Found problems: 85335
1998 Tournament Of Towns, 4
Twelve places have been arranged at a round table for members of the Jury, with a name tag at each place . Professor K. being absent-minded instead of occupying his place, sits down at the next place (clockwise) . Each of the other Jury members in turn either occupies the place assigned to this member or, if it has been already occupied, sits down at the first free place in the clockwise order. The resulting seating arrangement depends on the order in which the Jury members come to the table. How many different seating arrangements of this kind are possible?
(A Shapovalov)
1958 AMC 12/AHSME, 30
If $ xy \equal{} b$ and $ \frac{1}{x^2} \plus{} \frac{1}{y^2} \equal{} a$, then $ (x \plus{} y)^2$ equals:
$ \textbf{(A)}\ (a \plus{} 2b)^2\qquad
\textbf{(B)}\ a^2 \plus{} b^2\qquad
\textbf{(C)}\ b(ab \plus{} 2)\qquad
\textbf{(D)}\ ab(b \plus{} 2)\qquad
\textbf{(E)}\ \frac{1}{a} \plus{} 2b$
2010 Regional Olympiad of Mexico Northeast, 1
Sofia has $5$ pieces of paper on a table. He takes some of the pieces, cuts each one into $5$ little pieces, and puts them back on the table. She repeats this procedure several times until she gets tired. Could Sofia end up with $2010$ pieces on the table?
2014 Junior Balkan Team Selection Tests - Romania, 1
Let $a, b, c, d$ be positive real numbers so that $abc+bcd+cda+dab = 4$.
Prove that $a^2 + b^2 + c^2 + d^2 \ge 4$
2010 Contests, 3
Let $A$ be an infinite set of positive integers. Find all natural numbers $n$ such that for each $a \in A$,
\[a^n + a^{n-1} + \cdots + a^1 + 1 \mid a^{n!} + a^{(n-1)!} + \cdots + a^{1!} + 1.\]
[i]Proposed by Milos Milosavljevic[/i]
2021 Mediterranean Mathematics Olympiad, 2
For every sequence $p_1<p_2<\cdots<p_8$ of eight prime numbers, determine the largest integer $N$ for which the following equation has no solution in positive integers $x_1,\ldots,x_8$:
$$p_1\, p_2\, \cdots\, p_8
\left( \frac{x_1}{p_1}+ \frac{x_2}{p_2}+ ~\cdots~ +\frac{x_8}{p_8} \right)
~~=~~ N $$
[i]Proposed by Gerhard Woeginger, Austria[/i]
2022 MIG, 3
Real numbers $w$, $x$, $y$, and $z$ satisfy $w+x+y = 3$, $x+y+z = 4,$ and $w+x+y+z = 5$. What is the value of $x+y$?
$\textbf{(A) }-\frac{1}{2}\qquad\textbf{(B) }1\qquad\textbf{(C) }\frac{3}{2}\qquad\textbf{(D) }2\qquad\textbf{(E) }3$
1964 AMC 12/AHSME, 22
Given parallelogram $ABCD$ with $E$ the midpoint of diagonal $BD$. Point $E$ is connected to a point $F$ in $DA$ so that $DF=\frac{1}{3}DA$. What is the ratio of the area of triangle $DFE$ to the area of quadrilateral $ABEF$?
$ \textbf{(A)}\ 1:2 \qquad\textbf{(B)}\ 1:3 \qquad\textbf{(C)}\ 1:5 \qquad\textbf{(D)}\ 1:6 \qquad\textbf{(E)}\ 1:7 $
1996 VJIMC, Problem 2
Let $\{a_n\}^\infty_{n=0}$ be the sequence of integers such that $a_0=1$, $a_1=1$, $a_{n+2}=2a_{n+1}-2a_n$. Decide whether
$$a_n=\sum_{k=0}^{\left\lfloor\frac n2\right\rfloor}\binom n{2k}(-1)^k.$$
2010 Cono Sur Olympiad, 2
On a line, $44$ points are marked and numbered $1, 2, 3,…,44$ from left to right. Various crickets jump around the line. Each starts at point $1$, jumping on the marked points and ending up at point $44$. In addition, each cricket jumps from a marked point to another marked point with a greater number.
