Found problems: 85335
Every point with integer coordinates in the plane is the center of a disk with radius $1/1000$.
(1) Prove that there exists an equilateral triangle whose vertices lie in different discs.
(2) Prove that every equilateral triangle with vertices in different discs has side-length greater than $96$.
[i]Radu Gologan, Romania[/i]
[hide="Remark"]
The "> 96" in [b](b)[/b] can be strengthened to "> 124". By the way, part [b](a)[/b] of this problem is the place where I used [url=http://mathlinks.ro/viewtopic.php?t=5537]the well-known "Dedekind" theorem[/url].
[/hide]
[b]a)[/b] Let $ a,b $ two non-negative integers such that $ a^2>b. $ Show that the equation
$$ \left\lfloor\sqrt{x^2+2ax+b}\right\rfloor =x+a-1 $$
has an infinite number of solutions in the non-negative integers. Here, $ \lfloor\alpha\rfloor $ denotes the floor of $ \alpha. $
[b]b)[/b] Find the floor of $ m=\sqrt{2+\sqrt{2+\underbrace{\cdots}_{\text{n times}}+\sqrt{2}}} , $ where $ n $ is a natural number. Justify.
Consider the numbers $a,b, \alpha, \beta \in \mathbb{R}$ and the sets $$A=\left \{x \in \mathbb{R} : x^2+a|x|+b=0 \right \},$$ $$B=\left \{ x \in \mathbb{R} : \lfloor x \rfloor^2 + \alpha \lfloor x \rfloor + \beta = 0\right \}.$$
If $A \cap B$ has exactly three elements, prove that $a$ cannot be an integer.
3. In acute-angled triangle ABC point D is the perpendicular projection of C on the side AB. Point E is the perpendicular projection of D on the side BC. Point F lies on the side DE and:
$\frac{EF}{FD}=\frac{AD}{DB}$
Prove that $CF \bot AE$
Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ and relatively prime to $n$. Find all natural numbers $n$ and primes $p$ such that $\phi(n)=\phi(np)$.
Determine the largest integer $N$ for which there exists a table $T$ of integers with $N$ rows and $100$ columns that has the following properties:
$\text{(i)}$ Every row contains the numbers $1$, $2$, $\ldots$, $100$ in some order.
$\text{(ii)}$ For any two distinct rows $r$ and $s$, there is a column $c$ such that $|T(r,c) - T(s, c)|\geq 2$. (Here $T(r,c)$ is the entry in row $r$ and column $c$.)
An excircle centered at $I_{A}$ touches the side $BC$ of a triangle $ABC$ at point $D$. Prove that the pedal circles of $D$ with respect to the triangles $ABI_{A}$ and $ACI_{A}$ are congruent.
Proposed by:K.Belsky
Mark $4 \cdot k$ points in a circle and number them arbitrarily with numbers from $1$ to $4 \cdot k$. The chords cannot share common endpoints, also, the endpoints of these chords should be among the $4 \cdot k$ points.
[b]i.[/b] Prove that $2 \cdot k$ pairwisely non-intersecting chords can be drawn for each of whom its endpoints differ in at most $3 \cdot k - 1$.
[b]ii.[/b] Prove that the $3 \cdot k - 1$ cannot be improved.
Let $a, b, c$ be non-zero real numbers that satisfy $\frac{1}{abc} + \frac{1}{a} + \frac{1}{c} = \frac{1}{b}$. The expression $\frac{4}{a^2 + 1} + \frac{4}{b^2 + 1} + \frac{7}{c^2 + 1}$ has a maximum value $M$. Find the sum of the numerator and denominator of the reduced form of $M$.
A circle of radius 1 is placed in a corner of a room (i.e., it touches the horizontal floor and two vertical walls perpendicular to each other). Find the locus of the center of the band for all of its possible positions.
[b]Note.[/b] For the solution of this problem, it is useful to know the following Monge theorem: The locus of all points $P$, such that the two tangents from $P$ to the ellipse with equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ are perpendicular to each other, is a circle − a so-called Monge circle − with equation $x^2 + y^2 = a^2 + b^2$.
Oscar is drawing diagrams with trash can lids and sticks. He draws a triangle $ABC$ and a point $D$ such that $DB$ and $DC$ are tangent to the circumcircle of $ABC$. Let $B'$ be the reflection of $B$ over $AC$ and $C'$ be the reflection of $C$ over $AB$. If $O$ is the circumcenter of $DB'C'$, help Oscar prove that $AO$ is perpendicular to $BC$.
