Found problems: 85335
2007 Pre-Preparation Course Examination, 22
Prove that for any positive integer $n \geq 3$ there exist positive integers $a_1,a_2,\cdots , a_n$ such that
\[a_1a_2\cdots a_n \equiv a_i \pmod {a_i^2} \qquad \forall i \in \{1,2,\cdots ,n\}\]
1998 Harvard-MIT Mathematics Tournament, 7
Pyramid $EARLY$ has rectangular base $EARL$ and apex $Y$, and all of its edges are of integer length. The four edges from the apex have lengths $1, 4, 7, 8$ (in no particular order), and EY is perpendicular to $YR$. Find the area of rectangle $EARL$.
1993 APMO, 1
Let $ABCD$ be a quadrilateral such that all sides have equal length and $\angle{ABC} =60^o$. Let $l$ be a line passing through $D$ and not intersecting the quadrilateral (except at $D$). Let $E$ and $F$ be the points of intersection of $l$ with $AB$ and $BC$ respectively. Let $M$ be the point of intersection of $CE$ and $AF$.
Prove that $CA^2 = CM \times CE$.
2002 Italy TST, 1
A scalene triangle $ABC$ is inscribed in a circle $\Gamma$. The bisector of angle $A$ meets $BC$ at $E$. Let $M$ be the midpoint of the arc $BAC$. The line $ME$ intersects $\Gamma$ again at $D$. Show that the circumcentre of triangle $AED$ coincides with the intersection point of the tangent to $\Gamma$ at $D$ and the line $BC$.
2022 Taiwan TST Round 3, N
Let $a_1,a_2,a_3,\ldots$ be an infinite sequence of positive integers such that $a_{n+2m}$ divides $a_{n}+a_{n+m}$ for all positive integers $n$ and $m.$ Prove that this sequence is eventually periodic, i.e. there exist positive integers $N$ and $d$ such that $a_n=a_{n+d}$ for all $n>N.$
1990 Rioplatense Mathematical Olympiad, Level 3, 3
Let $ABCD$ be a trapezium with bases $AB$ and $CD$ such that $AB = 2 CD$. From $A$ the line $r$ is drawn perpendicular to $BC$ and from $B$ the line $t$ is drawn perpendicular to $AD$. Let $P$ be the intersection point of $r$ and $t$. From $C$ the line $s$ is drawn perpendicular to $BC$ and from $D$ the line $u$ perpendicular to $AD$. Let $Q$ be the intersection point of $s$ and $u$. If $R$ is the intersection point of the diagonals of the trapezium, prove that points $P, Q$ and $R$ are collinear.
2022 Durer Math Competition Finals, 10
The pair of positive integers $(a, b)$ is such that a does not divide $b$, $b$ does not divide a, both numbers are at most $100$, and they have the maximal possible number of common divisors. What is the largest possible value of $a \cdot· b$?
2002 Putnam, 2
Given any five points on a sphere, show that some four of them must lie on a closed hemisphere.
2001 Estonia Team Selection Test, 5
Find the exponent of $37$ in the representation of the number $111...... 11$ with $3\cdot 37^{2000}$ digits equals to $1$, as product of prime powers
I Soros Olympiad 1994-95 (Rus + Ukr), 10.3
Given a square board with dimensions $1 995 \times 1 995$. These cells are painted with black and white paints in checkerboard order like this. that the corner cells are black. Two black and one white cells were randomly cut out of the board. Prove that the rest of the board can be divided into rectangles of size $1 \times 2$ .
2007 May Olympiad, 4
A $7\times 7$ board has a lamp on each of its $49$ squares, which can be on or off.
The allowed operation is to choose $3$ consecutive cells of a row or a column that have two lamps neighboring each other on and the other off, and change the state of all three. Namely
[img]https://cdn.artofproblemsolving.com/attachments/e/b/28737b19c940ff5e1c98d05533c77069e990f5.png[/img]
Give a configuration of exactly $8$ lit lamps located in the first $4$ rows of the board such that, through a succession of permitted operations, a single lamp is lit on the board and that it is located in the last row. Show the sequence of operations used to achieve the goal.
2010 Malaysia National Olympiad, 4
A square $ABCD$ has side length $ 1$. A circle passes through the vertices of the square. Let $P, Q, R, S$ be the midpoints of the arcs which are symmetrical to the arcs $AB$, $BC$, $CD$, $DA$ when reflected on sides $AB$, $B$C, $CD$, $DA$, respectively. The area of square $PQRS$ is $a+b\sqrt2$, where $a$ and $ b$ are integers. Find the value of $a+b$.
[img]https://cdn.artofproblemsolving.com/attachments/4/3/fc9e1bd71b26cfd9ff076db7aa0a396ae64e72.png[/img]
2012 CHMMC Fall, 9
For a positive integer $n$, let $f(n)$ be equal to $n$ if there is an integer $x$ such that $x^2-n$ is divisible by $2^{12}$, and let $f(n)$ be $0$ otherwise. Determine the remainder when $$\sum^{2^{12}-1}_{n=0}f(n)$$ is divided by $2^{12}$.
