Found problems: 85335
1999 Balkan MO, 1
Let $O$ be the circumcenter of the triangle $ABC$. The segment $XY$ is the diameter of the circumcircle perpendicular to $BC$ and it meets $BC$ at $M$. The point $X$ is closer to $M$ than $Y$ and $Z$ is the point on $MY$ such that $MZ = MX$. The point $W$ is the midpoint of $AZ$.
a) Show that $W$ lies on the circle through the midpoints of the sides of $ABC$;
b) Show that $MW$ is perpendicular to $AY$.
2022 AMC 10, 17
How many three-digit positive integers $\underline{a}$ $\underline{b}$ $\underline{c}$ are there whose nonzero digits $a$, $b$, and $c$ satisfy
$$0.\overline{\underline{a}~\underline{b}~\underline{c}} = \frac{1}{3} (0.\overline{a} + 0.\overline{b} + 0.\overline{c})?$$
(The bar indicates repetition, thus $0.\overline{\underline{a}~\underline{b}~\underline{c}}$ in the infinite repeating decimal $0.\underline{a}~\underline{b}~\underline{c}~\underline{a}~\underline{b}~\underline{c}~\cdots$)
$\textbf{(A) }9\qquad\textbf{(B) }10\qquad\textbf{(C) }11\qquad\textbf{(D) }13\qquad\textbf{(E) }14$
1978 Bundeswettbewerb Mathematik, 3
For every positive integer $n$, define the remainder sum $r(n)$ as the sum of the remainders upon division of $n$ by each of the numbers $1$ through $n$. Prove that $r(2^{k}-1) =r(2^{k})$ for every $k\geq 1.$
1978 Germany Team Selection Test, 6
A lattice point in the plane is a point both of whose coordinates are integers. Each lattice point has four neighboring points: upper, lower, left, and right. Let $k$ be a circle with radius $r \geq 2$, that does not pass through any lattice point. An interior boundary point is a lattice point lying inside the circle $k$ that has a neighboring point lying outside $k$. Similarly, an exterior boundary point is a lattice point lying outside the circle $k$ that has a neighboring point lying inside $k$. Prove that there are four more exterior boundary points than interior boundary points.
2003 German National Olympiad, 4
From the midpoints of the sides of an acute-angled triangle, perpendiculars are drawn to the adjacent sides. The resulting six straight lines bound the hexagon. Prove that its area is half the area of the original triangle.
2004 Germany Team Selection Test, 1
Let n be a positive integer. Find all complex numbers $x_{1}$, $x_{2}$, ..., $x_{n}$ satisfying the following system of equations:
$x_{1}+2x_{2}+...+nx_{n}=0$,
$x_{1}^{2}+2x_{2}^{2}+...+nx_{n}^{2}=0$,
...
$x_{1}^{n}+2x_{2}^{n}+...+nx_{n}^{n}=0$.
2017 Purple Comet Problems, 10
Find the number of positive integers less than or equal to $2017$ that have at least one pair of adjacent digits that are both even. For example, count the numbers $24$, $1862$, and $2012$, but not $4$, $58$, or $1276$.
2005 National Olympiad First Round, 5
Let $M$ be the intersection of diagonals of the convex quadrilateral $ABCD$, where $m(\widehat{AMB})=60^\circ$. Let the points $O_1$, $O_2$, $O_3$, $O_4$ be the circumcenters of the triangles $ABM$, $BCM$, $CDM$, $DAM$, respectively. What is $Area(ABCD)/Area(O_1O_2O_3O_4)$?
$
\textbf{(A)}\ \dfrac 12
\qquad\textbf{(B)}\ \dfrac 32
\qquad\textbf{(C)}\ \dfrac {\sqrt 3}2
\qquad\textbf{(D)}\ \dfrac {1+2\sqrt 3}2
\qquad\textbf{(E)}\ \dfrac {1+\sqrt 3}2
$
2023 BAMO, E/3
In the following figure---not drawn to scale!---$E$ is the midpoint of $BC$, triangle $FEC$ has area $7$, and quadrilateral $DBEG$ has area $27$. Triangles $ADG$ and $GEF$ have the same area, $x$. Find $x$.
[asy]
unitsize(2cm);
pair A = (0,38/16);
pair B = (0,0);
pair C = (38/16,0);
pair D = (0,25/16);
pair E = (19/16,0);
pair F = .4*D+.6*C;
draw(D -- C -- B -- A -- E -- F);
label("$A$", A, W);
label("$B$", B, W);
label("$C$", C, S);
label("$D$", D, W);
label("$E$", E, S);
label("$F$", F, N);
label("$G$", (17*F-8*C)/9, NE);
[/asy]
2010 Stanford Mathematics Tournament, 11
What is the area of the regular hexagon with perimeter $60$?
1951 Miklós Schweitzer, 12
By number-theoretical functions, we will understand integer-valued functions defined on the set of all integers. Are there number-theoretical functions $ f_0(x),f_1(x),f_2(x),\dots$ such that every number theoretical function $ F(x)$ can be uniquely represented in the form
$ F(x)\equal{}\sum_{k\equal{}0}^{\infty}a_kf_k(x)$,
$ a_0,a_1,a_2,\dots$ being integers?