When all the crickets have finished jumping, it turns out that for pair $i, j$ with ${1}\leq{i}<{j}\leq{44}$, there was a cricket that jumped directly from point $i$ to point $j$, without visiting any of the points in between the two.
Determine the smallest number of crickets such that this is possible.
1996 Estonia Team Selection Test, 3
Each face of a cube is divided into $n^2$ equal squares. The vertices of the squares are called [i]nodes[/i], so each face has $(n+1)^2$ nodes.
$(a)$ If $n=2$,does there exist a closed polygonal line whose links are sids of the squares and which passes through each node exactly once?
$(b)$ Prove that, for each $n$, such a polygonal line divides the surface area of the cube into two equal parts
2004 IberoAmerican, 1
It is given a 1001*1001 board divided in 1*1 squares. We want to amrk m squares in such a way that:
1: if 2 squares are adjacent then one of them is marked.
2: if 6 squares lie consecutively in a row or column then two adjacent squares from them are marked.
Find the minimun number of squares we most mark.
2017 IMO Shortlist, A7
Let $a_0,a_1,a_2,\ldots$ be a sequence of integers and $b_0,b_1,b_2,\ldots$ be a sequence of [i]positive[/i] integers such that $a_0=0,a_1=1$, and
\[
a_{n+1} =
\begin{cases}
a_nb_n+a_{n-1} & \text{if $b_{n-1}=1$} \\
a_nb_n-a_{n-1} & \text{if $b_{n-1}>1$}
\end{cases}\qquad\text{for }n=1,2,\ldots.
\]
for $n=1,2,\ldots.$ Prove that at least one of the two numbers $a_{2017}$ and $a_{2018}$ must be greater than or equal to $2017$.
2022/2023 Tournament of Towns, P1
There are 2023 dice on the table. For 1 dollar, one can pick any dice and put it back on any of its four (other than top or bottom) side faces. How many dollars at a minimum will guarantee that all the dice have been repositioned to show equal number of dots on top faces?
[i]Egor Bakaev[/i]
DMM Team Rounds, 1998
[b][b]p1.[/b][/b] Find the perimeter of a regular hexagon with apothem $3$.
[b]p2.[/b] Concentric circles of radius $1$ and r are drawn on a circular dartboard of radius $5$. The probability that a randomly thrown dart lands between the two circles is $0.12$. Find $r$.
[b]p3.[/b] Find all ordered pairs of integers $(x, y)$ with $0 \le x \le 100$, $0 \le y \le 100$ satisfying $$xy = (x - 22) (y + 15) .$$
[b]p4.[/b] Points $A_1$,$A_2$,$...$,$A_{12}$ are evenly spaced around a circle of radius $1$, but not necessarily in order. Given that chords $A_1A_2$, $A_3A_4$, and $A_5A_6$ have length $2$ and chords $A_7A_8$ and $A_9A_{10}$ have length $2 sin (\pi / 12)$, find all possible lengths for chord $A_{11}A_{12}$.
[b]p5.[/b] Let $a$ be the number of digits of $2^{1998}$, and let $b$ be the number of digits in $5^{1998}$. Find $a + b$.
[b]p6.[/b] Find the volume of the solid in $R^3$ defined by the equations
$$x^2 + y^2 \le 2$$
$$x + y + |z| \le 3.$$
[b]p7.[/b] Positive integer $n$ is such that $3n$ has $28$ positive divisors and $4n$ has $36$ positive divisors. Find the number of positive divisors of $n$.
[b]p8.[/b] Define functions $f$ and $g$ by $f (x) = x +\sqrt{x}$ and $g (x) = x + 1/4$. Compute $$g(f(g(f(g(f(g(f(3)))))))).$$
(Your answer must be in the form $a + b \sqrt{ c}$ where $a$, $b$, and $c$ are rational.)
[b]p9.[/b] Sequence $(a_1, a_2,...)$ is defined recursively by $a_1 = 0$, $a_2 = 100$, and $a_n = 2a_{n-1}-a_{n-2}-3$. Find the greatest term in the sequence $(a_1, a_2,...)$.
