9 \times N$ be?
[b]p4.[/b] The sides and diagonals of a pentagon are colored either blue or red. You can choose three vertices and flip the colors of all three lines that join them. Can every possible coloring be turned all blue by a sequence of such moves?
[img]https://cdn.artofproblemsolving.com/attachments/5/a/644aa7dd995681fc1c813b41269f904283997b.png[/img]
[b]p5.[/b] You have $100$ pancakes, one with a single blueberry, one with two blueberries, one with three blueberries, and so on. The pancakes are stacked in a random order. Count the number of blueberries in the top pancake and call that number $N$. Pick up the stack of the top $N$ pancakes and flip it upside down. Prove that if you repeat this counting-and-flipping process, the pancake with one blueberry will eventually end up at the top of the stack.
[u]Round 2[/u]
[b]p6.[/b] A circus owner will arrange $100$ fleas on a long string of beads, each flea on her own bead. Once arranged, the fleas start jumping using the following rules. Every second, each flea chooses the closest bead occupied by one or more of the other fleas, and then all fleas jump simultaneously to their chosen beads. If there are two places where a flea could jump, she jumps to the right. At the start, the circus owner arranged the fleas so that, after some time, they all gather on just two beads. What is the shortest amount of time it could take for this to happen?
[b]p7.[/b] The faraway land of Noetheria has $2016$ cities. There is a nonstop flight between every pair of cities. The price of a nonstop ticket is the same in both directions, but flights between different pairs of cities have different prices. Prove that you can plan a route of $2015$ consecutive flights so that each flight is cheaper than the previous one. It is permissible to visit the same city several times along the way.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Let $a_1, a_2, . . ., a_n$ be a finite sequence of $0$ and $1$. Under any two consecutive terms of this sequence $0$ is written if the digits are equal and $1$ is written otherwise. This way a new sequence of length $n -1$ is obtained.
By repeating this procedure $n - 1$ times one obtains a triangular table of $0$ and $1$. Find the maximum possible number of ones that can appear on this table
Consider infinite sequences $a_1,a_2,\dots$ of positive integers satisfying $a_1=1$ and $$a_n \mid a_k+a_{k+1}+\dots+a_{k+n-1}$$ for all positive integers $k$ and $n.$ For a given positive integer $m,$ find the maximum possible value of $a_{2m}.$
[i]Proposed by Krit Boonsiriseth[/i]
(a). Given any natural number \(N\), prove that there exists a strictly increasing sequence of \(N\) positive integers in harmonic progression.
(b). Prove that there cannot exist a strictly increasing infinite sequence of positive integers which is in harmonic progression.
Let $n$ be a positive integer. Compute, in terms of $n$, the number of sequences $(x_1,\ldots,x_{2n})$ with each $x_i\in\{0,1,2,3,4\}$ such that $x_1^2+\dots+x_{2n}^2$ is divisible by $5$.
[i]2020 CCA Math Bonanza Individual Round #13[/i]
Consider a sequence of words , consisting of the letters $A$ and $B$ .
The first word in the sequence is "$A$" . The k-th word i s obtained from the $(k-1)$-th by means of the following transformation : each $A$ is substituted by $AAB$ , and each $B$ is substituted by $A$. It is easily seen that every word is an initial part of the next word. The initial parts of these words coincide to give a sequence of letters $AABAABAAA BAABAAB...$
(a) In which place of this sequence is the $1000$-th letter $A$?
(b ) Prove that this sequence is not periodic.
(V . Galperin , Moscows)
There is the sequence of numbers $1, a_2, a_3, ...$ such that satisfies $1 \cdot a_2 \cdot a_3 \cdot ... \cdot a_n = n^2$, for every integer $n> 2$. Determine the value of $a_3 + a_5$.
Let $A = \frac{1 \cdot 3 \cdot 5\cdot ... \cdot (2n-1)}{2 \cdot 4 \cdot 6 \cdot ... \cdot (2n)}$ Prove that in the infinite sequence $A, 2A, 4A, 8A, ..., 2^k A, ….$ only integers will be observed, eventually.
