Found problems: 85335
Decide if it is possible to consider $2011$ points in a plane such that the distance between every two of these points is different from $1$ and each unit circle centered at one of these points leaves exactly $1005$ points outside the circle.
Let $f(n)$ denote the sum of the distinct positive integer divisors of n. Evaluate:
\[f(1) + f(2) + f(3) + f(4) + f(5) + f(6) + f(7) + f(8) + f(9).\]
25) A planet orbits around a star S, as shown in the figure. The semi-major axis of the orbit is a. The perigee, namely the shortest distance between the planet and the star is 0.5a. When the planet passes point $P$ (on the line through the star and perpendicular to the major axis), its speed is $v_1$. What is its speed $v_2$ when it passes the perigee?
A) $v_2 = \frac{3}{\sqrt{5}}v_1$
B) $v_2 = \frac{3}{\sqrt{7}}v_1$
C) $v_2 = \frac{2}{\sqrt{3}}v_1$
D) $v_2 = \frac{\sqrt{7}}{\sqrt{3}}v_1$
E) $v_2 = 4v_1$
Let $ABC$ be a triangle with $AB = 42$, $AC = 39$, $BC = 45$. Let $E$, $F$ be on the sides $\overline{AC}$ and $\overline{AB}$ such that $AF = 21, AE = 13$. Let $\overline{CF}$ and $\overline{BE}$ intersect at $P$, and let ray $AP$ meet $\overline{BC}$ at $D$. Let $O$ denote the circumcenter of $\triangle DEF$, and $R$ its circumradius. Compute $CO^2-R^2$.
[i]Proposed by Yang Liu[/i]
Given a triangle $ABC$ with angle $A$ obtuse and with altitudes of length $h$ and $k$ as shown in the diagram, prove that $a+h\ge b+k$. Find under what conditions $a+h=b+k$.
[asy]
size(6cm);
pair A = dir(105), C = dir(170), B = dir(10), D = foot(B, A, C), E = foot(A, B, C);
draw(A--B--C--cycle);
draw(B--D--A--E);
dot(A); dot(B); dot(C); dot(D); dot(E);
label("$A$", A, dir(110));
label("$B$", B, B);
label("$C$", C, C);
label("$D$", D, D);
label("$E$", E, dir(45));
label("$h$", A--E, dir(0));
label("$k$", B--D, dir(45));
[/asy]
Let $ABC$ be an acute triangle inscribed in circle $(O)$, with orthocenter $H$. Median $AM$ of triangle $ABC$ intersects circle $(O)$ at $A$ and $N$. $AH$ intersects $(O)$ at $A$ and $K$. Three lines $KN, BC$ and line through $H$ and perpendicular to $AN$ intersect each other and form triangle $X Y Z$. Prove that the circumcircle of triangle $X Y Z$ is tangent to $(O)$.
I start with a sequence of letters $A_1 A_2 \cdots A_{2021} A_1 A_2 \cdots A_{2021} A_1 A_2 \cdots A_{2021}$. I go through $i = 1, 2, 3, \cdots, 6062$ in order, and for each $i$, I can choose to swap letters $i$ and $i+1$. Let $N$ be the number of distinct strings I can end up with. What is the remainder when $N$ is divided by $2017$?
Compute the number of squares of positive area whose vertices all are points on the grid shown below.
[asy]
unitsize(1cm);
dot((0,0));
dot((1,0));
dot((2,0));
dot((3,0));
dot((0,1));
dot((1,1));
dot((2,1));
dot((3,1));
dot((0,2));
dot((1,2));
dot((2,2));
dot((3,2));
dot((0,3));
dot((1,3));
dot((2,3));
dot((3,3));
[/asy]
A circle with radius $5$ is inscribed in a right triangle with hypotenuse $34$ as shown below. What is the area of the triangle? Note that the diagram is not to scale.
In Fig. , $PA$ and $QB$ are tangents to the circle at $A$ and $B$ respectively. The line $AB$ is extended to meet $PQ$ at $S$. Suppose that $PA = QB$. Prove that $QS = SP$.
[img]https://cdn.artofproblemsolving.com/attachments/6/f/f21c0c70b37768f3e80e9ee909ef34c57635d5.png[/img]
Given a cube and two colours. Two players paint in turn a triple of arbitrary unpainted edges with his colour. (Everyone makes two moves.) The first wins if he has painted all the edges of some face with his colour. Can he always win?
