Found problems: 5923
We are given an infinite sequence of $ 1$'s and $ 2$'s with the following properties:
(1) The first element of the sequence is $ 1$.
(2) There are no two consecutive $ 2$'s or three consecutive $ 1$'s.
(3) If we replace consecutive $ 1$'s by a single $ 2$, leave the single $ 1$'s alone, and delete the original $ 2$'s, then we recover the original sequence.
How many $ 2$'s are there among the first $ n$ elements of the sequence?
[i]P. P. Palfy[/i]
Prove that there exists an unique sequence $ \left( c_n \right)_{n\ge 1} $ of real numbers from the interval $ (0,1) $ such that$$ \int_0^1 \frac{dx}{1+x^m} =\frac{1}{1+c_m^m } , $$ for all natural numbers $ m, $ and calculate $ \lim_{k\to\infty } kc_k^k. $
[i]Radu Pop[/i]
[b]p1.[/b] Compute $1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55$.
[b]p2.[/b] Given that $a$, $b$, and $c$ are positive integers such that $a+b = 9$ and $bc = 30$, find the minimum possible value of $a + c$.
[b]p3.[/b] Points $X$ and $Y$ lie outside regular pentagon $ABCDE$ such that $ABX$ and $DEY$ are equilateral triangles. Find the degree measure of $\angle XCY$ .
[b]p4.[/b] Let $N$ be the product of the positive integer divisors of $8!$, including itself. The largest integer power of $2$ that divides $N$ is $2^k$. Compute $k$.
[b]p5.[/b] Let $A=(-20, 22)$, $B = (k, 0)$, and $C = (202, 2)$ be points on the coordinate plane. Given that $\angle ABC = 90^o$, find the sum of all possible values of $k$.
[b]p6.[/b] Tej is typing a string of $L$s and $O$s that consists of exactly $7$ $L$s and $4$ $O$s. How many different strings can he type that do not contain the substring ‘$LOL$’ anywhere? A substring is a sequence of consecutive letters contained within the original string.
[b]p7.[/b] How many ordered triples of integers $(a, b, c)$ satisfy both $a+b-c = 12$ and $a^2+b^2-c^2 = 24$?
[b]p8.[/b] For how many three-digit base-$7$ numbers $\overline{ABC}_7$ does $\overline{ABC}_7$ divide $\overline{ABC}_{10}$? (Note: $\overline{ABC}_D$ refers to the number whose digits in base $D$ are, from left to right, $A$, $B$, and $C$; for example, $\overline{123}_4$ equals $27$ in base ten).
[b]p9.[/b] Natasha is sitting on one of the $35$ squares of a $5$-by-$7$ grid of squares. Wanda wants to walk through every square on the board exactly once except the one Natasha is on, starting and ending on any $2$ squares she chooses, such that from any square she can only go to an adjacent square (two squares are adjacent if they share an edge). How many squares can Natasha choose to sit on such that Wanda cannot go on her walk?
[b]p10.[/b] In triangle $ABC$, $AB = 13$, $BC = 14$, and $CA = 15$. Point $P$ lies inside $ABC$ and points $D,E$, and $F$ lie on sides $BC$, $CA$, and $AB$, respectively, so that $PD \perp BC$, $PE \perp CA$, and $PF \perp AB$. Given that $PD$, $PE$, and $PF$ are all integers, find the sum of all possible distinct values of $PD \cdot PE \cdot PF$.
[b]p11.[/b] A palindrome is a positive integer which is the same when read forwards or backwards. Find the sum of the two smallest palindromes that are multiples of $137$.
[b]p12.[/b] Let $P(x) = x^2+px+q$ be a quadratic polynomial with positive integer coefficients. Compute the least possible value of p such that 220 divides p and the equation $P(x^3) = P(x)$ has at least four distinct integer solutions.
[b]p13.[/b] Everyone at a math club is either a truth-teller, a liar, or a piggybacker. A truth-teller always tells the truth, a liar always lies, and a piggybacker will answer in the style of the previous person who spoke (i.e., if the person before told the truth, they will tell the truth, and if the person before lied, then they will lie). If a piggybacker is the first one to talk, they will randomly either tell the truth or lie. Four seniors in the math club were interviewed and here was their conversation:
Neil: There are two liars among us.
Lucy: Neil is a piggybacker.
Kevin: Excluding me, there are more truth-tellers than liars here.
Neil: Actually, there are more liars than truth-tellers if we exclude Kevin.
Jacob: One plus one equals three.
Define the base-$4$ number $M = \overline{NLKJ}_4$, where each digit is $1$ for a truth-teller, $2$ for a piggybacker, and $3$ for a liar ($N$ corresponds to Neil, $L$ to Lucy, $K$ corresponds to Kevin, and $J$ corresponds to Jacob). What is the sum of all possible values of $M$, expressed in base $10$?