Michael and James are playing a game where they alternate throwing darts at a simplified dartboard. Each dart throw is worth either 25 points or 50 points. They track the sequence of scores per throw (which is shared between them), and on the first time the three most recent scores sum to 125, the person who threw the last dart wins. On each throw, a given player has a $2/3$ chance of getting the score they aim for, and a $1/3$ chance of getting the other score. Suppose Michael goes first, and the first two throws are both 25. If both players use an optimal strategy, what is the probability Michael wins?
[i]Proposed by Michael Duncan[/i]
A grasshopper starts at the origin in the coordinate plane and makes a sequence of hops. Each hop has length $5$, and after each hop the grasshopper is at a point whose coordinates are both integers; thus, there are $12$ possible locations for the grasshopper after the first hop. What is the smallest number of hops needed for the grasshopper to reach the point $(2021,2021)$?
Let $h$ be a positive integer. The sequence $a_n$ is defined by $a_0 = 1$ and
\[a_{n+1} = \{\begin{array}{c} \frac{a_n}{2} \text{ if } a_n \text{ is even }\\\\a_n+h \text{ otherwise }.\end{array}\]
For example, $h = 27$ yields $a_1=28, a_2 = 14, a_3 = 7, a_4 = 34$ etc. For which $h$ is there an $n > 0$ with $a_n = 1$?
Let $a_0, a_1, a_2, ...$ be a sequence such that: $a_0 = 2$; each $a_n = 2$ or $3; a_n =$the number of $3$s between the $n$th and $n+1$th $2$ in the sequence. So the sequence starts: $233233323332332 ...$ . Show that we can find $\alpha$ such that $a_n = 2$ iff $n = [\alpha m]$ for some integer $m \geq 0$.
A regular polygon with $1000$ sides has the vertices colored in red, yellow or blue. A move consists in choosing to adjiacent vertices colored differently and coloring them in the third color. Prove that there is a sequence of moves after which all the vertices of the polygon will have the same color.
Marius Ghergu
An equilateral triangle is divided by lines, parallel to its sides, into equilateral triangles, all of the same size. One of the smaller triangles is black while the others are white. It is permitted to intersect simultaneously some small triangles with a line parallel to any side of the original triangle and to change the colour of each intersected small triangle from one colour to the other . Is it always possible to find a sequence of such operations so that the smaller triangles all become white?
For all $n \geq 1$, define $a_{n}$ to be the fraction $\frac{k}{2^n}$ such that $a_{n}$ is the closest to $\frac{1}{3}$ over all integer values of $k$.
Prove that the sequence $a_{1}, a_{2}, \cdots $satisfies the equation $2a_{i+2} = a_{i+1} + a_{i}$ for all $i \geq 1$.
Find the number of strictly increasing sequences of nonnegative integers with the following properties:
• The first term is $0$ and the last term is $12$. In particular, the sequence has at least two terms.
• Among any two consecutive terms, exactly one of them is even.
Let $a_1,a_2,a_3,\ldots$ be an infinite sequence where for all positive integers $i$, $a_i$ is chosen to be a random positive integer between $1$ and $2016$, inclusive. Let $S$ be the set of all positive integers $k$ such that for all positive integers $j<k$, $a_j\neq a_k$. (So $1\in S$; $2\in S$ if and only if $a_1\neq a_2$; $3\in S$ if and only if $a_1\neq a_3$ and $a_2\neq a_3$; and so on.) In simplest form, let $\dfrac{p}{q}$ be the expected number of positive integers $m$ such that $m$ and $m+1$ are in $S$. Compute $pq$.
Any positive integer $n$ can be written in the form $n = 2^aq$, where $a \ge 0$ and $q$ is odd. We call $q$ the [i]odd part[/i] of $n$. Define the sequence $a_0,a_1,...$ as follows: $a_0 = 2^{2011}-1$ and for $m > 0, a_{m+i}$ is the odd part of $3a_m + 1$. Find $a_{2011}$.
Given $n\ge 3$. consider a sequence $a_1,a_2,...,a_n$, if $(a_i,a_j,a_k)$ with i+k=2j (i<j<k) and $a_i+a_k\ne 2a_j$, we call such a triple a $NOT-AP$ triple. If a sequence has at least one $NOT-AP$ triple, find the least possible number of the $NOT-AP$ triple it contains.