1992 AMC 12/AHSME, 22
Ten points are selected on the positive x-axis, $X^{+}$, and fives points are selected on the positive y-axis, $Y^{+}$. The fifty segments connecting the ten selected points on $X^{+}$ to the five selected points on $Y^{+}$ are drawn. What is the maximum possible number of points of intersection of these fifty segments that could lie in the interior of the first quadrant?
$ \textbf{(A)}\ 250\qquad\textbf{(B)}\ 450\qquad\textbf{(C)}\ 500\qquad\textbf{(D)}\ 1250\qquad\textbf{(E)}\ 2500 $
2011 Saint Petersburg Mathematical Olympiad, 1
$f(x),g(x)$ - two square trinomials and $a,b,c,d$ - some reals. $f(a)=2,f(b)=3,f(c)=7,f(d)=10$ and $g(a)=16,g(b)=15,g(c)=11$ Find $g(d)$
1993 IMO Shortlist, 7
Let $n > 1$ be an integer and let $f(x) = x^n + 5 \cdot x^{n-1} + 3.$ Prove that there do not exist polynomials $g(x),h(x),$ each having integer coefficients and degree at least one, such that $f(x) = g(x) \cdot h(x).$
2021 CHMMC Winter (2021-22), 9
Find the largest prime divisor of $$\sum^{30}_{n=3} {{n \choose 3} \choose 2}$$
2019 239 Open Mathematical Olympiad, 2
Is it true that there are $130$ consecutive natural numbers, such that each of them has exactly $900$ natural divisors?
2008 China Team Selection Test, 3
Let $ 0 < x_{1}\leq\frac {x_{2}}{2}\leq\cdots\leq\frac {x_{n}}{n}, 0 < y_{n}\leq y_{n \minus{} 1}\leq\cdots\leq y_{1},$ Prove that $ (\sum_{k \equal{} 1}^{n}x_{k}y_{k})^2\leq(\sum_{k \equal{} 1}^{n}y_{k})(\sum_{k \equal{} 1}^{n}(x_{k}^2 \minus{} \frac {1}{4}x_{k}x_{k \minus{} 1})y_{k}).$ where $ x_{0} \equal{} 0.$
2016 BMT Spring, 2
Jennifer wants to do origami, and she has a square of side length $ 1$. However, she would prefer to use a regular octagon for her origami, so she decides to cut the four corners of the square to get a regular octagon. Once she does so, what will be the side length of the octagon Jennifer obtains?
2016 Flanders Math Olympiad, 1
In the quadrilateral $ABCD$ is $AD \parallel BC$ and the angles $\angle A$ and $\angle D$ are acute. The diagonals intersect in $P$. The circumscribed circles of $\vartriangle ABP$ and $\vartriangle CDP$ intersect the line $AD$ again at $S$ and $T$ respectively. Call $M$ the midpoint of $[ST]$. Prove that $\vartriangle BCM$ is isosceles.
[img]https://1.bp.blogspot.com/-C5MqC0RTqwY/Xy1fAavi_aI/AAAAAAAAMSM/2MXMlwb13McCYTrOHm1ZzWc0nkaR1J6zQCLcBGAsYHQ/s0/flanders%2B2016%2Bp1.png[/img]
2014 Junior Regional Olympiad - FBH, 3
If $BK$ is an angle bisector of $\angle ABC$ in triangle $ABC$. Find angles of triangle $ABC$ if $BK=KC=2AK$
2007 Tournament Of Towns, 5
A triangular pie has the same shape as its box, except that they are mirror images of each other. We wish to cut the pie in two pieces which can t together in the box without turning either piece over. How can this be done if
[list][b](a)[/b] one angle of the triangle is three times as big as another;
[b](b)[/b] one angle of the triangle is obtuse and is twice as big as one of the acute angles?[/list]
2013 Pan African, 2
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(x)f(y)+f(x+y)=xy$ for all real numbers $x$ and $y$.
1992 IMO Longlists, 67
In a triangle, a symmedian is a line through a vertex that is symmetric to the median with the respect to the internal bisector (all relative to the same vertex). In the triangle $ABC$, the median $m_a$ meets $BC$ at $A'$ and the circumcircle again at $A_1$. The symmedian $s_a$ meets $BC$ at $M$ and the circumcircle again at $A_2$. Given that the line $A_1A_2$ contains the circumcenter $O$ of the triangle, prove that:
[i](a) [/i]$\frac{AA'}{AM} = \frac{b^2+c^2}{2bc} ;$
[i](b) [/i]$1+4b^2c^2 = a^2(b^2+c^2)$
2010 Contests, 1
Solve the equation
\[ x^3+2y^3-4x-5y+z^2=2012, \]
in the set of integers.
2024 German National Olympiad, 1
The five real numbers $v,w,x,y,s$ satisfy the system of equations
\begin{align*}
v&=wx+ys,\\
v^2&=w^2x+y^2s,\\
v^3&=w^3x+y^3s.
\end{align*}
Show that at least two of them are equal.