Found problems: 85335
2010 IMO Shortlist, 8
Given six positive numbers $a,b,c,d,e,f$ such that $a < b < c < d < e < f.$ Let $a+c+e=S$ and $b+d+f=T.$ Prove that
\[2ST > \sqrt{3(S+T)\left(S(bd + bf + df) + T(ac + ae + ce) \right)}.\]
[i]Proposed by Sung Yun Kim, South Korea[/i]
2022 Baltic Way, 11
Let $ABC$ be a triangle with circumcircle $\Gamma$ and circumcentre $O$. The circle with centre on the line $AB$ and passing through the points $A$ and $O$ intersects $\Gamma$ again in $D$. Similarly, the circle with centre on the line $AC$ and passing through the points $A$ and $O$ intersects $\Gamma$ again in $E$. Prove that $BD$ is parallel with $CE$.
2005 Brazil National Olympiad, 5
Let $ABC$ be a triangle with all angles $\leq 120^{\circ}$. Let $F$ be the Fermat point of triangle $ABC$, that is, the interior point of $ABC$ such that $\angle AFB = \angle BFC = \angle CFA = 120^\circ$. For each one of the three triangles $BFC$, $CFA$ and $AFB$, draw its Euler line - that is, the line connecting its circumcenter and its centroid.
Prove that these three Euler lines pass through one common point.
[i]Remark.[/i] The Fermat point $F$ is also known as the [b]first Fermat point[/b] or the [b]first Toricelli point[/b] of triangle $ABC$.
[i]Floor van Lamoen[/i]
2017 Iran Team Selection Test, 6
In the unit squares of a transparent $1 \times 100$ tape, numbers $1,2,\cdots,100$ are written in the ascending order.We fold this tape on it's lines with arbitrary order and arbitrary directions until we reach a $1 \times1$ tape with $100$ layers.A permutation of the numbers $1,2,\cdots,100$ can be seen on the tape, from the top to the bottom.
Prove that the number of possible permutations is between $2^{100}$ and $4^{100}$.
([i]e.g.[/i] We can produce all permutations of numbers $1,2,3$ with a $1\times3$ tape)
[i]Proposed by Morteza Saghafian[/i]
2012 AMC 10, 10
How many ordered pairs of positive integers $(M,N)$ staisfy the equation $\frac{M}{6}=\frac{6}{N}$?
$ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10 $
2015 BMT Spring, 5
Let $x$ and $y$ be real numbers satisfying the equation $x^2-4x+y^2+3=0$. If the maximum and minimum values of $x^2+y^2$ are $M$ and $m$ respectively, compute the numerical value of $M-m$.
1994 Romania TST for IMO, 2:
Let $n$ be a positive integer. Find the number of polynomials $P(x)$ with coefficients in $\{0, 1, 2, 3\}$ for which $P(2) = n$.
2012 NIMO Summer Contest, 6
When Eva counts, she skips all numbers containing a digit divisible by 3. For example, the first ten numbers she counts are 1, 2, 4, 5, 7, 8, 11, 12, 14, 15. What is the $100^{\text{th}}$ number she counts?
[i]Proposed by Eugene Chen[/i]
2012 Albania National Olympiad, 5
Let $ABC$ be a triangle where $AC\neq BC$. Let $P$ be the foot of the altitude taken from $C$ to $AB$; and let $V$ be the orthocentre, $O$ the circumcentre of $ABC$, and $D$ the point of intersection between the radius $OC$ and the side $AB$. The midpoint of $CD$ is $E$.
a) Prove that the reflection $V'$ of $V$ in $AB$ is on the circumcircle of the triangle $ABC$.
b) In what ratio does the segment $EP$ divide the segment $OV$?
2005 MOP Homework, 7
Let $x_{1,1}$, $x_{2,1}$, ..., $x_{n,1}$, $n \ge 2$, be a sequence of integers and assume that not all $x_{i,1}$ are equal. For $k \ge 2$, if sequence $\{x_{i,k}\}^n_{i=1}$ is defined, we define sequence $\{x_{i,k+1}\}^n_{i=1}$ as \[x_{i,k+1}=\frac{1}{2}(x_{i,k}+x_{i+1,k}),\] for $i=1, 2, ..., n$, (where $x_{n+1,k}=x_{1,k}$). Show that if $n$ is odd then there exist indices $j$ and $k$ such that $x_{j,k}$ is not an integer.
1966 IMO Longlists, 53
Prove that in every convex hexagon of area $S$ one can draw a diagonal that cuts off a triangle of area not exceeding $\frac{1}{6}S.$
JBMO Geometry Collection, 2014
Consider an acute triangle $ABC$ of area $S$. Let $CD \perp AB$ ($D \in AB$), $DM \perp AC$ ($M \in AC$) and $DN \perp BC$ ($N \in BC$). Denote by $H_1$ and $H_2$ the orthocentres of the triangles $MNC$, respectively $MND$. Find the area of the quadrilateral $AH_1BH_2$ in terms of $S$.
2005 Oral Moscow Geometry Olympiad, 2
On a circle with diameter $AB$, lie points $C$ and $D$. $XY$ is the diameter passing through the midpoint $K$ of the chord $CD$. Point $M$ is the projection of point $X$ onto line $AC$, and point $N$ is the projection of point $Y$ on line $BD$. Prove that points $M, N$ and $K$ are collinear.