Suppose a doubly infinite sequence of real numbers $. . . , a_{-2}, a_{-1}, a_0, a_1, a_2, . . .$ has the property that $$a_{n+3} =\frac{a_n + a_{n+1} + a_{n+2}}{3},$$ for all integers $n .$ Show that if this sequence is bounded (i.e., if there exists a number $R$ such that $|a_n| \leq R$ for all $n$), then $a_n$ has the same value for all $n.$
Consider infinite sequences $a_1,a_2,\dots$ of positive integers satisfying $a_1=1$ and $$a_n \mid a_k+a_{k+1}+\dots+a_{k+n-1}$$ for all positive integers $k$ and $n.$ For a given positive integer $m,$ find the maximum possible value of $a_{2m}.$
[i]Proposed by Krit Boonsiriseth[/i]
Determine the maximal length $L$ of a sequence $a_1,\dots,a_L$ of positive integers satisfying both the following properties:
[list=disc]
[*]every term in the sequence is less than or equal to $2^{2023}$, and
[*]there does not exist a consecutive subsequence $a_i,a_{i+1},\dots,a_j$ (where $1\le i\le j\le L$) with a choice of signs $s_i,s_{i+1},\dots,s_j\in\{1,-1\}$ for which \[s_ia_i+s_{i+1}a_{i+1}+\dots+s_ja_j=0.\]
[/list]
Let $n \ge 2$ be an integer and $f_1(x), f_2(x), \ldots, f_{n}(x)$ a sequence of polynomials with integer coefficients. One is allowed to make moves $M_1, M_2, \ldots $ as follows: in the $k$-th move $M_k$ one chooses an element $f(x)$ of the sequence with degree of $f$ at least $2$ and replaces it with $(f(x) - f(k))/(x-k)$. The process stops when all the elements of the sequence are of degree $1$. If $f_1(x) = f_2(x) = \cdots = f_n(x) = x^n + 1$, determine whether or not it is possible to make appropriate moves such that the process stops with a sequence of $n$ identical polynomials of degree 1.
Let $N$ be a positive integer, and consider an $N \times N$ grid. A [i]right-down path[/i] is a sequence of grid cells such that each cell is either one cell to the right of or one cell below the previous cell in the sequence. A [i]right-up path[/i] is a sequence of grid cells such that each cell is either one cell to the right of or one cell above the previous cell in the sequence.
Prove that the cells of the $N \times N$ grid cannot be partitioned into less than $N$ right-down or right-up paths. For example, the following partition of the $5 \times 5$ grid uses $5$ paths.
[asy]
size(4cm);
draw((5,-1)--(0,-1)--(0,-2)--(5,-2)--(5,-3)--(0,-3)--(0,-4)--(5,-4),gray+linewidth(0.5)+miterjoin);
draw((1,-5)--(1,0)--(2,0)--(2,-5)--(3,-5)--(3,0)--(4,0)--(4,-5),gray+linewidth(0.5)+miterjoin);
draw((0,0)--(5,0)--(5,-5)--(0,-5)--cycle,black+linewidth(2.5)+miterjoin);
draw((0,-1)--(3,-1)--(3,-2)--(1,-2)--(1,-4)--(4,-4)--(4,-3)--(2,-3)--(2,-2),black+linewidth(2.5)+miterjoin);
draw((3,0)--(3,-1),black+linewidth(2.5)+miterjoin);
draw((1,-4)--(1,-5),black+linewidth(2.5)+miterjoin);
draw((4,-3)--(4,-1)--(5,-1),black+linewidth(2.5)+miterjoin);
[/asy]
[i]Proposed by Zixiang Zhou, Canada[/i]
Let $S$ be the set of sequences of length 2018 whose terms are in the set $\{1, 2, 3, 4, 5, 6, 10\}$ and sum to 3860. Prove that the cardinality of $S$ is at most
\[2^{3860} \cdot \left(\frac{2018}{2048}\right)^{2018}.\]