[b]p10.[/b] Points $X = (3/5, 0)$ and $Y = (0, 4/5)$ are located on a Cartesian coordinate system. Consider all line segments which (like $\overline{XY}$ ) are of length 1 and have one endpoint on each axis. Find the coordinates of the unique point $P$ on $\overline{XY}$ such that none of these line segments (except $\overline{XY}$ itself) pass through $P$.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2019 Peru Cono Sur TST, P3
Let $A$ be the number of ways in which the set $\{ 1, 2, . . . , n\}$ can be partitioned into non-empty subsets. Let $B$ be the number of ways in which the set $\{ 1, 2, . . . , n, n + 1 \}$ can be partitioned into non-empty subsets such that consecutive numbers belong to distinct subsets. Partitions that differ only in the order of the subsets are considered equal. Prove that $A = B$.
2012 Today's Calculation Of Integral, 823
Let $C$ be the curve expressed by $x=\sin t,\ y=\sin 2t\ \left(0\leq t\leq \frac{\pi}{2}\right).$
(1) Express $y$ in terms of $x$.
(2) Find the area of the figure $D$ enclosed by the $x$-axis and $C$.
(3) Find the volume of the solid generated by a rotation of $D$ about the $y$-axis.
2011 ELMO Shortlist, 8
Let $n>1$ be an integer and $a,b,c$ be three complex numbers such that $a+b+c=0$ and $a^n+b^n+c^n=0$. Prove that two of $a,b,c$ have the same magnitude.
[i]Evan O'Dorney.[/i]
2025 Harvard-MIT Mathematics Tournament, 7
Compute the number of ways to arrange $3$ copies of each of the $26$ lowercase letters of the English alphabet such that for any two distinct letters $x_1$ and $x_2,$ the number of $x_2$’s between the first and second occurrences of $x_1$ equals the number of $x_2$’s between the second and third occurrences of $x_1.$
2024 New Zealand MO, 2
Prove the following inequality $$\dfrac{6}{2024^3} < \left(1-\dfrac{3}{4}\right)\left(1-\dfrac{3}{5}\right)\left(1-\dfrac{3}{6}\right)\left(1-\dfrac{3}{7}\right)\ldots\left(1-\dfrac{3}{2025}\right).$$
EMCC Team Rounds, 2013
[b]p1.[/b] Determine the number of ways to place $4$ rooks on a $4 \times 4$ chessboard such that:
(a) no two rooks attack one another, and
(b) the main diagonal (the set of squares marked $X$ below) does not contain any rooks.
[img]https://cdn.artofproblemsolving.com/attachments/e/e/e3aa96de6c8ed468c6ef3837e66a0bce360d36.png[/img]
The rooks are indistinguishable and the board cannot be rotated. (Two rooks attack each other if they are in the same row or column.)
[b]p2.[/b] Seven students, numbered $1$ to $7$ in counter-clockwise order, are seated in a circle. Fresh Mann has 100 erasers, and he wants to distribute them to the students, albeit unfairly. Starting with person $ 1$ and proceeding counter-clockwise, Fresh Mann gives $i$ erasers to student $i$; for example, he gives $ 1$ eraser to student $ 1$, then $2$ erasers to student $2$, et cetera. He continues around the circle until he does not have enough erasers to give to the next person. At this point, determine the number of erasers that Fresh Mann has.
[b]p3.[/b] Let $ABC$ be a triangle with $AB = AC = 17$ and $BC = 24$. Approximate $\angle ABC$ to the nearest multiple of $10$ degrees.
[b]p4.[/b] Define a sequence of rational numbers $\{x_n\}$ by $x_1 =\frac35$ and for $n \ge 1$, $x_{n+1} = 2 - \frac{1}{x_n}$ . Compute the product $x_1x_2x_3... x_{2013}$.
[b]p5.[/b] In equilateral triangle $ABC$, points $P$ and $R$ lie on segment $AB$, points $I$ and $M$ lie on segment $BC$, and points $E$ and $S$ lie on segment $CA$ such that $PRIMES$ is a equiangular hexagon. Given that $AB = 11$, $PR = 2$, $IM = 3$, and $ES = 5$, compute the area of hexagon $PRIMES$.
