Found problems: 85335
2020 BMT Fall, 9
For any point $(x, y)$ with $0\le x < 1$ and $0 \le y < 1$, Jenny can perform a shuffle on that point, which takes the point to $(\{3x + y\} ,\{x + 2y\})$ where $\{a\}$ denotes the fractional or decimal part of $a$ (so for example,$\{\pi\} = \pi - 3 = 0.1415...$). How many points $p$ are there such that after $3$ shuffles on $p$, $p$ ends up in its original position?
2010 AMC 10, 12
Logan is constructing a scaled model of his town. The city's water tower stands $ 40$ meters high, and the top portion is a sphere that holds $ 100,000$ liters of water. Logan's miniature water tower holds $ 0.1$ liters. How tall, in meters, should Logan make his tower?
$ \textbf{(A)}\ 0.04\qquad \textbf{(B)}\ \frac{0.4}{\pi}\qquad \textbf{(C)}\ 0.4\qquad \textbf{(D)}\ \frac{4}{\pi}\qquad \textbf{(E)}\ 4$
2012 Indonesia TST, 3
Let $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$ be positive reals such that
\[a_1 + b_1 = a_2 + b_2 = \ldots + a_n + b_n\]
and
\[\sqrt[n]{\dfrac{a_1a_2\ldots a_n}{b_1b_2\ldots b_n}} \ge n.\]
Prove that
\[\sqrt[n]{\dfrac{a_1a_2\ldots a_n}{b_1b_2\ldots b_n}} \ge \dfrac{a_1+a_2+\ldots+a_n}{b_1+b_2+\ldots+b_n}.\]
2021 USAMO, 1
Rectangles $BCC_1B_2,$ $CAA_1C_2,$ and $ABB_1A_2$ are erected outside an acute triangle $ABC.$ Suppose that \[\angle BC_1C+\angle CA_1A+\angle AB_1B=180^{\circ}.\] Prove that lines $B_1C_2,$ $C_1A_2,$ and $A_1B_2$ are concurrent.
2007 Hanoi Open Mathematics Competitions, 11
How many possible values are there for the sum
a + b + c + d if a; b; c; d are positive integers and abcd = 2007:
2024 Iran MO (2nd Round), 3
In a triangle $ABC$ the incenter, the $B$-excenter and the $C$-excenter are $I, K$ and $L$, respectively. The perpendiculars at $B$ and $C$ to $BC$ intersect the lines $AC$ and $AB$ at $E$ and $F$, respectively. Prove that the circumcircles of $AEF, FIL, EIK$ concur.
2010 Contests, 4
Let $n$ be a positive integer. Find the smallest positive integer $k$ with the property that for any colouring nof the squares of a $2n$ by $k$ chessboard with $n$ colours, there are $2$ columns and $2$ rows such that the $4$ squares in their intersections have the same colour.
2018 India Regional Mathematical Olympiad, 5
In a cyclic quadrilateral $ABCD$ with circumcenter $O$, the diagonals $AC$ and $BD$ intersect at $X$. Let the circumcircles of triangles $AXD$ and $BXC$ intersect at $Y$. Let the circumcircles of triangles $AXB$ and $CXD$ intersect at $Z$. If $O$ lies inside $ABCD$ and if the points $O,X,Y,Z$ are all distinct, prove that $O,X,Y,Z$ lie on a circle.
2012 Moldova Team Selection Test, 12
Let $k \in \mathbb{N}$. Prove that \[ \binom{k}{0} \cdot (x+k)^k - \binom{k}{1} \cdot (x+k-1)^k+...+(-1)^k \cdot \binom{k}{k} \cdot x^k=k! ,\forall k \in \mathbb{R}\]
IV Soros Olympiad 1997 - 98 (Russia), 10.5
Three rays with a common origin are drawn on the plane, dividing the plane into three angles. One point is marked inside each corner. Using one ruler, construct a triangle whose vertices lie on the given rays and whose sides contain the given points.
1993 USAMO, 2
Let $\, ABCD \,$ be a convex quadrilateral such that diagonals $\, AC \,$ and $\, BD \,$ intersect at right angles, and let $\, E \,$ be their intersection. Prove that the reflections of $\, E \,$ across $\, AB, \, BC, \, CD, \, DA \,$ are concyclic.
