Found problems: 85335
Is it possible for some natural number $k$ to divide all natural numbers from $1$ to $k$ into two groups and write down the numbers in each group in a row in some order so that you get two the same numbers?
[hide=original wording beacuse it doesn't make much sense]Можно ли при каком-то натуральном k разбить все натуральные числа от 1 до k на две группы и выписать числа в каждой группе подряд в некотором порядке так, чтобы получились два одинаковых числа?[/hide]
When the binomial coefficient $\binom{125}{64}$ is written out in base 10, how many zeros are at the rightmost end?
Let $F:(1,\infty) \rightarrow \mathbb{R}$ be the function defined by
$$F(x)=\int_{x}^{x^{2}} \frac{dt}{\ln(t)}.$$
Show that $F$ is injective and find the set of values of $F$.
Suppose $S= \{x|x=a^2+ab+b^2,a,b \in Z\}$. Prove that:
(1) If $m \in S$, $3|m$ , then $\frac{m}{3} \in S$
(2) If $m,n \in S$ , then $mn\in S$.
In the set of all positive real numbers define the operation $a * b = a^b$ .
Find all positive rational numbers for which $a * b = b * a$.
Circles $A$, $B$, and $C$ are externally tangent circles. Line $PQ$ is drawn such that $PQ$ is tangent to $A$ at $P$, tangent to $B$ at $Q$, and does not intersect with $C$. Circle $D$ is drawn such that it passes through the centers of $A$, $B$, and $C$. Let $R$ be the point on $D$ furthest from $PQ$. If $A$, $B$, and $C$ have radii $3$, $2$, and $1$, respectively, the area of triangle $PQR$ can be expressed in the form of $a+b\sqrt{c}$, where $a$, $b$, and $c$ are integers with $c$ not divisible by any prime square. What is $a + b + c$?
Find all positive integers $d$ that can be written in the form
$$ d = \gcd(|x^2 - y| , |y^2 - z| , |z^2 - x|), $$
where $x, y, z$ are pairwise coprime positive integers such that $x^2 \neq y$, $y^2 \neq z$, and $z^2 \neq x$.
Is there a closed self-intersecting broken line in space that intersects each of its links exactly once, and in its midpoint ?
Find all pairs $(x, y)$ of positive rational numbers such that $x^{2}+3y^{2}=1$.
Define the sequence $(x_{n})$: $x_{1}=\frac{1}{3}$ and $x_{n+1}=x_{n}^{2}+x_{n}$. Find $\left[\frac{1}{x_{1}+1}+\frac{1}{x_{2}+1}+\dots+\frac{1}{x_{2007}+1}\right]$, wehere $[$ $]$ denotes the integer part.
Let $ABC$ be an acute-angled triangle. The line through $C$ perpendicular to $AC$ meets the external angle bisector of $\angle ABC$ at $D$. Let $H$ be the foot of the perpendicular from $D$ onto $BC$. The point $K$ is chosen on $AB$ so that $KH \parallel AC$. Let $M$ be the midpoint of $AK$. Prove that $MC = MB + BH$.
Giorgi Arabidze, Georgia,
Let $ABC$ be a triangle inscribed in circle $(I)$ that is tangent to the sides $BC,CA,AB$ at points $D,E, F$ respectively. Assume that $L$ is the intersection of $BE$ and $CF,G$ is the centroid of triangle $DEF,K$ is the symmetric point of $L$ about $G$. If $DK$ meets $EF$ at $P, Q$ is on $EF$ such that $QF = PE$, prove that $\angle DGE + \angle FGQ = 180^o$.
Nguyễn Minh Hà
We say that a finite set $\mathcal{S}$ of points in the plane is [i]balanced[/i] if, for any two different points $A$ and $B$ in $\mathcal{S}$, there is a point $C$ in $\mathcal{S}$ such that $AC=BC$. We say that $\mathcal{S}$ is [i]centre-free[/i] if for any three different points $A$, $B$ and $C$ in $\mathcal{S}$, there is no points $P$ in $\mathcal{S}$ such that $PA=PB=PC$.
(a) Show that for all integers $n\ge 3$, there exists a balanced set consisting of $n$ points.
(b) Determine all integers $n\ge 3$ for which there exists a balanced centre-free set consisting of $n$ points.
