Found problems: 85335
1998 Singapore Senior Math Olympiad, 2
Let $C$ be a circle in the plane. Let $C_1$ and $C_2$ be two non-intersecting circles touching $C$ internally at points $A$ and $B$ respectively (Fig. ). Suppose that $D$ and $E$ are two points on $C_1$ and $C_2$ respectively such that $DE$ is a common tangent of $C_1$ and $C_2$, and both $C_1$ and C2 are on the same side of $DE$. Let $F$ be the intersection point of $AD$ and $BE$. Prove that $F$ lies on $C$.
[img]https://cdn.artofproblemsolving.com/attachments/f/c/5c733db462ef8ec3d3f82bbb762f7f087fbd3d.png[/img]
2005 Tournament of Towns, 3
Baron Münchhausen’s watch works properly, but has no markings on its face. The hour, minute and second hands have distinct lengths, and they move uniformly. The Baron claims that since none of the mutual positions of the hands is repeats twice in the period between 8:00 and 19:59, he can use his watch to tell the time during the day. Is his assertion true?
[i](5 points)[/i]
2021 Ukraine National Mathematical Olympiad, 8
Given a natural number $n$. Prove that you can choose $ \phi (n)+1 $ (not necessarily different) divisors $n$ with the sum $n$.
Here $ \phi (n)$ denotes the number of natural numbers less than $n$ that are coprime with $n$.
(Fedir Yudin)
1998 Harvard-MIT Mathematics Tournament, 3
$MD$ is a chord of length $2$ in a circle of radius $1,$ and $L$ is chosen on the circle so that the area of $\triangle MLD$ is the maximized. Find $\angle MLD.$
2005 iTest, 22
A regular $n$-gon has $135$ diagonals. What is the measure of its exterior angle, in degrees? (An exterior angle is the supplement of an interior angle.)
1955 AMC 12/AHSME, 1
Which one of the following is not equivalent to $ 0.000000375$?
$ \textbf{(A)}\ 3.75 \times 10^{\minus{}7} \qquad
\textbf{(B)}\ 3 \frac{3}{4} \times 10^{\minus{}7} \qquad
\textbf{(C)}\ 375 \times 10^{\minus{}9} \\
\textbf{(D)}\ \frac{3}{8} \times 10^{\minus{}7} \qquad
\textbf{(E)}\ \frac{3}{80000000}$
2020 Balkan MO Shortlist, N3
Given an integer $k\geq 2$, determine all functions $f$ from the positive integers into themselves such that $f(x_1)!+f(x_2)!+\cdots f(x_k)!$ is divisibe by $x_1!+x_2!+\cdots x_k!$ for all positive integers $x_1,x_2,\cdots x_k$.
$Albania$
1986 Brazil National Olympiad, 2
Find the number of ways that a positive integer $n$ can be represented as a sum of one or more consecutive positive integers.
2023 Harvard-MIT Mathematics Tournament, 14
Acute triangle $ABC$ has circumcenter $O.$ The bisector of $ABC$ and the altitude from $C$ to side $AB$ intersect at $X.$ Suppose that there is a circle passing through $B, O, X,$ and $C.$ If $\angle BAC = n^\circ,$ where $n$ is a positive integer, compute the largest possible value of $n.$
2022 Romania National Olympiad, P1
Let $\mathcal{F}$ be the set of functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(2x)=f(x)$ for all $x\in\mathbb{R}.$
[list=a]
[*]Determine all functions $f\in\mathcal{F}$ which admit antiderivatives on $\mathbb{R}.$
[*]Give an example of a non-constant function $f\in\mathcal{F}$ which is integrable on any interval $[a,b]\subset\mathbb{R}$ and satisfies \[\int_a^bf(x) \ dx=0\]for all real numbers $a$ and $b.$
[/list][i]Mihai Piticari and Sorin Rădulescu[/i]
2013 IMO Shortlist, G1
Let $ABC$ be an acute triangle with orthocenter $H$, and let $W$ be a point on the side $BC$, lying strictly between $B$ and $C$. The points $M$ and $N$ are the feet of the altitudes from $B$ and $C$, respectively. Denote by $\omega_1$ is the circumcircle of $BWN$, and let $X$ be the point on $\omega_1$ such that $WX$ is a diameter of $\omega_1$. Analogously, denote by $\omega_2$ the circumcircle of triangle $CWM$, and let $Y$ be the point such that $WY$ is a diameter of $\omega_2$. Prove that $X,Y$ and $H$ are collinear.
[i]Proposed by Warut Suksompong and Potcharapol Suteparuk, Thailand[/i]
Oliforum Contest II 2009, 3
Find all $ (x,y,z) \in \mathbb{Z}^3$ such that $ x^3 \minus{} 5x \equal{} 1728^{y}\cdot 1733^z \minus{} 17$.
[i](Paolo Leonetti)[/i]
2004 Thailand Mathematical Olympiad, 21
The ratio between the circumradius and the inradius of a given triangle is $7 : 2$. If the length of two sides of the triangle are $3$ and $7$, and the length of the remaining side is also an integer, what is the length of the remaining side?
