Found problems: 5923
Let $a_n$ be the number of unordered sets of three distinct bijections $f, g, h : \{1, 2, ..., n\} \to \{1, 2, ..., n\}$ such that the composition of any two of the bijections equals the third. What is the largest value in the sequence $a_1, a_2, ...$ which is less than $2021$?
Let $m$ be a fixed integer greater than $1$. The sequence $x_0$, $x_1$, $x_2$, $\ldots$ is defined as follows:
\[x_i = \begin{cases}2^i&\text{if }0\leq i \leq m - 1;\\\sum_{j=1}^mx_{i-j}&\text{if }i\geq m.\end{cases}\]
Find the greatest $k$ for which the sequence contains $k$ consecutive terms divisible by $m$ .
[i]Proposed by Marcin Kuczma, Poland[/i]
Define the sequence $\{x_i\}_{i \ge 0}$ by $x_0 = x_1 = x_2 = 1$ and $x_k = \frac{x_{k-1}+x_{k-2}+1}{x_{k-3}}$ for $k > 2$. Find $x_{2013}$.
Two sequences $(x_n)_{n\in N}$ and $(y_n)_{n\in N}$ are defined recursively as follows:
$x_0 = 2015$ and $x_{n+1} =\left \lfloor x_n \cdot \frac{y_{n+1}}{y_{n-1}} \right \rfloor$ for all $n \ge 0$,
$y_0 = 307$ and $y_{n+1} = y_n + 1$ for all $n \ge 0$.
Compute $\lim_{n\to \infty} \frac{x_n}{(y_n)^2}$.
Let $y_{1}, y_{2}, y_{3},\ldots$ be a sequence such that $y_{1}=1$ and, for $k>0,$ is defined by the relationship:
\[y_{2k}=\begin{cases}2y_{k}& \text{if}~k~ \text{is even}\\ 2y_{k}+1 & \text{if}~k~ \text{is odd}\end{cases}\]\[y_{2k+1}=\begin{cases}2y_{k}& \text{if}~k~ \text{is odd}\\ 2y_{k}+1 & \text{if}~k~ \text{is even}\end{cases}\]Show that the sequence takes on every positive integer value exactly once.
Prove that there exists a real number $C > 1$ with the following property.
Whenever $n > 1$ and $a_0 < a_1 < a_2 <\cdots < a_n$ are positive integers such that $\frac{1}{a_0},\frac{1}{a_1} \cdots \frac{1}{a_n}$ form an arithmetic progression, then $a_0 > C^n$.
Let $ a_0$, $ a_1$, $ a_2$, $ \ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $ \gcd (a_i, a_{i \plus{} 1}) > a_{i \minus{} 1}$. Prove that $ a_n\ge 2^n$ for all $ n\ge 0$.
[i]Proposed by Morteza Saghafian, Iran[/i]
The internal angles of quadrilateral $ABCD$ form an arithmetic progression. Triangles $ABD$ and $DCB$ are similar with $\angle DBA=\angle DCB$ and $\angle ADB=\angle CBD$. Moreover, the angles in each of these two triangles also form an arithmetic progression. In degrees, what is the largest possible sum of the two largest angles of $ABCD$?
${\textbf{(A)}\ 210\qquad\textbf{(B)}\ 220\qquad\textbf{(C)}\ 230\qquad\textbf{(D}}\ 240\qquad\textbf{(E)}\ 250$
Let us choose arbitrarily $n$ vertices of a regular $2n$-gon and color them red. The remaining vertices are colored blue. We arrange all red-red distances into a non-decreasing sequence and do the same with the blue-blue distances. Prove that the sequences are equal.
For any real $ x$, let $ \lfloor x\rfloor$ be the largest integer that is not more than $ x$. Given a sequence of positive integers $ a_1,a_2,a_3,\ldots$ such that $ a_1>1$ and
\[ \left\lfloor\frac{a_1\plus{}1}{a_2}\right\rfloor\equal{}\left\lfloor\frac{a_2\plus{}1}{a_3}\right\rfloor\equal{}\left\lfloor\frac{a_3\plus{}1}{a_4}\right\rfloor\equal{}\cdots\]
Prove that
\[ \left\lfloor\frac{a_n\plus{}1}{a_{n\plus{}1}}\right\rfloor\leq1\]
holds for every positive integer $ n$.