2003 National Olympiad First Round, 31
Positive integers are written into squares of a infinite chessboard such that a number $n$ is written $n$ times. If the absolute differences of numbers written into any two squares having a common side is not greater than $k$, what is the least possible value of $k$?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ \text{None of the preceding}
$
2016 Harvard-MIT Mathematics Tournament, 22
On the Cartesian plane $\mathbb{R}^2$, a circle is said to be \textit{nice}
if its center is at the origin $(0,0)$ and it passes through at least one lattice point
(i.e. a point with integer coordinates).
Define the points $A = (20,15)$ and $B = (20,16)$. How many nice circles intersect the open segment $AB$?
For reference, the numbers $601$, $607$, $613$, $617$, $619$, $631$, $641$, $643$, $647$, $653$, $659$, $661$, $673$, $677$, $683$, $691$ are the only prime numbers between $600$ and $700$.
1992 Polish MO Finals, 3
Show that for real numbers $x_1, x_2, ... , x_n$ we have:
\[ \sum\limits_{i=1}^n \sum\limits_{j=1}^n \dfrac{x_ix_j}{i+j} \geq 0 \]
When do we have equality?
V Soros Olympiad 1998 - 99 (Russia), 9.1
It is known that each of the equations $x^2 + ax + b = 0$ and $x^2 + bx + a = 0$ has two different roots and these four roots form an arithmetic progression in some order. Find $a$ and $b$.
1945 Moscow Mathematical Olympiad, 092
Prove that for any positive integer $n\ge 2$ the following inequality holds:
$$\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{2n}>\frac{1}{2}$$
1974 IMO Longlists, 21
Let $M$ be a nonempty subset of $\mathbb Z^+$ such that for every element $x$ in $M,$ the numbers $4x$ and $\lfloor \sqrt x \rfloor$ also belong to $M.$ Prove that $M = \mathbb Z^+.$
2020 AMC 8 -, 10
Zara has a collection of $4$ marbles: an Aggie, a Bumblebee, a Steelie, and a Tiger. She wants to display them in a row on a shelf, but does not want to put the Steelie and the Tiger next to one another. In how many ways can she do this?
$\textbf{(A) }6 \qquad \textbf{(B) }8 \qquad \textbf{(C) }12 \qquad \textbf{(D) }18 \qquad \textbf{(E) }24$
2023 Poland - Second Round, 3
Given positive integers $k,n$ and a real number $\ell$, where $k,n \geq 1$. Given are also pairwise different positive real numbers $a_1,a_2,\ldots, a_k$. Let $S = \{a_1,a_2,\ldots,a_k, -a_1, -a_2,\ldots, -a_k\}$.
Let $A$ be the number of solutions of the equation
$$x_1 + x_2 + \ldots + x_{2n} = 0,$$
where $x_1,x_2,\ldots, x_{2n} \in S$. Let $B$ be the number of solutions of the equation
$$x_1 + x_2 + \ldots + x_{2n} = \ell,$$
where $x_1,x_2,\ldots,x_{2n} \in S$. Prove that $A \geq B$.
Solutions of an equation with only difference in the permutation are different.
2019 Romania National Olympiad, 4
A piece of rectangular paper $20 \times 19$, divided into four units, is cut into several square pieces, the cuts being along the sides of the unit squares. Such a square piece is called odd square if the length of its side is an odd number.
a) What is the minimum possible number of odd squares?
b) What is the smallest value that the sum of the perimeters of the odd squares can take?
1941 Putnam, B5
A car is being driven so that its wheels, all of radius $a$ feet, have an angular velocity of $\omega$ radians per second.
A particle is thrown off from the tire of one of these wheels, where it is supposed that $a \omega^{2} >g$. Neglecting the resistance of the air, show that the maximum height above the roadway which the particle can reach is
$$\frac{(a \omega+g \omega^{-1})^{2}}{2g}.$$
2006 Estonia Team Selection Test, 6
Denote by $d(n)$ the number of divisors of the positive integer $n$. A positive integer $n$ is called highly divisible if $d(n) > d(m)$ for all positive integers $m < n$.
Two highly divisible integers $m$ and $n$ with $m < n$ are called consecutive if there exists no highly divisible integer $s$ satisfying $m < s < n$.
(a) Show that there are only finitely many pairs of consecutive highly divisible
integers of the form $(a, b)$ with $a\mid b$.
(b) Show that for every prime number $p$ there exist infinitely many positive highly divisible integers $r$ such that $pr$ is also highly divisible.
1973 Polish MO Finals, 1
Prove that every polynomial is a difference of two increasing polynomials.