Found problems: 401
1984 AIME Problems, 11
A gardener plants three maple trees, four oak trees, and five birch trees in a row. He plants them in random order, each arrangement being equally likely. Let $\frac{m}{n}$ in lowest terms be the probability that no two birch trees are next to one another. Find $m + n$.
2017 District Olympiad, 4
An ant can move from a vertex of a cube to the opposite side of its diagonal only on the edges and the diagonals of the faces such that it doesn’t trespass a second time through the path made. Find the distance of the maximum journey this ant can make.
1990 IMO Shortlist, 13
An eccentric mathematician has a ladder with $ n$ rungs that he always ascends and descends in the following way: When he ascends, each step he takes covers $ a$ rungs of the ladder, and when he descends, each step he takes covers $ b$ rungs of the ladder, where $ a$ and $ b$ are fixed positive integers. By a sequence of ascending and descending steps he can climb from ground level to the top rung of the ladder and come back down to ground level again. Find, with proof, the minimum value of $ n,$ expressed in terms of $ a$ and $ b.$
1992 IMO Longlists, 15
Prove that there exist $78$ lines in the plane such that they have exactly $1992$ points of intersection.
2022-23 IOQM India, 19
Consider a string of $n$ $1's$. We wish to place some $+$ signs in between so that the sum is $1000$. For instance, if $n=190$, one may put $+$ signs so as to get $11$ ninety times and $1$ ten times , and get the sum $1000$. If $a$ is the number of positive integers $n$ for which it is possible to place $+$ signs so as to get the sum $1000$, then find the sum of digits of $a$.
2021 Bangladeshi National Mathematical Olympiad, 10
A positive integer $n$ is called [i]nice[/i] if it has at least $3$ proper divisors and it is equal to the sum of its three largest proper divisors. For example, $6$ is [i]nice[/i] because its largest three proper divisors are $3,2,1$ and $6=3+2+1$. Find the number of [i]nice[/i] integers not greater than $3000$.
2016 India Regional Mathematical Olympiad, 6
$ABC$ is an equilateral triangle with side length $11$ units. Consider the points $P_1,P_2, \dots, P_10$ dividing segment $BC$ into $11$ parts of unit length. Similarly, define $Q_1, Q_2, \dots, Q_10$ for the side $CA$ and $R_1,R_2,\dots, R_10$ for the side $AB$. Find the number of triples $(i,j,k)$ with $i,j,k \in \{1,2,\dots,10\}$ such that the centroids of triangles $ABC$ and $P_iQ_jR_k$ coincide.
2016 Fall CHMMC, 6
How many binary strings of length $10$ do not contain the substrings $101$ or $010$?
Russian TST 2014, P4
For a natural number $n{},$ determine the number of ordered pairs $(S,T)$ of subsets of $\{1,2,\ldots,n\}$ for which $s>|T|$ for any element $s\in S$ and $t>|S|$ for any element $t\in T.$
1970 IMO, 3
Given $100$ coplanar points, no three collinear, prove that at most $70\%$ of the triangles formed by the points have all angles acute.
2007 Stanford Mathematics Tournament, 9
Let $d_n$ denote the number of derangements of the integers $1, 2, \ldots n$ so that no integer $i$ is in the $i^{th}$ position. It is possible to write a recurrence relation $d_{n}=f(n)d_{n-1}+g(n)d_{n-2}$; what is $f(n)+g(n)$?
1973 IMO Shortlist, 4
Let $P$ be a set of $7$ different prime numbers and $C$ a set of $28$ different composite numbers each of which is a product of two (not necessarily different) numbers from $P$. The set $C$ is divided into $7$ disjoint four-element subsets such that each of the numbers in one set has a common prime divisor with at least two other numbers in that set. How many such partitions of $C$ are there ?
2004 Switzerland Team Selection Test, 6
Find all finite sequences $(x_0, x_1, \ldots,x_n)$ such that for every $j$, $0 \leq j \leq n$, $x_j$ equals the number of times $j$ appears in the sequence.
1987 IMO Longlists, 48
Find the number of partitions of the set $\{1, 2, \cdots, n\}$ into three subsets $A_1,A_2,A_3$, some of which may be empty, such that the following conditions are satisfied:
$(i)$ After the elements of every subset have been put in ascending order, every two consecutive elements of any subset have different parity.
$(ii)$ If $A_1,A_2,A_3$ are all nonempty, then in exactly one of them the minimal number is even .
[i]Proposed by Poland.[/i]
2021 Indonesia TST, C
Let $p$ be an odd prime. Determine the number of nonempty subsets from $\{1, 2, \dots, p - 1\}$ for which the sum of its elements is divisible by $p$.