[b]p14.[/b] An equilateral triangle of side length $8$ is tiled by $64$ equilateral triangles of unit side length to form a triangular grid. Initially, each triangular cell is either living or dead. The grid evolves over time under the following rule: every minute, if a dead cell is edge-adjacent to at least two living cells, then that cell becomes living, and any living cell remains living. Given that every cell in the grid eventually evolves to be living, what is the minimum possible number of living cells in the initial grid?
[b]p15.[/b] In triangle $ABC$, $AB = 7$, $BC = 11$, and $CA = 13$. Let $\Gamma$ be the circumcircle of $ABC$ and let $M$, $N$, and $P$ be the midpoints of minor arcs $BC$ , $CA$, and $AB$ of $\Gamma$, respectively. Given that $K$ denotes the area of $ABC$ and $L$ denotes the area of the intersection of $ABC$ and $MNP$, the ratio $L/K$ can be written as $a/b$ , where $a$ and $b$ are relatively prime positive integers. Compute $a + b$.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Suppose that $a$ is real number. Sequence $a_1,a_2,a_3,....$ satisfies
$$a_1=a, a_{n+1} =
\begin{cases}
a_n - \frac{1}{a_n}, & a_n\ne 0 \\
0, & a_n=0
\end{cases}
(n=1,2,3,..)$$
Find all possible values of $a$ such that $|a_n|<1$ for all positive integer $n$.
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]
We are given $n$ mass points of equal mass in space. We define a sequence of points $O_1,O_2,O_3,\ldots $ as follows: $O_1$ is an arbitrary point (within the unit distance of at least one of the $n$ points); $O_2$ is the centre of gravity of all the $n$ given points that are inside the unit sphere centred at $O_1$;$O_3$ is the centre of gravity of all of the $n$ given points that are inside the unit sphere centred at $O_2$; etc. Prove that starting from some $m$, all points $O_m,O_{m+1},O_{m+2},\ldots$ coincide.
Let $(a_n)_{n\ge 1}$ and $(b_n)_{n\ge 1}$ be two sequences of natural numbers such that $a_{n+1} = na_n + 1, b_{n+1} = nb_n - 1$ for every $n\ge 1$. Show that these two sequences can have only a finite number of terms in common.
Let $s,t$ be given nonzero integers, $(x,y)$ be any (ordered) pair of integers. A sequence of moves is performed as follows: per move $(x,y)$ changes to $(x+t, y-s)$. The pair (x,y) is said to be [i]good [/i] if after some (may be, zero) number of moves described a pair of integers arises that are not relatively prime.
a) Determine whether $(s,t)$ is itself a good pair;
bj Prove that for any nonzero $s$ and $t$ there is a pair $(x,y)$ which is not good.
Given the polynomial sequence $(p_{n}(x))$ satisfying $p_{1}(x)=x^{2}-1$, $p_{2}(x)=2x(x^{2}-1)$, and $p_{n+1}(x)p_{n-1}(x)=(p_{n}(x)^{2}-(x^{2}-1)^{2}$, for $n\geq 2$, let $s_{n}$ be the sum of the absolute values of the coefficients of $p_{n}(x)$. For each $n$, find a non-negative integer $k_{n}$ such that $2^{-k_{n}}s_n$ is odd.
The sum of all terms of a finite arithmetical progression of integers is a power of two. Prove that the number of terms is also a power of two.
Let $ a,$ $ b,$ $ c,$ $ d,$ and $ e$ be five consecutive terms in an arithmetic sequence, and suppose that $ a \plus{} b \plus{} c \plus{} d \plus{} e \equal{} 30.$ Which of the following can be found?
$ \textbf{(A)}\ a \qquad \textbf{(B)}\ b \qquad \textbf{(C)}\ c \qquad \textbf{(D)}\ d \qquad \textbf{(E)}\ e$
Let $a$ be a non-negative real number and a sequence $(u_n)$ defined as:
$u_1=6,u_{n+1} = \frac{2n+a}{n} + \sqrt{\frac{n+a}{n}u_n+4}, \forall n \ge 1$
a) With $a=0$, prove that there exist a finite limit of $(u_n)$ and find that limit
b) With $a \ge 0$, prove that there exist a finite limit of $(u_n)$
The sequences $ a_0, a_1, \ldots$ and $ b_0, b_1, \ldots$ are defined for $ n \equal{} 0, 1, 2, \ldots$ by the equalities
\[ a_0 \equal{} \frac {\sqrt {2}}{2}, \quad a_{n \plus{} 1} \equal{} \frac {\sqrt {2}}{2} \cdot \sqrt {1 \minus{} \sqrt {1 \minus{} a^2_n}}
\]
and
\[ b_0 \equal{} 1, \quad b_{n \plus{} 1} \equal{} \frac {\sqrt {1 \plus{} b^2_n} \minus{} 1}{b_n}
\]
Prove the inequalities for every $ n \equal{} 0, 1, 2, \ldots$
\[ 2^{n \plus{} 2} a_n < \pi < 2^{n \plus{} 2} b_n.