(A. Zaslavsky)
Russian TST 2018, P2
Let $\mathcal{F}$ be a finite family of subsets of some set $X{}$. It is known that for any two elements $x,y\in X$ there exists a permutation $\pi$ of the set $X$ such that $\pi(x)=y$, and for any $A\in\mathcal{F}$ \[\pi(A):=\{\pi(a):a\in A\}\in\mathcal{F}.\]A bear and crocodile play a game. At a move, a player paints one or more elements of the set $X$ in his own color: brown for the bear, green for the crocodile. The first player to fully paint one of the sets in $\mathcal{F}$ in his own color loses. If this does not happen and all the elements of $X$ have been painted, it is a draw. The bear goes first. Prove that he doesn't have a winning strategy.
2020 LMT Fall, A17
There are $n$ ordered tuples of positive integers $(a,b,c,d)$ that satisfy $$a^2+ b^2+ c^2+ d^2=13 \cdot 2^{13}.$$ Let these ordered tuples be $(a_1,b_1,c_1,d_1), (a_2,b_2,c_2,d_2), \dots, (a_n,b_n,c_n,d_n)$. Compute $\sum_{i=1}^{n}(a_i+b_i+c_i+d_i)$.
[i]Proposed by Kaylee Ji[/i]
2023 AIME, 4
The sum of all positive integers $m$ for which $\tfrac{13!}{m}$ is a perfect square can be written as $2^{a}3^{b}5^{c}7^{d}11^{e}13^{f}$, where $a, b, c, d, e,$ and $f$ are positive integers. Find $a+b+c+d+e+f$.
VII Soros Olympiad 2000 - 01, 11.1
Prove that for any $a$ the function $y (x) = \cos (\cos x) + a \cdot \sin (\sin x)$ is periodic.
Find its smallest period in terms of $a$.
2022 Harvard-MIT Mathematics Tournament, 2
Rectangle $R_0$ has sides of lengths $3$ and $4$. Rectangles $R_1$, $R_2$, and $R_3$ are formed such that:
$\bullet$ all four rectangles share a common vertex $P$,
$\bullet$ for each $n = 1, 2, 3$, one side of $R_n$ is a diagonal of $R_{n-1}$,
$\bullet$ for each $n = 1, 2, 3$, the opposite side of $R_n$ passes through a vertex of $R_{n-1}$ such that the center of $R_n$ is located counterclockwise of the center of $R_{n-1}$ with respect to $P$.
Compute the total area covered by the union of the four rectangles.
[img]https://cdn.artofproblemsolving.com/attachments/3/1/e9edd39e60e4a4defdb127b93b19ab0d0f443c.png[/img]
2006 IberoAmerican, 2
For n real numbers $a_{1},\, a_{2},\, \ldots\, , a_{n},$ let $d$ denote the difference between the greatest and smallest of them and $S = \sum_{i<j}\left |a_i-a_j \right|.$ Prove that \[(n-1)d\le S\le\frac{n^{2}}{4}d\] and find when each equality holds.
2024 Mexican Girls' Contest, 3
Let \( ABC \) be a triangle and \( D \) the foot of the altitude from \( A \). Let \( M \) be a point such that \( MB = MC \). Let \( E \) and \( F \) be the intersections of the circumcircle of \( BMD \) and \( CMD \) with \( AD \). Let \( G \) and \( H \) be the intersections of \( MB \) and \( MC \) with \( AD \). Prove that \( EG = FH \).
2015 AIME Problems, 1
The expressions $A=1\times2+3\times4+5\times6+\cdots+37\times38+39$ and $B=1+2\times3+4\times5+\cdots+36\times37+38\times39$ are obtained by writing multiplication and addition operators in an alternating pattern between successive integers. Find the positive difference between integers $A$ and $B$.
2020 Bangladesh Mathematical Olympiad National, Problem 3
Let $R$ be the set of all rectangles centered at the origin and with perimeter $1$ (the center of a rectangle is the intersection point of its two diagonals). Let $S$ be a region that contains all of the rectangles in $R$ (region $A$ contains region $B$, if $B$ is completely inside of $A$). The minimum possible area of $S$ has the form $\pi a$, where $a$ is a real number. Find $1/a$.
2021 MOAA, 1
What is $2021+20+21+2+0+2+1$?
[i]Proposed by Nathan Xiong[/i]
2017 Brazil Undergrad MO, 2
Let $a$ and $b$ be fixed positive integers. Show that the set of primes that divide at least one of the terms of the sequence $a_n = a \cdot 2017^n + b \cdot 2016^n$ is infinite.
2017-2018 SDML (Middle School), 2
A circle and a square are drawn on the plane so that they overlap. Together, the two shapes cover an area of $329$ square units. The area common to both shapes is $101$ square units. The area of the circle is $234$ square units. What is the perimeter of the square in units?
$\mathrm{(A) \ } 14 \qquad \mathrm{(B) \ } 48 \qquad \mathrm {(C) \ } 56 \qquad \mathrm{(D) \ } 64 \qquad \mathrm{(E) \ } 196$