Found problems: 85335
Let $ABC$ be an acute-angled scalene triangle with circumcircle $\Gamma$ and circumcenter $O$. Suppose $AB < AC$. Let $H$ be the orthocenter and $I$ be the incenter of triangle $ABC$. Let $F$ be the midpoint of the arc $BC$ of the circumcircle of triangle $BHC$, containing $H$.
Let $X$ be a point on the arc $AB$ of $\Gamma$ not containing $C$, such that $\angle AXH = \angle AFH$. Let $K$ be the circumcenter of triangle $XIA$. Prove that the lines $AO$ and $KI$ meet on $\Gamma$.
[i]Proposed by Anant Mudgal[/i]
Compute the sum of the positive divisors (including $1$) of $9!$ that have units digit $1.$
Aaron takes a square sheet of paper, with one corner labeled $A$. Point $P$ is chosen at random inside of the square and Aaron folds the paper so that points $A$ and $P$ coincide. He cuts the sheet along the crease and discards the piece containing $A$. Let $p$ be the probability that the remaining piece is a pentagon. Find the integer nearest to $100p$.
[i]Proposed by Aaron Lin[/i]
In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.
All three sides of an equilateral triangle are assumed to be reflective (except in the vertices), in such a way that they reflect the rays of light located in their plane, that fall on them and that come out of an interior point of the triangle.
Determine the path of a ray of light that, starting from a vertex of the triangle reach another vertex of the same after reflecting successively on the three sides. Calculate the length of the path followed by the light assuming that the side of the triangle measures $1$ m.
Consider an equilateral triangle $\vartriangle ABC$ with side length $1$. Let $D$ and $E$ lie on segments $AB$ and $AC$ respectively such that $\angle ADE = 30^o$ and $DE$ is tangent to the incircle of $\vartriangle ABC$. Compute the perimeter of $\vartriangle ADE$.
If $S = 6 \times10 000 +5\times 1000+ 4 \times 10+ 3 \times 1$, what is $S$?
$\textbf{(A)}\ 6543 \qquad \textbf{(B)}\ 65043 \qquad \textbf{(C)}\ 65431 \qquad \textbf{(D)}\ 65403 \qquad \textbf{(E)}\ 60541$
Alphabet $A$ contains $n$ letters. $S$ is a set of words of finite length composed of letters of $A$. It is known that every infinite sequence of letters of $A$ begins with one and only one word of $S$.
Prove that the set $S$ is finite.
[i]Proposed by F. Bakharev[/i]
Find all integer solutions to the equation
\[ x^2 + y^2 + z^2 = 2xyz \]
Consider a polynomial $P(x) = \prod^9_{j=1}(x+d_j),$ where $d_1, d_2, \ldots d_9$ are nine distinct integers. Prove that there exists an integer $N,$ such that for all integers $x \geq N$ the number $P(x)$ is divisible by a prime number greater than 20.
[i]Proposed by Luxembourg[/i]
Determine whether there exists a polynomial $f(x_1, x_2)$ with two variables, with integer coefficients, and two points $A=(a_1, a_2)$ and $B=(b_1, b_2)$ in the plane, satisfying the following conditions:
(i) $A$ is an integer point (i.e $a_1$ and $a_2$ are integers);
(ii) $|a_1-b_1|+|a_2-b_2|=2010$;
(iii) $f(n_1, n_2)>f(a_1, a_2)$ for all integer points $(n_1, n_2)$ in the plane other than $A$;
(iv) $f(x_1, x_2)>f(b_1, b_2)$ for all integer points $(x_1, x_2)$ in the plane other than $B$.
[i]Massimo Gobbino, Italy[/i]
A random rectangle (not necessarily a square) with positive integer dimensions is selected from the $2\times4$ grid below. The probability that the selected rectangle contains only white squares can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a+b$.
[asy]
fill((2,0)--(3,0)--(3,1)--(2,1)--cycle,blue);
fill((2,1)--(3,1)--(3,2)--(2,2)--cycle,blue);
draw((0,0)--(4,0),black);
draw((0,0)--(0,2),black);
draw((4,0)--(4,2),black);
draw((4,2)--(0,2),black);
draw((0,1)--(4,1),black);
draw((1,0)--(1,2),black);
draw((2,0)--(2,2),black);
draw((3,0)--(3,2),black);
[/asy]
A man born in the first half of the nineteenth century was $ x$ years old in the year $ x^2$. He was born in:
$ \textbf{(A)}\ 1849 \qquad
\textbf{(B)}\ 1825 \qquad
\textbf{(C)}\ 1812 \qquad
\textbf{(D)}\ 1836 \qquad
\textbf{(E)}\ 1806$
Thomas is to read $225$ pages of a book over the summer. He decides to read one page the first day, three pages the second day, five pages the third day, and so on, each day reading two more pages than the previous. How many days will it take Thomas to finish reading the book?
$\textbf{(A) } 11\qquad\textbf{(B) } 12\qquad\textbf{(C) } 13\qquad\textbf{(D) } 14\qquad\textbf{(E) } 15$
Let $a$ be a real number. Determine all functions $f: R \to R$ with $f (f (x) + y) = f (x^2 - y) + af (x) y$ for all $x, y \in R$.
(Walther Janous)
Investigate topologically the Riemann surface of the function
\[w=\arctan z\]
Find the minimum value of $x^2+2x+ \frac{24}{x}$ for $x > 0$.
Let $ABCD$ be a trapezoid with bases $ AB$ and $CD$ $(AB>CD)$. Diagonals $AC$ and $BD$ intersect in point $ N$ and lines $AD$ and $BC$ intersect in point $ M$. The circumscribed circles of $ADN$ and $BCN$ intersect in point $ P$, different from point $ N$. Prove that the angles $AMP$ and $BMN$ are equal.
Nine people sit down for dinner where there are three choices of meals. Three people order the beef meal, three order the chicken meal, and three order the fish meal. The waiter serves the nine meals in random order. Find the number of ways in which the waiter could serve the meal types to the nine people such that exactly one person receives the type of meal ordered by that person.
Is it possible that a set consisting of $23$ real numbers has a property that the number of the nonempty subsets whose product of the elements is rational number is exactly $2422$?
In a group of $n$ people, each one knows exactly three others. They are seated around a table. We say that the seating is $perfect$ if everyone knows the two sitting by their sides. Show that, if there is a perfect seating $S$ for the group, then there is always another perfect seating which cannot be obtained from $S$ by rotation or reflection.
Four spheres, each of radius $r$, lie inside a regular tetrahedron with side length $1$ such that each sphere is tangent to three faces of the tetrahedron and to the other three spheres. Find $r$.
A quadrilateral with sides $a,b,c,d$ is inscribed in a circle of radius $R$. Prove that if $a^2+b^2+c^2+d^2=8R^2$, then either one of the angles of the quadrilateral is right or the diagonals of the quadrilateral are perpendicular.
Find all functions $f: \mathbb{Z}^+\to \mathbb{R}$, which satisfies $f(n+1)\geq f(n)$ for all $n\geq 1$ and $f(mn)=f(m)f(n)$ for all $(m,n)=1$.
Let $p>3$ and $p+2$ are prime numbers,and define sequence
$$a_{1}=2,a_{n}=a_{n-1}+\lfloor \dfrac{pa_{n-1}}{n}\rfloor$$
show that:for any $n=3,4,\cdots,p-1$ have $$n|pa_{n-1}+1$$