\]
Consider the function $ f(x) \equal{} \frac {2x^3 \minus{} 3}{3x^2 \minus{} 1}$.
$ 1.$ Prove that there is a continuous function $ g(x)$ on $ \mathbb{R}$ satisfying $ f(g(x)) \equal{} x$ and $ g(x) > x$ for all real $ x$.
$ 2.$ Show that there exists a real number $ a > 1$ such that the sequence $ \{a_n\}$, $ n \equal{} 1, 2, \ldots$, defined as follows $ a_0 \equal{} a$, $ a_{n \plus{} 1} \equal{} f(a_n)$, $ \forall n\in\mathbb{N}$ is periodic with the smallest period $ 1995$.
Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.
The princess wishes to have a bracelet with $r$ rubies and $s$ emeralds arranged in such order that there exist two jewels on the bracelet such that starting with these and enumerating the jewels in the same direction she would obtain identical sequences of jewels. Prove that it is possible to fulfill the princess’s wish if and only if $r$ and $s$ have a common divisor.
Let $ABC$ be a triangle which it not right-angled. De fine a sequence of triangles $A_iB_iC_i$, with $i \ge 0$, as follows: $A_0B_0C_0$ is the triangle $ABC$ and, for $i \ge 0$, $A_{i+1},B_{i+1},C_{i+1}$ are the reflections of the orthocentre of triangle $A_iB_iC_i$ in the sides $B_iC_i$,$C_iA_i$,$A_iB_i$, respectively.
Assume that $\angle A_m = \angle A_n$ for some distinct natural numbers $m,n$. Prove that $\angle A = 60^{\circ}$.
Let $a_0, a_1,...$ be a sequence of non-negative integers and $b_0, b_1,... $ be a sequence of non-negative integers defined by the following rule:
$b_i=gcd(a_i, a_{i+1})$ for every $i=>0$
Is it possible every positive integer to occur exactly once in the sequence $b_0, b_1,... $
Given a positive real number $t$ , determine the sets $A$ of real numbers containing $t$ , for which there exists a set $B$ of real numbers depending on $A$ , $|B| \geq 4$ , such that the elements of the set $AB =\{ ab \mid a\in A , b \in B \}$ form a finite arithmetic progression .
Let $(a_n)_{n=1}^{\infty}$ be an integer sequence such that $a_{n+48} \equiv a_n \pmod {35}$ for every $n \geq 1$. Let $i$ and $j$ be the least numbers satisfying the conditions $a_{n+i} \equiv a_n \pmod {5}$ and $a_{n+j} \equiv a_n \pmod {7}$ for every $n\geq 1$. Which one below cannot be an $(i,j)$ pair?
$
\textbf{(A)}\ (16,4)
\qquad\textbf{(B)}\ (3,16)
\qquad\textbf{(C)}\ (8,6)
\qquad\textbf{(D)}\ (1,48)
\qquad\textbf{(E)}\ (16,18)
$
Let $N \geq 2$ be an integer, and let $\mathbf a$ $= (a_1, \ldots, a_N)$ and $\mathbf b$ $= (b_1, \ldots b_N)$ be sequences of non-negative integers. For each integer $i \not \in \{1, \ldots, N\}$, let $a_i = a_k$ and $b_i = b_k$, where $k \in \{1, \ldots, N\}$ is the integer such that $i-k$ is divisible by $n$. We say $\mathbf a$ is $\mathbf b$-[i]harmonic[/i] if each $a_i$ equals the following arithmetic mean: \[a_i = \frac{1}{2b_i+1} \sum_{s=-b_i}^{b_i} a_{i+s}.\]
Suppose that neither $\mathbf a $ nor $\mathbf b$ is a constant sequence, and that both $\mathbf a$ is $\mathbf b$-[i]harmonic[/i] and $\mathbf b$ is $\mathbf a$-[i]harmonic[/i].
Prove that at least $N+1$ of the numbers $a_1, \ldots, a_N,b_1, \ldots, b_N$ are zero.
Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the
sequence $1$, $2$, $\dots$ , $n$ satisfying
$$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$.
Proposed by United Kingdom
[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 $M = \{(a,b,c,d)|a,b,c,d \in \{1,2,3,4\} \text{ and } abcd > 1\}$. For each $n\in \{1,2,\dots, 254\}$, the sequence $(a_1, b_1, c_1, d_1)$, $(a_2, b_2, c_2, d_2)$, $\dots$, $(a_{255}, b_{255},c_{255},d_{255})$ contains each element of $M$ exactly once and the equality \[|a_{n+1} - a_n|+|b_{n+1} - b_n|+|c_{n+1} - c_n|+|d_{n+1} - d_n| = 1\] holds. If $c_1 = d_1 = 1$, find all possible values of the pair $(a_1,b_1)$.
The increasing sequence $3, 15, 24, 48, \ldots$ consists of those positive multiples of 3 that are one less than a perfect square. What is the remainder when the 1994th term of the sequence is divided by 1000?