Found problems: 85335
JBMO Geometry Collection, 2017
Let $ABC $ be an acute triangle such that $AB\neq AC$ ,with circumcircle $ \Gamma$ and circumcenter $O$. Let $M$ be the midpoint of $BC$ and $D$ be a point on $ \Gamma$ such that $AD \perp BC$. let $T$ be a point such that $BDCT$ is a parallelogram and $Q$ a point on the same side of $BC$ as $A$ such that $\angle{BQM}=\angle{BCA}$ and $\angle{CQM}=\angle{CBA}$. Let the line $AO$ intersect $ \Gamma$ at $E$ $(E\neq A)$ and let the circumcircle of $\triangle ETQ$ intersect $ \Gamma$ at point $X\neq E$. Prove that the point $A,M$ and $X$ are collinear.
2009 National Olympiad First Round, 29
$ P$ is the intersection point of diagonals of cyclic $ ABCD$. The circumcenters of $ \triangle APB$ and $ \triangle CPD$ lie on circumcircle of $ ABCD$. If $ AC \plus{} BD \equal{} 18$, then area of $ ABCD$ is ?
$\textbf{(A)}\ 36 \qquad\textbf{(B)}\ \frac {81}{2} \qquad\textbf{(C)}\ \frac {36\sqrt 3}{2} \qquad\textbf{(D)}\ \frac {81\sqrt 3}{4} \qquad\textbf{(E)}\ \text{None}$
2022 Pan-African, 3
Let $n$ be a positive integer, and $a_1, a_2, \dots, a_{2n}$ be a sequence of positive real numbers whose product is equal to $2$. For $k = 1, 2, \dots, 2n$, set $a_{2n + k} = a_k$, and define
$$
A_k = \frac{1 + a_k + a_k a_{k + 1} + \dots + a_k a_{k + 1} \cdots a_{k + n - 2}}{1 + a_k + a_k a_{k + 1} + \dots + a_k a_{k + 1} \cdots a_{k + 2n - 2}}.
$$
Suppose that $A_1, A_2, \dots, A_{2n}$ are pairwise distinct; show that exactly half of them are less than $\sqrt{2} - 1$.
2019 Bundeswettbewerb Mathematik, 4
Prove that for no integer $k \ge 2$, between $10k$ and $10k + 100$ there are more than $23$ prime numbers.
2010 Contests, 3
Suppose that $a_1,...,a_{15}$ are prime numbers forming an arithmetic progression with common difference $d > 0$ if $a_1 > 15$ show that $d > 30000$
1996 All-Russian Olympiad Regional Round, 11.8
Is there an infinite periodic sequence consisting of the letters $a$ and$ b$, such that if all letters are replaced simultaneously $a$ to $aba$ and letters $b$ to $bba$ does it transform into itself (possibly with a shift)? (A sequence is called periodic if there is such natural number $n$, which for every $i = 1, 2, . . . i$-th member of this sequence is equal to the ($i + n$)- th.)
1997 IMO Shortlist, 13
In town $ A,$ there are $ n$ girls and $ n$ boys, and each girl knows each boy. In town $ B,$ there are $ n$ girls $ g_1, g_2, \ldots, g_n$ and $ 2n \minus{} 1$ boys $ b_1, b_2, \ldots, b_{2n\minus{}1}.$ The girl $ g_i,$ $ i \equal{} 1, 2, \ldots, n,$ knows the boys $ b_1, b_2, \ldots, b_{2i\minus{}1},$ and no others. For all $ r \equal{} 1, 2, \ldots, n,$ denote by $ A(r),B(r)$ the number of different ways in which $ r$ girls from town $ A,$ respectively town $ B,$ can dance with $ r$ boys from their own town, forming $ r$ pairs, each girl with a boy she knows. Prove that $ A(r) \equal{} B(r)$ for each $ r \equal{} 1, 2, \ldots, n.$
II Soros Olympiad 1995 - 96 (Russia), 9.1
Solve the inequality
$$(x-1)(x^2-1)(x^3-1)\cdot ...\cdot (x^{100}-1)(x^{101}-1)\ge 0$$
2018 Junior Regional Olympiad - FBH, 1
Four buddies bought a ball. First one paid half of the ball price. Second one gave one third of money that other three gave. Third one paid a quarter of sum paid by other three. Fourth paid $5\$$. How much did the ball cost?
2010 Rioplatense Mathematical Olympiad, Level 3, 3
Find all the functions $f:\mathbb{N}\to\mathbb{R}$ that satisfy
\[ f(x+y)=f(x)+f(y) \] for all $x,y\in\mathbb{N}$ satisfying $10^6-\frac{1}{10^6} < \frac{x}{y} < 10^6+\frac{1}{10^6}$.
Note: $\mathbb{N}$ denotes the set of positive integers and $\mathbb{R}$ denotes the set of real numbers.
1991 AMC 12/AHSME, 15
A circular table has exactly 60 chairs around it. There are $N$ people seated at this table in such a way that the next person to be seated must sit next to someone. The smallest possible value of $N$ is
$ \textbf{(A)}\ 15\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 30\qquad\textbf{(D)}\ 40\qquad\textbf{(E)}\ 58 $
1997 AMC 8, 16
Penni Precisely buys $\$100$ worth of stock in each of three companies: Alabama Almonds, Boston Beans, and California Cauliflower. After one year, AA was up $20\%$, BB was down $25\%$, and CC was unchanged. For the second year, AA was down $20\%$ from the previous year, BB was up $25\%$ from the previous year, and CC was unchanged. If A, B, and C are the final values of the stock, then
$\textbf{(A)}\ A=B=C \qquad \textbf{(B)}\ A=B<C \qquad \textbf{(C)}\ C<B=A$
$\textbf{(D)}\ A<B<C \qquad \textbf{(E)}\ B<A<C$
2002 AMC 12/AHSME, 11
Let $t_n=\dfrac{n(n+1)}2$ be the $n$th triangular number. Find
\[\dfrac1{t_1}+\dfrac1{t_2}+\dfrac1{t_3}+\cdots+\dfrac1{t_{2002}}.\]
$\textbf{(A) }\dfrac{4003}{2003}\qquad\textbf{(B) }\dfrac{2001}{1001}\qquad\textbf{(C) }\dfrac{4004}{2003}\qquad\textbf{(D) }\dfrac{4001}{2001}\qquad\textbf{(E) }2$
1993 All-Russian Olympiad Regional Round, 11.1
Find all natural numbers $n$ for which the sum of digits of $5^n$ equals $2^n$.
2017 AMC 10, 13
There are $20$ students participating in an after-school program offering classes in yoga, bridge, and painting. Each student must take at least one of these three classes, but may take two or all three. There are $10$ students taking yoga, $13$ taking bridge, and $9$ taking painting. There are $9$ students taking at least two classes. How many students are taking all three classes?
$\textbf{(A) } 1
\qquad\textbf{(B) } 2
\qquad\textbf{(C) } 3
\qquad\textbf{(D) } 4
\qquad\textbf{(E) } 5
$
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 $60$?