Found problems: 85335
The following picture illustrates the model of the Tháp Rùa (The Central Tower in Hanoi), which consists of $3$ levels. For the first and second levels, each has $10$ doorways among which $3$ doorways are located at the front, $3$ at the back, $2$ on the right side and $2$ on the left side. The third level is on the top of the tower model and has no doorways. The front of the tower model is signified by a circle symbol on the top level (Figure).
We paint the tower model with three colors: Blue, Yellow and Brown by fulfilling the following requirements:
(a) The top level is painted with only one color.
(b) The $3$ doorways at the front on the second level are painted with the same color.
(c) The $3$ doorways at the front on the first level are painted with the same color.
(d) Each of the remaining $14$ doorways is painted with one of the three colors in such a way that any two adjacent doorways with a common side on the same level, including the pairs at the same corners, are painted with different colors.
How many ways are there to paint the first level?
How many ways are there to paint the entire tower model?
[img]https://cdn.artofproblemsolving.com/attachments/f/9/2249f8595a8efe711680f3dfb8ff959c140a21.png[/img]
Let $f:[0,1]\to\mathbb R$ be a continuous function such that $\int^1_0x^mf(x)dx=0$ for $m=0,1,\ldots,1999$. Prove that $f$ has at least $2000$ zeroes on the segment $[0,1]$.
Four circles in a plane have a common center. Their radii form a strictly increasing arithmetic progression. Prove that there is no square with each vertex lying on a different circle.
Let $s (n)$ denote the sum of digits of a positive integer $n$. Using six different digits, we formed three 2-digits $p, q, r$ such that $$p \cdot q \cdot s(r) = p\cdot s(q) \cdot r = s (p) \cdot q \cdot r.$$ Find all such numbers $p, q, r$.
In a triangle $ABC$, the inner and outer bisectors of the $\angle A$ meet the line $BC$ at $D$ and $E$, respectively. Let $d$ be a common tangent of the circumcircle $(O)$ of $\triangle ABC$ and the circle with diameter $DE$ and center $F$. The projections of the tangency points onto $FO$ are denoted by $P$ and $Q$, and the length of their common chord is denoted by $m$. Prove that $PQ = m$
Let $I$ be the incenter of triangle $ABC$, and $K$ be the common point of $BC$ with the external bisector of angle $A$. The line $KI$ meets the external bisectors of angles $B$ and $C$ at points $X$ and $Y$ . Prove that $\angle BAX = \angle CAY$
Let $ G$ be a connected graph with $ n$ vertices and $ m$ edges such that each edge is contained in at least one triangle. Find the minimum value of $ m$.
A triangle $ABC$ is given. Let $\omega_1$, $\omega_2$, $\omega_3$, $\omega_4$ be circles centered at points $X$, $Y$, $Z$, $T$ respectively such that each of lines $BC$, $CA$, $AB$ cuts off on them four equal chords. Prove that the centroid of $ABC$ divides the segment joining $X$ and the radical center of $\omega_2$, $\omega_3$, $\omega_4$ in the ratio $2:1$ from $X$.
A park is in the shape of a regular hexagon $2$ km on a side. Starting at a corner, Alice walks along the perimeter of the park for a distance of $5$ km. How many kilometers is she from her starting point?
$ \textbf{(A)}\ \sqrt{13}\qquad\textbf{(B)}\ \sqrt{14}\qquad\textbf{(C)}\ \sqrt{15}\qquad\textbf{(D)}\ \sqrt{16}\qquad\textbf{(E)}\ \sqrt{17}$
The bisector of $ \angle BAC $ of a triangle $ ABC $ meet the segment $ BC $ at $ D. $ Through the midpoint of $ AD $ passes aline that intersects $ AB,AC $ at $ M,N, $ respectively. Show that:
$$ \frac{1}{MA}+\frac{1}{NA} =2\left( \frac{1}{AB} +\frac{1}{AC} \right) $$
[i]Toni Mihalcea[/i]
We are given a convex quadrilateral $ABCD$ whose angles are not right. Assume there are points $P, Q, R, S$ on its sides $AB, BC, CD, DA$, respectively, such that $PS \parallel BD$, $SQ \perp BC$, $PR \perp CD$. Furthermore, assume that the lines $PR, SQ$, and $AC$ are concurrent. Prove thatthe points $P, Q, R, S$ are concyclic.