[i]James Lin[/i]
Alex has a piece of cheese. He chooses a positive number $a\neq 1$ and cuts the piece into several pieces one by one. Every time he chooses a piece and cuts it in the same ratio $1:a.$ His goal is to divide the cheese into two piles of equal masses. Can he do it?
Find four positive integers, each not exceeding $70000$ and each having more than $100$ divisors.
A herder has forgotten the number of cows she has, and does not want to count them all of them. She remembers these four facts about the number of cows:
[list]
[*]It has $3$ digits.
[*]It is a palindrome.
[*]The middle digit is a multiple of $4$.
[*]It is divisible by $11$.
[/list]
What is the sum of all possible numbers of cows that the herder has?
$\textbf{(A) }343\qquad\textbf{(B) }494\qquad\textbf{(C) }615\qquad\textbf{(D) }635\qquad\textbf{(E) }726$
Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.
Suppose that $n > m \geq 1$ are integers such that the string of digits $143$ occurs somewhere in the decimal representation of the fraction $\frac{m}{n}$. Prove that $n > 125.$
Find all pairs of positive integers $ m$ and $ n$ that satisfy (both) following conditions:
(i) $ m^{2}\minus{}n$ divides $ m\plus{}n^{2}$
(ii) $ n^{2}\minus{}m$ divides $ n\plus{}m^{2}$
Let $\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\overline{IA}\perp\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\cdot AC$.
Two monic quadratic trinomials $f(x)$ and $g(x)$ take negative values on disjoint intervals. Prove that there exist positive numbers $\alpha$ and $\beta$ such that $\alpha f(x) + \beta g(x) > 0$ for all real $x$.
How many subsets of $\{1,2,3,4,\dots,12\}$ contain exactly one prime number?
Jacob rolls two fair six-sided dice. If the outcomes of these dice rolls are the same, he rolls a third fair six-sided die. Compute the probability that the sum of the outcomes of all the dice he rolls is even.
Aidan rolls a pair of fair, six sided dice. Let$ n$ be the probability that the product of the two numbers at the top is prime. Given that $n$ can be written as $a/b$ , where $a$ and $b$ are relatively prime positive integers, find $a +b$.
[i]Proposed by Aidan Duncan[/i]
Suppose $a,b,c,d> 0$ and $x = \sqrt{a^2+b^2}, y = \sqrt{c^2+d^2}$. Prove that $xy \ge ac + bd$.
A unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure 1. The cube is then cut in the same manner along the dashed lines shown in Figure 2. This creates nine pieces. What is the volume of the piece that contains vertex $ W$?
[asy]import three;
size(200);
defaultpen(linewidth(.8pt)+fontsize(10pt));
currentprojection=oblique;
path3 p1=(0,2,2)--(0,2,0)--(2,2,0)--(2,2,2)--(0,2,2)--(0,0,2)--(2,0,2)--(2,2,2);
path3 p2=(2,2,0)--(2,0,0)--(2,0,2);
path3 p3=(0,0,2)--(0,2,1)--(2,2,1)--(2,0,2);
path3 p4=(2,2,1)--(2,0,0);
pen finedashed=linetype("4 4");
draw(p1^^p2^^p3^^p4);
draw(shift((4,0,0))*p1);
draw(shift((4,0,0))*p2);
draw(shift((4,0,0))*p3);
draw(shift((4,0,0))*p4);
draw((4,0,2)--(5,2,2)--(6,0,2),finedashed);
draw((5,2,2)--(5,2,0)--(6,0,0),finedashed);
label("$W$",(3,0,2));
draw((2.7,.3,2)--(2.1,1.9,2),linewidth(.6pt));
draw((3.4,.3,2)--(5.9,1.9,2),linewidth(.6pt));
label("Figure 1",(1,-0.5,2));
label("Figure 2",(5,-0.5,2));[/asy]$ \textbf{(A)}\ \frac {1}{12}\qquad \textbf{(B)}\ \frac {1}{9}\qquad \textbf{(C)}\ \frac {1}{8}\qquad \textbf{(D)}\ \frac {1}{6}\qquad \textbf{(E)}\ \frac {1}{4}$
The polynomial $f(x)=x^{2007}+17x^{2006}+1$ has distinct zeroes $r_1,\ldots,r_{2007}$. A polynomial $P$ of degree $2007$ has the property that $P\left(r_j+\dfrac{1}{r_j}\right)=0$ for $j=1,\ldots,2007$. Determine the value of $P(1)/P(-1)$.