Given a line $\ell$ and a ray $p$ on a plane with its origin on this line. Two fixed circles (not necessarily equal) are constructed, inscribed in the two formed angles. On ray $p$, point $A$ is taken so that the tangents from $A$ to the given circles, different from $p$, intersect line $\ell$ at points $B$ and $C$, and at the same time triangle $ABC$ contains the given circles. Find the locus of the centers of the circles inscribed in triangle $ABC$ (as $A$ moves).
Let $ \left( c_n \right)_{n\ge 1} $ be a sequence of real numbers. Prove that the sequences $ \left( c_n\sin n \right)_{n\ge 1} ,\left( c_n\cos n \right)_{n\ge 1} $ are both convergent if and only if $ \left( c_n \right)_{n\ge 1} $ converges to $ 0. $
[i]Mihai Piticari[/i] and [i]Vladimir Cerbu[/i]
A cube consisting of $(2N)^3$ unit cubes is pierced by several needles parallel to the edges of the cube (each needle pierces exactly $2N$ unit cubes). Each unit cube is pierced by at least one needle. Let us call any subset of these needles “regular” if there are no two needles in this subset that pierce the same unit cube.
a) Prove that there exists a regular subset consisting of $2N^2$ needles such that all of them have either the same direction or two different directions.
b) What is the maximum size of a regular subset that does exist for sure?
(Nikita Gladkov, Alexandr Zimin)
Find the number of positive integers that are equal to or equal to 1000 that have exactly 6 divisors that are perfect squares
a magical $3\times3$ square is a $3\times3$ matrix containing all number from 1 to 9, and of which the sum of every row, every column, every diagonal, are all equal.
Determine all magical $3\times3$ square
Consider a Riemannian metric on the vector space ${\Bbb{R}^n}$ which satisfies the property that for each two points ${a,b}$ there is a single distance minimising geodesic segment ${g(a,b)}$. Suppose that for all ${a \in \Bbb{R}^n}$, the Riemannian distance with respect to ${a}, {\rho_a : \Bbb{R}^n \rightarrow \Bbb{R}}$ is convex and differentiable outside of ${a}$. Prove that if for a point ${x \neq a,b}$ we have
\[ \displaystyle \partial_i \rho_a(x)=-\partial_i \rho_b(x),\ i=1,\cdots, n\]
then ${x}$ is a point on ${g(a,b)}$ and conversely.
[i]Proposed by Lajos Tamássy and Dávid Kertész[/i]
Compute \[\displaystyle\int_{0}^{\pi}\frac{2\sin\theta+3\cos\theta-3}{13\cos\theta-5}\mathrm{d}\theta.\]
Prove that the sequence defined by:
$$ y_ {n + 1} = \frac {1} {2} (3y_ {n} + \sqrt {5y_ {n} ^ {2} -4}) , \,\, \forall n \ge 0$$ with $ y_ {0} = 1$ consists only of integers.
Find all functions $f:\mathbb{R^{+}} \rightarrow \mathbb{R^{+}}$, such that $$xf(1+xf(y))=f(f(x)+f(y))$$ for all positive reals $x, y$.
A mouse is sitting in a toy car on a negligibly small turntable. The car cannot turn on its own, but the mouse can control when the car is launched and when the car stops (the car has brakes). When the mouse chooses to launch, the car will immediately leave the turntable on a straight trajectory at $1$ meter per second. Suddenly someone turns on the turntable; it spins at $30$ rpm. Consider the set $S$ of points the mouse can reach in his car within $1$ second after the turntable is set in motion. What is the area of $S$, in square meters?
Find all functions $f$ defined on the set of positive reals which take positive real values and satisfy: $f(xf(y))=yf(x)$ for all $x,y$; and $f(x)\to0$ as $x\to\infty$.
Find all positive integers $n$ such that there are $k \geq 2$ positive rational numbers $a_1, a_2, \ldots, a_k$ satisfying $a_1 + a_2 + \ldots + a_k = a_1 \cdot a_2 \cdots a_k = n.$
There are $n$ straight lines in the plane such that each intersects exactly $1999$ of the others . Find all posssible values of $n$.
(R Zhenodarov)
Let $f$ be a continuously differentiable function on $[0, 1]$ and $m \in \mathbb{N}$. Let $A = f(1)$ and let $B=\int \limits_{0}^1 x^{-\frac{1}{m}}f(x)dx$. Calculate $$\lim \limits_{n \to \infty} n\left(\int \limits_{0}^1 f(x)dx-\sum \limits_{k=1}^n \left(\frac{k^m}{n^m}-\frac{(k-1)^m}{n^m}\right)f\left(\frac{(k-1)^m}{n^m}\right)\right)$$in terms of $A$ and $B$.