Right triangle $\triangle{}ABC$ has $\angle{}C=90^{\circ{}}$. A fly is trapped inside $\triangle{}ABC$. It starts at point $D$, the foot of the altitude from $C$ to $\overline{AB}$, and then makes a (finite) sequence of moves. In each move, it flies in a direction parallel to either $\overline{AC}$ or $\overline{BC}$; upon reaching a leg of the triangle, it then flies to a point on $\overline{AB}$ in a direction parallel to $\overline{CD}$. For example, on its first move, the fly can move to either of the points $Y_1$ or $Y_2$, as shown.
[asy]
pair C = (0,0);
pair A = (0,4);
pair B = (5,0);
draw(C--A);
draw(C--B);
draw(B--A);
dot(A);
dot(B);
dot(C);
label("$A$",A,NW);
label("$C$",C,SW);
label("$B$",B,SE);
pair D = foot(C,A,B);
draw(C--D,dotted);
label("$D$",D,NE);
dot(D);
draw(rightanglemark(A,C,B));
pair B1 = foot(D,C,B);
draw(D--B1,dotted);
pair A1 = foot(D,A,C);
draw(D--A1,dotted);
pair Y1 = foot(A1,A,D);
draw(A1--Y1,dotted);
dot(Y1);
label("$Y_1$",Y1,NE);
pair Y2 = foot(B1,D,B);
draw(B1--Y2,dotted);
dot(Y2);
label("$Y_2$",Y2,NE);
draw(rightanglemark(C,A1,D));
draw(rightanglemark(C,B1,D));
draw(rightanglemark(B1,Y2,D));
draw(rightanglemark(A1,Y1,D));
draw(rightanglemark(C,D,A));
[/asy]
Let $P$ and $Q$ be distinct points on $\overline{AB}$. Show that the fly can reach some point on $\overline{PQ}$.
Let $n$ be a positive integer.
Consider sequences $a_0, a_1, ..., a_k$ and $b_0, b_1,,..,b_k$ such that $a_0 = b_0 = 1$ and $a_k = b_k = n$ and such that for all $i$ such that $1 \le i \le k $, we have that $(a_i, b_i)$ is either equal to $(1 + a_{i-1}, b_{i-1})$ or $(a_{i-1}; 1 + b_{i-1})$.
Consider for $1 \le i \le k$ the number $c_i = \begin{cases} a_i \,\,\, if \,\,\, a_i = a_{i-1} \\
b_i \,\,\, if \,\,\, b_i = b_{i-1}\end{cases}$
Show that $c_1 + c_2 + ... + c_k = n^2 - 1$.
Given that $n> 2$ is a positive integer and a sequence of positive integers $a_1 <a_2 <...<a_n$. In
the subsets of the set $\{1,2,..., n\} $, there a subset $X$ such that $| \sum_{i \notin X} a_i -\sum_{i \in X} a_i |$ is the smallest .
Prove that there exists a sequence of positive integers $0<b_1 <b_2 <...<b_n$ such that $\sum_{i \notin X} b_i= \sum_{i \in X} b_i$.
In case this doesn't make sense, have a look at [url=https://drive.google.com/file/d/1xoBhJlG0xHwn6zAAA7AZDoaAqzZue-73/view]original wording in Vietnamese[/url].
[b]a)[/b] Provide an example of a sequence $ \left( a_n \right)_{n\ge 1} $ of positive real numbers whose series converges, and has the property that each member (sequence) of the family of sequences $ \left(\left( n^{\alpha } a_n \right)_{n\ge 1}\right)_{\alpha >0} $ is unbounded.
