Found problems: 85335
DMM Individual Rounds, 2013 (-14)
[b]p1.[/b] $p, q, r$ are prime numbers such that $p^q + 1 = r$. Find $p + q + r$.
[b]p2.[/b] $2014$ apples are distributed among a number of children such that each child gets a different number of apples. Every child gets at least one apple. What is the maximum possible number of children who receive apples?
[b]p3.[/b] Cathy has a jar containing jelly beans. At the beginning of each minute he takes jelly beans out of the jar. At the $n$-th minute, if $n$ is odd, he takes out $5$ jellies. If n is even he takes out $n$ jellies. After the $46$th minute there are only $4$ jellies in the jar. How many jellies were in the jar in the beginning?
[b]p4.[/b] David is traveling to Budapest from Paris without a cellphone and he needs to use a public payphone. He only has two coins with him. There are three pay-phones - one that never works, one that works half of the time, and one that always works. The first phone that David tries does not work. Assuming that he does not use the same phone again, what is the probability that the second phone that he uses will work?
[b]p5.[/b] Let $a, b, c, d$ be positive real numbers such that
$$a^2 + b^2 = 1$$
$$c^2 + d^2 = 1;$$
$$ad - bc =\frac17$$
Find $ac + bd$.
[b]p6.[/b] Three circles $C_A,C_B,C_C$ of radius $1$ are centered at points $A,B,C$ such that $A$ lies on $C_B$ and $C_C$, $B$ lies on $C_C$ and $C_A$, and $C$ lies on $C_A$ and $C_B$. Find the area of the region where $C_A$, $C_B$, and $C_C$ all overlap.
[b]p7.[/b] Two distinct numbers $a$ and $b$ are randomly and uniformly chosen from the set $\{3, 8, 16, 18, 24\}$. What is the probability that there exist integers $c$ and $d$ such that $ac + bd = 6$?
[b]p8.[/b] Let $S$ be the set of integers $1 \le N \le 2^{20}$ such that $N = 2^i + 2^j$ where $i, j$ are distinct integers. What is the probability that a randomly chosen element of $S$ will be divisible by $9$?
[b]p9.[/b] Given a two-pan balance, what is the minimum number of weights you must have to weigh any object that weighs an integer number of kilograms not exceeding $100$ kilograms?
[b]p10.[/b] Alex, Michael and Will write $2$-digit perfect squares $A,M,W$ on the board. They notice that the $6$-digit number $10000A + 100M +W$ is also a perfect square. Given that $A < W$, find the square root of the $6$-digit number.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2012 Flanders Math Olympiad, 1
Our class decides to have a alpha - beta - gamma tournament. This party game is always played in groups of three. Any possible combination of three players (three students or two students and the teacher) plays the game $1$ time. The player who wins gets $1$ point. The two losers get no points. At the end of the tournament, miraculously, all students have as many points. The teacher has $3$ points. How many students are there in our class?
2018 SIMO, Q1
Find all functions $f:\mathbb{N}\setminus\{1\} \rightarrow\mathbb{N}$ such that for all distinct $x,y\in \mathbb{N}$ with $y\ge 2018$, $$\gcd(f(x),y)\cdot \mathrm{lcm}(x,f(y))=f(x)f(y).$$
1967 IMO Shortlist, 5
Let $n$ be a positive integer. Find the maximal number of non-congruent triangles whose sides lengths are integers $\leq n.$
2004 Baltic Way, 17
Consider a rectangle with sidelengths 3 and 4, pick an arbitrary inner point on each side of this rectangle. Let $x, y, z$ and $u$ denote the side lengths of the quadrilateral spanned by these four points. Prove that $25 \leq x^2+y^2+z^2+u^2 \leq 50$.
1999 May Olympiad, 3
The first row of this table is filled with the numbers $1$ through $10$, in that order.
The second row is filled with the numbers from $1$ to $10$, in any order.
In each box of the third row the sum of the two numbers written above is written.
Is there a way to fill in the second row so that the ones digits of the numbers in the third row are all different?
[img]https://cdn.artofproblemsolving.com/attachments/8/5/41117d105cc880bf452fa46132c20f2167aa5b.png[/img]
2023 Chile Classification NMO Seniors, 2
There are 7 numbers on a board. The product of any four of them is divisible by 2023.
Prove that at least one of the numbers on the board is divisible by 119.
2015 Balkan MO Shortlist, C3
A chessboard $1000 \times 1000$ is covered by dominoes $1 \times 10$ that can be rotated. We don't know which is the cover, but we are looking for it. For this reason, we choose a few $N$ cells of the chessboard, for which we know the position of the dominoes that cover them.
Which is the minimum $N$ such that after the choice of $N$ and knowing the dominoed that cover them, we can be sure and for the rest of the cover?