2019 Junior Balkan Team Selection Tests - Romania, 2
Let $a, b, c, d \ge 0$ such that $a^2 + b^2 + c^2 + d^2 = 4$. Prove that $$\frac{a + b + c + d}{2} \ge 1 + \sqrt{abcd}$$ When does the equality hold?
Leonard Giugiuc and Valmir B. Krasniqi
2001 AMC 10, 1
The median of the list
\[ n, n \plus{} 3, n \plus{} 4, n \plus{} 5, n \plus{} 6, n \plus{} 8, n \plus{} 10, n \plus{} 12, n \plus{} 15
\]is $ 10$. What is the mean?
$ \textbf{(A) }4\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }10\qquad\textbf{(E) }11$
2016 Belarus Team Selection Test, 3
Solve the equation $2^a-5^b=3$ in positive integers $a,b$.
1993 National High School Mathematics League, 10
The last two digits of number of $\left[\frac{10^{93}}{10^{31}+1}\right]$ is________.
Geometry Mathley 2011-12, 1.2
Let $ABC$ be an acute triangle with its altitudes $BE,CF$. $M$ is the midpoint of $BC$. $N$ is the intersection of $AM$ and $EF. X$ is the projection of $N$ on $BC$. $Y,Z$ are respectively the projections of $X$ onto $AB,AC$. Prove that $N$ is the orthocenter of triangle $AYZ$.
Nguyễn Minh Hà
2024 Korea Junior Math Olympiad (First Round), 7.
There are four collinear spots: $ A,B,C,D $
$ \bar{AB}=\bar{BC}=\frac{\bar{CD}}{4}=\sqrt{5} $
There are two circles; One which has $ \bar{AC} $ as a diameter, and the other having $ \bar{BD} $ as a diameter.
Let's put $ \odot (AC) \cap \odot (BD) = E,F $
Let's put the area of $ EAFD $ $ S $
Find $ S^2 $.
2017 Sharygin Geometry Olympiad, 5
A square $ABCD$ is given. Two circles are inscribed into angles $A$ and $B$, and the sum of their diameters is equal to the sidelength of the square. Prove that one of their common tangents passes through the midpoint of $AB$.
2022 Argentina National Olympiad, 2
Determine all positive integers $n$ such that numbers from $1$ to $n$ can be sorted in some order $x_1,x_2,...,x_n$ with the property that the number $x_1+x_2+...+x_k$ is divisible by $k$, for all $1\le k\le n$., that is $1$ is divides $x_1$, $2$ divides $x_1+x_2$, $3$ divides $x_1+x_2+x_3$, and so on until $n$ divides $x_1+x_2+...+x_n$.
1998 Harvard-MIT Mathematics Tournament, 4
Find the range of $ f(A)=\frac{\sin A(3\cos^{2}A+\cos^{4}A+3\sin^{2}A+\sin^{2}A\cos^{2}A)}{\tan A (\sec A-\sin A\tan A)} $ if $A\neq \dfrac{n\pi}{2}$.
2013 MTRP Senior, 1
Find how many committees with a chairman can be chosen from a set of n persons. Hence or otherwise prove that
$${n \choose 1} + 2{n \choose 2} + 3{n \choose 3} + ...... + n{n \choose n} = n2^{n-1}$$
1987 Austrian-Polish Competition, 9
Let $M$ be the set of all points $(x,y)$ in the cartesian plane, with integer coordinates satisfying $1 \le x \le 12$ and $1 \le y \le 13$.
(a) Prove that every $49$-element subset of $M$ contains four vertices of a rectangle with sides parallel to the coordinate axes.
(b) Give an example of a $48$-element subset of $M$ without this property.
2017 ASDAN Math Tournament, 7
For real numbers $x,y$ satisfying $x^2+y^2-4x-2y+4=0$, what is the greatest value of
$$16\cos^2\sqrt{x^2+y^2}+24\sin\sqrt{x^2+y^2}?$$
III Soros Olympiad 1996 - 97 (Russia), 9.7
Find the side of the smallest regular triangle that can be inscribed in a right triangle with an acute angle of $30^o$ and a hypotenuse of $2$. (All vertices of the required regular triangle must be located on different sides of this right triangle.)
2012 Princeton University Math Competition, B3
Evaluate $\sqrt[3]{26 + 15\sqrt3} + \sqrt[3]{26 - 15\sqrt3}$