[b]b)[/b] Let $ \left( b_n \right)_{n\ge 1} $ be a sequence of positive real numbers, having the property that
$$ nb_{n+1}\leqslant b_1+b_2+\cdots +b_n, $$
for any natural number $ n. $ Prove that the following relations are equivalent:
$\text{(i)} $ there exists a convergent member (series) of the family of series $ \left( \sum_{i=1}^{\infty } b_i^{\beta } \right)_{\beta >0} $
$ \text{(ii)} $ there exists a member (sequence) of the family of sequences $ \left(\left( n^{\beta } b_n \right)_{n\ge 1}\right)_{\beta >0} $ that is convergent to $ 0. $
[i]Eugen Păltănea[/i]
Given a polynomial $P(x)=a_{d}x^{d}+ \ldots +a_{2}x^{2}+a_{0}$ with positive integers for coefficients and degree $d\geq 2$. Consider the sequence defined by $$b_{1}=a_{0} ,b_{n+1}=P(b_{n}) $$ for $n \geq 1$ . Prove that for all $n \geq 2$ there exists a prime $p$ such that $p$ divides $b_{n}$ but does not divide $b_{1}b_{2} \ldots b_{n-1}$.
Let the sequence $(a_{n})$ be defined by $a_{1} = t$ and $a_{n+1} = 4a_{n}(1 - a_{n})$ for $n \geq 1$. How many possible values of t are there, if $a_{1998} = 0$?
Calculate $ \lim_{n\to\infty } \left( e^{1+1/2+1/3+\cdots +1/n+1/(n+1)} -e^{1+1/2+1/3+\cdots +1/n} \right) . $
[u]Round 1[/u]
[b]p1.[/b] In a chemical lab there are three vials: one that can hold $1$ oz of fluid, another that can hold $2$ oz, and a third that can hold $3$ oz. The first is filled with grape juice, the second with sulfuric acid, and the third with water. There are also $3$ empty vials in the cupboard, also of sizes $1$ oz, $2$ oz, and $3$ oz. In order to save the world with grape-flavored acid, James Bond must make three full bottles, one of each size, filled with a mixture of all three liquids so that each bottle has the same ratio of juice to acid to water. How can he do this, considering he was silly enough not to bring any equipment?
[b]p2.[/b] Twelve people, some are knights and some are knaves, are sitting around a table. Knaves always lie and knights always tell the truth. At some point they start up a conversation. The first person says, “There are no knights around this table.” The second says, “There is at most one knight at this table.” The third – “There are at most two knights at the table.” And so on until the $12$th says, “There are at most eleven knights at the table.” How many knights are at the table? Justify your answer.
[b]p3.[/b] Aquaman has a barrel divided up into six sections, and he has placed a red herring in each. Aquaman can command any fish of his choice to either ‘jump counterclockwise to the next sector’ or ‘jump clockwise to the next sector.’ Using a sequence of exactly $30$ of these commands, can he relocate all the red herrings to one sector? If yes, show how. If no, explain why not.
[img]https://cdn.artofproblemsolving.com/attachments/0/f/956f64e346bae82dee5cbd1326b0d1789100f3.png[/img]
[b]p4.[/b] Is it possible to place $13$ integers around a circle so that the sum of any $3$ adjacent numbers is exactly $13$?
[b]p5.[/b] Two girls are playing a game. The first player writes the letters $A$ or $B$ in a row, left to right, adding one letter on her turn. The second player switches any two letters after each move by the first player (the letters do not have to be adjacent), or does nothing, which also counts as a move. The game is over when each player has made $2011$ moves. Can the second player plan her moves so that the resulting letters form a palindrome? (A palindrome is a sequence that reads the same forward and backwards, e.g. $AABABAA$.)
[u]Round 2[/u]
[b]p6.[/b] Eight students participated in a math competition. There were eight problems to solve. Each problem was solved by exactly five people. Show that there are two students who solved all eight problems between them.
[b]p7.[/b] There are $3n$ checkers of three different colors: $n$ red, $n$ green and $n$ blue. They were used to randomly fill a board with $3$ rows and $n$ columns so that each square of the board has one checker on it. Prove that it is possible to reshuffle the checkers within each row so that in each column there are checkers of all three colors. Moving checkers to a different row is not allowed.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Define the sequence $a_1, a_2, a_3, \ldots$ by $a_1 = 1$ and, for $n > 1$,
\[a_n = a_{\lfloor n/2 \rfloor} + a_{\lfloor n/3 \rfloor} + \ldots + a_{\lfloor n/n \rfloor} + 1.\]
Prove that there are infinitely many $n$ such that $a_n \equiv n \pmod{2^{2010}}$.