(Bulgaria)
2012 Turkey Team Selection Test, 1
In a triangle $ABC,$ incircle touches the sides $BC, CA, AB$ at $D, E, F,$ respectively. A circle $\omega$ passing through $A$ and tangent to line $BC$ at $D$ intersects the line segments $BF$ and $CE$ at $K$ and $L,$ respectively. The line passing through $E$ and parallel to $DL$ intersects the line passing through $F$ and parallel to $DK$ at $P.$ If $R_1, R_2, R_3, R_4$ denotes the circumradius of the triangles $AFD, AED, FPD, EPD,$ respectively, prove that $R_1R_4=R_2R_3.$
2014 Online Math Open Problems, 10
Find the sum of the decimal digits of \[ \left\lfloor \frac{51525354555657\dots979899}{50} \right\rfloor. \] Here $\left\lfloor x \right\rfloor$ is the greatest integer not exceeding $x$.
[i]Proposed by Evan Chen[/i]
2019 Purple Comet Problems, 26
Let $D$ be a regular dodecahedron, which is a polyhedron with $20$ vertices, $30$ edges, and $12$ regular pentagon faces. A tetrahedron is a polyhedron with $4$ vertices, $6$ edges, and $4$ triangular faces. Find the number of tetrahedra with positive volume whose vertices are vertices of $D$.
[img]https://cdn.artofproblemsolving.com/attachments/c/d/44d11fa3326780941d0b6756fb2e5989c2dc5a.png[/img]
1962 Bulgaria National Olympiad, Problem 4
There are given a triangle and some internal point $P$. $x,y,z$ are distances from $P$ to the vertices $A,B$ and $C$. $p,q,r$ are distances from $P$ to the sides $BC,CA,AB$ respectively. Prove that:
$$xyz\ge(q+r)(r+p)(p+q).$$
2022 Iran MO (3rd Round), 6
Prove that among any $9$ distinct real numbers, there exist $4$ distinct numbers $a,b,c,d$ such that
$$(ac+bd)^2\ge\frac{9}{10}(a^2+b^2)(c^2+d^2)$$
2002 Tournament Of Towns, 4
The spectators are seated in a row with no empty places. Each is in a seat which does not match the spectator's ticket. An usher can order two spectators in adjacent seats to trade places unless one of them is already seated correctly. Is it true that from any initial arrangement, the spectators can be brought to their correct seats?
1985 IMO Shortlist, 6
Let $x_n = \sqrt[2]{2+\sqrt[3]{3+\cdots+\sqrt[n]{n}}}.$ Prove that
\[x_{n+1}-x_n <\frac{1}{n!} \quad n=2,3,\cdots\]
1975 Swedish Mathematical Competition, 1
$A$ is the point $(1,0)$, $L$ is the line $y = kx$ (where $k > 0$). For which points $P(t,0)$ can we find a point $Q$ on $L$ such that $AQ$ and $QP$ are perpendicular?
2001 Czech And Slovak Olympiad IIIA, 4
In a certain language there are $n$ letters. A sequence of letters is a word, if there are no two equal letters between two other equal letters. Find the number of words of the maximum length.
1936 Moscow Mathematical Olympiad, 024
Represent an arbitrary positive integer as an expression involving only $3$ twos and any mathematical signs.
(P. Dirac)
PEN P Problems, 13
Let $a_{1}=1$, $a_{2}=2$, $a_{3}$, $a_{4}$, $\cdots$ be the sequence of positive integers of the form $2^{\alpha}3^{\beta}$, where $\alpha$ and $\beta$ are nonnegative integers. Prove that every positive integer is expressible in the form \[a_{i_{1}}+a_{i_{2}}+\cdots+a_{i_{n}},\] where no summand is a multiple of any other.
2016 Azerbaijan Junior Mathematical Olympiad, 3
$65$ distinct natural numbers not exceeding $2016$ are given. Prove that among these numbers we can find four $a,b,c,d$ such that $a+b-c-d$ is divisible by $2016.$
2018 PUMaC Combinatorics A, 5
How many ways are there to color the $8$ regions of a three-set Venn Diagram with $3$ colors such that each color is used at least once? Two colorings are considered the same if one can be reached from the other by rotation and/or reflection.
2005 Junior Balkan Team Selection Tests - Romania, 16
Let $AB$ and $BC$ be two consecutive sides of a regular polygon with 9 vertices inscribed in a circle of center $O$. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of the radius perpendicular to $BC$. Find the measure of the angle $\angle OMN$.
2001 IMO Shortlist, 2
Let $a_0, a_1, a_2, \ldots$ be an arbitrary infinite sequence of positive numbers. Show that the inequality $1 + a_n > a_{n-1} \sqrt[n]{2}$ holds for infinitely many positive integers $n$.
1962 AMC 12/AHSME, 2
The expression $ \sqrt{\frac{4}{3}} - \sqrt{\frac{3}{4}}$ is equal to:
$ \textbf{(A)}\ \frac{\sqrt{3}}{6} \qquad
\textbf{(B)}\ \frac{-\sqrt{3}}{6} \qquad
\textbf{(C)}\ \frac{\sqrt{-3}}{6} \qquad
\textbf{(D)}\ \frac{5 \sqrt{3}}{6} \qquad
\textbf{(E)}\ 1$
2000 CentroAmerican, 1
Find all three-digit numbers $ abc$ (with $ a \neq 0$) such that $ a^{2}+b^{2}+c^{2}$ is a divisor of 26.