1994 National High School Mathematics League, 6
In rectangular coordinate system, the equation $\frac{|x+y|}{2a}+\frac{|x-y|}{2b}=1$ ($a,b$ are different positive numbers) refers to
$\text{(A)}$ a triangle
$\text{(B)}$ a square
$\text{(C)}$ rectangle, not square
$\text{(D)}$ rhombus, not square
1981 AMC 12/AHSME, 17
The function $f$ is not defined for $x=0$, but, for all non-zero real numbers $x$, $f(x)+2f\left( \frac1x \right)=3x$. The equation $f(x)=f(-x)$ is satisfied by
$\text{(A)} ~\text{exactly one real number}$
$\text{(B)}~\text{exactly two real numbers}$
$\text{(C)} ~\text{no real numbers}$
$\text{(D)} ~\text{infinitely many, but not all, non-zero real numbers}$
$\text{(E)} ~\text{all non-zero real numbers}$
2022 Germany Team Selection Test, 3
A hunter and an invisible rabbit play a game on an infinite square grid. First the hunter fixes a colouring of the cells with finitely many colours. The rabbit then secretly chooses a cell to start in. Every minute, the rabbit reports the colour of its current cell to the hunter, and then secretly moves to an adjacent cell that it has not visited before (two cells are adjacent if they share an edge). The hunter wins if after some finite time either:[list][*]the rabbit cannot move; or
[*]the hunter can determine the cell in which the rabbit started.[/list]Decide whether there exists a winning strategy for the hunter.
[i]Proposed by Aron Thomas[/i]
1991 Hungary-Israel Binational, 3
Let $ \mathcal{H}_n$ be the set of all numbers of the form $ 2 \pm\sqrt{2 \pm\sqrt{2 \pm\ldots\pm\sqrt 2}}$ where "root signs" appear $ n$ times.
(a) Prove that all the elements of $ \mathcal{H}_n$ are real.
(b) Computer the product of the elements of $ \mathcal{H}_n$.
(c) The elements of $ \mathcal{H}_{11}$ are arranged in a row, and are sorted by size in an ascending order. Find the position in that row, of the elements of $ \mathcal{H}_{11}$ that corresponds to the following combination of $ \pm$ signs: \[ \plus{}\plus{}\plus{}\plus{}\plus{}\minus{}\plus{}\plus{}\minus{}\plus{}\minus{}\]
2024 Mexican Girls' Contest, 8
Find all positive integers \(n\) such that among the \(n\) numbers
\[ 2n + 1, \, 2^2 n + 1, \, \ldots, \, 2^n n + 1 \]
there are \(n\), \(n - 1\), or \(n - 2\) primes.
2005 Cono Sur Olympiad, 1
Let $a_n$ be the last digit of the sum of the digits of $20052005...2005$, where the $2005$ block occurs $n$ times. Find $a_1 +a_2 + \dots +a_{2005}$.
2023 Moldova EGMO TST, 5
Find all pairs of real numbers $(x, y)$, that satisfy the system of equations: $$\left\{\begin{matrix} 6(1-x)^2=\dfrac{1}{y} \\ \\6(1-y)^2=\dfrac{1}{x}.\end{matrix}\right.$$
1999 Ukraine Team Selection Test, 7
Let $P_1P_2...P_n$ be an oriented closed polygonal line with no three segments passing through a single point. Each point $P_i$ is assinged the angle $180^o - \angle P_{i-1}P_iP_{i+1} \ge 0$ if $P_{i+1}$ lies on the left from the ray $P_{i-1}P_i$, and the angle $-(180^o -\angle P_{i-1}P_iP_{i+1}) < 0$ if $P_{i+1}$ lies on the right. Prove that if the sum of all the assigned angles is a multiple of $720^o$, then the number of self-intersections of the polygonal line is odd
1952 Putnam, A5
Let $a_j (j = 1, 2, \ldots, n)$ be entirely arbitrary numbers except that no one is equal to unity. Prove \[ a_1 + \sum^n_{i=2} a_i \prod^{i-1}_{j=1} (1 - a_j) = 1 - \prod^n_{j=1} (1 - a_j).\]
1998 Harvard-MIT Mathematics Tournament, 4
A cube with side length $100cm$ is filled with water and has a hole through which the water drains into a cylinder of radius $100cm.$ If the water level in the cube is falling at a rate of $1 \frac{cm}{s} ,$ how fast is the water level in the cylinder rising?
1983 AMC 12/AHSME, 2
Point $P$ is outside circle $C$ on the plane. At most how many points on $C$ are $3 \text{cm}$ from $P$?
$\text{(A)} \ 1 \qquad \text{(B)} \ 2 \qquad \text{(C)} \ 3 \qquad \text{(D)} \ 4 \qquad \text{(E)} \ 8$
1992 IMO Longlists, 38
Let $\,S\,$ be a finite set of points in three-dimensional space. Let $\,S_{x},\,S_{y},\,S_{z}\,$ be the sets consisting of the orthogonal projections of the points of $\,S\,$ onto the $yz$-plane, $zx$-plane, $xy$-plane, respectively. Prove that \[ \vert S\vert^{2}\leq \vert S_{x} \vert \cdot \vert S_{y} \vert \cdot \vert S_{z} \vert, \] where $\vert A \vert$ denotes the number of elements in the finite set $A$.
[hide="Note"] Note: The orthogonal projection of a point onto a plane is the foot of the perpendicular from that point to the plane. [/hide]