Let $(x_n)_{n\geq1}$ be a sequence that verifies: $$x_1=1, \quad x_2=7, \quad x_{n+1}=x_n+3x_{n-1}, \forall n \geq 2.$$
Prove that for every prime number $p$ the number $x_p-1$ is divisible by $3p.$
For any given integers $m,n$ such that $2\leq m<n$ and $(m,n)=1$. Determine the smallest positive integer $k$ satisfying the following condition: for any $m$-element subset $I$ of $\{1,2,\cdots,n\}$ if $\sum_{i\in I}i> k$, then there exists a sequence of $n$ real numbers $a_1\leq a_2 \leq \cdots \leq a_n$ such that
$$\frac1m\sum_{i\in I} a_i>\frac1n\sum_{i=1}^na_i$$
[b]p1.[/b] (a) Find three positive integers $a, b, c$ whose sum is $407$, and whose product (when written in base $10$) ends in six $0$'s.
(b) Prove that there do NOT exist positive integers $a, b, c$ whose sum is $407$ and whose product ends in seven $0$'s.
[b]p2.[/b] Three circles, each of radius $r$, are placed on a plane so that the center of each circle lies on a point of intersection of the other two circles. The region $R$ consists of all points inside or on at least one of these three circles. Find the area of $R$.
[b]p3.[/b] Let $f_1(x) = a_1x^2+b_1x+c_1$, $f_2(x) = a_2x^2+b_2x+c_2$ and $f_3(x) = a_3x^2+b_3x+c_3$ be the equations of three parabolas such that $a_1 > a_2 > a-3$. Prove that if each pair of parabolas intersects in exactly one point, then all three parabolas intersect in a common point.
[b]p4.[/b] Gigafirm is a large corporation with many employees.
(a) Show that the number of employees with an odd number of acquaintances is even.
(b) Suppose that each employee with an even number of acquaintances sends a letter to each of these acquaintances. Each employee with an odd number of acquaintances sends a letter to each non-acquaintance. So far, Leslie has received $99$ letters. Prove that Leslie will receive at least one more letter.
(Notes: "acquaintance" and "non-acquaintance" refer to employees of Gigaform. If $A$ is acquainted with $B$, then $B$ is acquainted with $A$. However, no one is acquainted with himself.)
[b]p5.[/b] (a) Prove that for every positive integer $N$, if $A$ is a subset of the numbers $\{1, 2, ...,N\}$ and $A$ has size at least $2N/3 + 1$, then $A$ contains a three-term arithmetic progression (i.e., there are positive integers $a$ and $b$ so that all three of the numbers $a$,$a + b$, and $a + 2b$ are elements of $A$).
(b) Show that if $A$ is a subset of $\{1, 2, ..., 3500\}$ and $A$ has size at least $2003$, then $A$ contains a three-term arithmetic progression.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Let $a_1,a_2,...$ be a sequence of positive integers with $a_1=2$. For each $n \ge 1$, $a_{n+1}$ is the biggest prime divisor of $a_1a_2...a_n+1$.
Prove that the sequence does not contain numbers $5$ and $11$.
Let $ q_{0}, q_{1}, \cdots$ be a sequence of integers such that
a) for any $ m > n$, $ m \minus{} n$ is a factor of $ q_{m} \minus{} q_{n}$,
b) item $ |q_n| \le n^{10}$ for all integers $ n \ge 0$.
Show that there exists a polynomial $ Q(x)$ satisfying $ q_{n} \equal{} Q(n)$ for all $ n$.
Let $a_1<a_2< \cdots$ be a strictly increasing sequence of positive integers. Suppose there exist $N$ such that for all $n>N$, $$a_{n+1}\mid a_1+a_2+\cdots+a_n$$ Prove that there exist $M$ such that $a_{m+1}=2a_m$ for all $m>M$.
[i]Proposed by Ivan Chan Kai Chin[/i]