2020 AMC 10, 5
How many distinguishable arrangements are there of $1$ brown tile, $1$ purple tile, $2$ green tiles, and $3$ yellow tiles in a row from left to right? (Tiles of the same color are indistinguishable)
$\textbf{(A)}\ 210\qquad\textbf{(B)}\ 420\qquad\textbf{(C)}\ 630\qquad\textbf{(D)}\ 840\qquad\textbf{(E)}\ 1050$
2013 Purple Comet Problems, 14
How many triangles appear in the diagram below?
[asy]
import graph;
size(4.4cm);
real labelscalefactor = 0.5;
pen dotstyle = black;
draw((-2,5)--(-2,1));
draw((-2,5)--(2,5));
draw((2,5)--(2,1));
draw((-2,1)--(2,1));
draw((0,5)--(0,1));
draw((-2,3)--(2,3));
draw((-1,5)--(-1,1));
draw((1,5)--(1,1));
draw((-2,2)--(2,2));
draw((-2,4)--(2,4));
draw((1,5)--(-2,2));
draw((-2,2)--(-1,1));
draw((-1,1)--(2,4));
draw((2,4)--(1,5));
draw((-1,5)--(-2,4));
draw((-2,4)--(1,1));
draw((1,1)--(2,2));
draw((2,2)--(-1,5));
[/asy]
2004 Germany Team Selection Test, 3
Let $f(k)$ be the number of integers $n$ satisfying the following conditions:
(i) $0\leq n < 10^k$ so $n$ has exactly $k$ digits (in decimal notation), with leading zeroes allowed;
(ii) the digits of $n$ can be permuted in such a way that they yield an integer divisible by $11$.
Prove that $f(2m) = 10f(2m-1)$ for every positive integer $m$.
[i]Proposed by Dirk Laurie, South Africa[/i]
2013 Harvard-MIT Mathematics Tournament, 8
In a game, there are three indistinguishable boxes; one box contains two red balls, one contains two blue balls, and the last contains one ball of each color. To play, Raj first predicts whether he will draw two balls of the same color or two of different colors. Then, he picks a box, draws a ball at random,
looks at the color, and replaces the ball in the same box. Finally, he repeats this; however, the boxes are not shuffled between draws, so he can determine whether he wants to draw again from the same box. Raj wins if he predicts correctly; if he plays optimally, what is the probability that he will win?
2019 India PRMO, 27
We will say that a rearrangement of the letters of a word has no [i]fixed letters[/i] if, when the rearrangement is placed directly below the word, no column has the same letter repeated. For instance $HBRATA$ is a rearragnement with no fixed letter of $BHARAT$. How many distinguishable rearrangements with no fixed letters does $BHARAT$ have? (The two $A$s are considered identical.)
2022-23 IOQM India, 24
Let $N$ be the number of ways of distributing $52$ identical balls into $4$ distinguishable boxes such that no box is empty and the difference between the number of balls in any two of the boxes is not a multiple of $6$ If $N=100a+b$, where $a,b$ are positive integers less than $100$, find $a+b.$
2021 Balkan MO Shortlist, C1
Let $\mathcal{A}_n$ be the set of $n$-tuples $x = (x_1, ..., x_n)$ with $x_i \in \{0, 1, 2\}$. A triple $x, y, z$ of distinct elements of $\mathcal{A}_n$ is called [i]good[/i] if there is some $i$ such that $\{x_i, y_i, z_i\} = \{0, 1, 2\}$. A subset $A$ of $\mathcal{A}_n$ is called [i]good[/i] if every three distinct elements of $A$ form a good triple.
Prove that every good subset of $\mathcal{A}_n$ has at most $2(\frac{3}{2})^n$ elements.
2015 Moldova Team Selection Test, 4
Consider a positive integer $n$ and $A = \{ 1,2,...,n \}$. Call a subset $X \subseteq A$ [i][b]perfect[/b][/i] if $|X| \in X$. Call a perfect subset $X$ [i][b]minimal[/b][/i] if it doesn't contain another perfect subset. Find the number of minimal subsets of $A$.
ICMC 4, 1
Let \(S\) be a set with 10 distinct elements. A set \(T\) of subsets of \(S\) (possibly containing the empty set) is called [i]union-closed[/i] if, for all \(A, B \in T\), it is true that \(A \cup B \in T\). Show that the number of union-closed sets \(T\) is less than \(2^{1023}\).
[i]Proposed by Tony Wang[/i]
2007 Purple Comet Problems, 12
If you alphabetize all of the distinguishable rearrangements of the letters in the word [b]PURPLE[/b], find the number $n$ such that the word [b]PURPLE [/b]is the $n$th item in the list.