Proposed by Netherlands
Let $ ABCD$ be a quadrilateral inscribed in a circle of radius $ AB$ such that $ BC \equal{} a, CD \equal{} b,$ $ DA \equal{} \frac{3 \sqrt{3} \minus{} 1}{2} \cdot a$ For each point $ M$ on the semicircle with radius $ AB$ not containing $ C$ and $ D,$ denote by $ h_1, h_2, h_3$ the distances from $ M$ to the straight lines (sides) $ BC, CD,$ and $ DA.$ Find the maximum of $ h_1 \plus{} h_2 \plus{} h_3.$
Let $ABCD$ be a trapezoid with base sides $AB$ and $CD$ and let $AB=a$, $BC=b$, $CD=c$, $DA=d$, $AC=m$ and $BD=n$. We know that $m^2+n^2=(a+c)^2$
$a)$ Prove that lines $AC$ and $BD$ are perpendicular
$b)$ Prove that $ac<bd$
Let $ a_1$, $ a_2$, ..., $ a_n$ and $ b_1$, $ b_2$, ..., $ b_n$ be $ 2 \cdot n$ real numbers. Prove that the following two statements are equivalent:
[b]i)[/b] For any $ n$ real numbers $ x_1$, $ x_2$, ..., $ x_n$ satisfying $ x_1 \leq x_2 \leq \ldots \leq x_ n$, we have $ \sum^{n}_{k \equal{} 1} a_k \cdot x_k \leq \sum^{n}_{k \equal{} 1} b_k \cdot x_k,$
[b]ii)[/b] We have $ \sum^{s}_{k \equal{} 1} a_k \leq \sum^{s}_{k \equal{} 1} b_k$ for every $ s\in\left\{1,2,...,n\minus{}1\right\}$ and $ \sum^{n}_{k \equal{} 1} a_k \equal{} \sum^{n}_{k \equal{} 1} b_k$.
Let $m, n$ be natural numbers and let $m\cdot n$ be a multiple of $4$. A chessboard with $m \times n$ fields are covered with $1 \times 2$ large dominoes without gaps and without overlapping. Show that the number of dominoes that are parallel to a edge of the chess board is fixed .
[hide=original wording] Seien m, n natu¨rliche Zahlen und sei m · n ein Vielfaches von 4. Ein Schachbrett mit m × n Feldern sei mit 1 × 2 großen Dominosteinen lu¨ckenlos und u¨berlappungsfrei u¨berdeckt. Zeige, dass die Anzahl der Dominosteine, die zu einer fest gew¨ahlten Kante des Schachbrettes parallel sind, gerade ist. [/hide]
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function with the following property: for all $\alpha \in \mathbb{R}_{>0}$, the sequence $(a_n)_{n \in \mathbb{N}}$ defined as $a_n = f(n\alpha)$ satisfies $\lim_{n \to \infty} a_n = 0$. Is it necessarily true that $\lim_{x \to +\infty} f(x) = 0$?
Sum of $ 100 $ natural distinct numbers is $ 9999 $. Prove that $ 2006 $ divide their product.
10.5 Let $BB'$, $CC'$ be the altitudes of an acute triangle $ABC$. Two circles through $A$ and $C'$ are tangent to $BC$ at points $P$ and $Q$. Prove that $A, B', P, Q$ are concyclic.
A chord $AB$ is drawn in the circle, on which the point $P$ is selected in such a way that $AP = 2PB$. The chord $DE$ is perpendicular to the chord $AB $ and passes through the point $P$. Prove that the midpoint of the segment $AP$ is the orthocener of the triangle $AED$.
Point $O$ is selected in equilateral $\triangle ABC$ such that the sum of the distances from $O$ to each side of $ABC$ is $15$. Compute the area of $ABC$.
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$\textbf{(A) } 15\sqrt3\qquad\textbf{(B) } 30\sqrt3\qquad\textbf{(C) } 50\sqrt3\qquad\textbf{(D) } 75\sqrt3\qquad\textbf{(E) } 225\sqrt3$
Let $\omega$ be a circle with center $O$, and let $l$ be a line not intersecting $\omega$. $E$ is a point on $l$ such that $OE$ is perpendicular with $l$. Let $M$ be an arbitrary point on $M$ different from $E$. Let $A$ and $B$ be distinct points on the circle $\omega$ such that $MA$ and $MB$ are tangents to $\omega$. Let $C$ and $D$ be the foot of perpendiculars from $E$ to $MA$ and $MB$ respectively. Let $F$ be the intersection of $CD$ and $OE$. As $M$ moves, determine the locus of $F$.
Show that any subset of $ A=\{ 1,2,...,2007\} $ having $ 27 $ elements contains three distinct numbers such that the greatest common divisor of two of them divides the other one.
[i]Dan Schwarz[/i]
Let $a_{1}={11}^{11}$, $a_{2}={12}^{12}$, $a_{3}={13}^{13}$, and \[a_{n}= \vert a_{n-1}-a_{n-2}\vert+\vert a_{n-2}-a_{n-3}\vert, n \ge 4.\] Determine $a_{{14}^{14}}$.