[b]p6.[/b] Let $f(a, b) = \frac{a^2}{a+b}$ . Let $A$ denote the sum of $f(i, j)$ over all pairs of integers $(i, j)$ with $1 \le i < j \le 10$; that is,
$$A = (f(1, 2) + f(1, 3) + ...+ f(1, 10)) + (f(2, 3) + f(2, 4) +... + f(2, 10)) +... + f(9, 10).$$
Similarly, let $B$ denote the sum of $f(i, j)$ over all pairs of integers $(i, j)$ with $1 \le j < i \le 10$, that is, $$B = (f(2, 1) + f(3, 1) + ... + f(10, 1)) + (f(3, 2) + f(4, 2) +... + f(10, 2)) +... + f(10, 9).$$ Compute $B - A$.
[b]p7.[/b] Fresh Mann has a pile of seven rocks with weights $1, 1, 2, 4, 8, 16$, and $32$ pounds and some integer X between $1$ and $64$, inclusive. He would like to choose a set of the rocks whose total weight is exactly $X$ pounds. Given that he can do so in more than one way, determine the sum of all possible values of $X$. (The two $1$-pound rocks are indistinguishable.)
[b]p8.[/b] Let $ABCD$ be a convex quadrilateral with $AB = BC = CA$. Suppose that point $P$ lies inside the quadrilateral with $AP = PD = DA$ and $\angle PCD = 30^o$. Given that $CP = 2$ and $CD = 3$, compute $CA$.
[b]p9.[/b] Define a sequence of rational numbers $\{x_n\}$ by $x_1 = 2$, $x_2 = \frac{13}{2}$ , and for $n \ge 1$, $x_{n+2} = 3 -\frac{3}{x_{n+1}}+\frac{1}{x_nx_{n+1}}$. Compute $x_{100}$.
[b]p10.[/b] Ten prisoners are standing in a line. A prison guard wants to place a hat on each prisoner. He has two colors of hats, red and blue, and he has $10$ hats of each color. Determine the number of ways in which the prison guard can place hats such that among any set of consecutive prisoners, the number of prisoners with red hats and the number of prisoners with blue hats differ by at most $2$.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2008 Putnam, A5
Let $ n\ge 3$ be an integer. Let $ f(x)$ and $ g(x)$ be polynomials with real coefficients such that the points $ (f(1),g(1)),(f(2),g(2)),\dots,(f(n),g(n))$ in $ \mathbb{R}^2$ are the vertices of a regular $ n$-gon in counterclockwise order. Prove that at least one of $ f(x)$ and $ g(x)$ has degree greater than or equal to $ n\minus{}1.$
2010 Saudi Arabia BMO TST, 3
Let $(a_n )_{n \ge o}$ and $(b_n )_{n \ge o}$ be sequences defined by $a_{n+2} = a_{n+1}+ a_n$ , $n = 0 , 1 , . .. $, $a_0 = 1$, $a_1 = 2$, and $b_{n+2} = b_{n+1} + b_n$ , $n = 0 , 1 , . . .$, $b_0 = 2$, $b_1 = 1$. How many integers do the sequences have in common?
2011 Portugal MO, 3
A set of $n$ lights, numbered $1$ to $n$, are initially off. At every moment, it is possible to perform one of the following operations:
$\bullet$ change the state of lamp $1$,
$\bullet$ change the state of lamp $2$, as long as lamp $1$ is on,
$\bullet$ change the state of lamp $k > 2$, as long as lamp $k - 1$ is on and all lamps $1, . . . , k - 2$ are off.
It shows that it is possible, after a certain number of operations, to have only the lamp left on.
2018 India Regional Mathematical Olympiad, 6
Let $ABC$ be an acute-angled triangle with $AB<AC$. Let $I$ be the incentre of triangle $ABC$, and let $D,E,F$ be the points where the incircle touches the sides $BC,CA,AB,$ respectively. Let $BI,CI$ meet the line $EF$ at $Y,X$ respectively. Further assume that both $X$ and $Y$ are outside the triangle $ABC$. Prove that
$\text{(i)}$ $B,C,Y,X$ are concyclic.
$\text{(ii)}$ $I$ is also the incentre of triangle $DYX$.