Find all ordered 6-tuples satisfy following system of modular equation:
$ab+a'b' \equiv 1 $(mod 15)
$bc+b'c' \equiv 1 $(mod 15)
$ca+c'a' \equiv 1 $(mod 15)
Given that $a,b,c,a',b',c' \epsilon (0;1;2;...;14)$
Suppose that every point in the plane is coloured either black or white. Must there be an equilateral triangle such that all of its vertices are the same colour?
Given a point $P$ lying on a line $g,$ and given a circle $K.$ Construct a circle passing through the point $P$ and touching the circle $K$ and the line $g.$
The points of a circle of unit radius are colored in two colors. Prove that $3$ points of the same color can be chosen such that the area of the triangle they define is at least $\frac{9}{10}$.
How many $10$-digit positive integers can be written by using four $0$s, five $1$s, and one $2$?
$
\textbf{(A)}\ 1260
\qquad\textbf{(B)}\ 1134
\qquad\textbf{(C)}\ 756
\qquad\textbf{(D)}\ 630
\qquad\textbf{(E)}\ \text{None of above}
$
Let $a$ be a positive real number. Then prove that the polynomial
\[ p(x)=a^3x^3+a^2x^2+ax+a \]
has integer roots if and only if $a=1$ and determine those roots.
For every $n$ in the set $\mathrm{N} = \{1,2,\dots \}$ of positive integers, let $r_n$ be the minimum value of $|c-d\sqrt{3}|$ for all nonnegative integers $c$ and $d$ with $c+d=n$. Find, with proof, the smallest positive real number $g$ with $r_n \leq g$ for all $n \in \mathbb{N}$.
Find the limit as below.
\[\lim_{n\to\infty} \frac{(1^2+2^2+\cdots +n^2)(1^3+2^3+\cdots +n^3)(1^4+2^4+\cdots +n^4)}{(1^5+2^5+\cdots +n^5)^2}\]
There are many opposition societies in the city of N.
Each society consists of $10$ members. It is known that for every $2004$ societies there is a person belonging to at least $11$ of them.
Prove that the government can arrest $2003$ people so that at least one member of each society is arrested.
[i]Proposed by V.Dolnikov, D.Karpov[/i]
Prove that for positive real numbers $a,b,c$ such that $a+b+c=1$,
$$a\sqrt{2b+1}+b\sqrt{2c+1}+c\sqrt{2a+1}\le \sqrt{2-(a^2+b^2+c^2)}.$$
A right triangle has the property that it's sides are pairwise relatively prime positive integers and that the ratio of it's area to it's perimeter is a perfect square. Find the minimum possible area of this triangle.
Let $n > 1$ be an integer.
Determine the greatest common divisor of the set of numbers $\left\{ \left( \begin{matrix}
2n \\
2i+1 \\
\end{matrix} \right):0 \le i \le n-1 \right\}$
i.e. the largest positive integer, dividing $\left( \begin{matrix}
2n \\
2i+1 \\
\end{matrix} \right)$ without remainder for every $i = 0, 1, ..., n–1$ .
(Here $\left( \begin{matrix}
m \\
l \\
\end{matrix} \right)=\text{C}_{m}^{l}=\frac{m\text{!}}{l\text{!}\left( m-l \right)\text{!}}$ is binomial coefficient.)
A sequence $(a_n)$ is defined recursively by $a_1=0, a_2=1$ and for $n\ge 3$,
\[a_n=\frac12na_{n-1}+\frac12n(n-1)a_{n-2}+(-1)^n\left(1-\frac{n}{2}\right).\]
Find a closed-form expression for $f_n=a_n+2\binom{n}{1}a_{n-1}+3\binom{n}{2}a_{n-2}+\ldots +(n-1)\binom{n}{n-2}a_2+n\binom{n}{n-1}a_1$.
Each vertex of a regular dodecagon (12-gon) is to be colored either red or blue, and thus there are $2^{12}$ possible colorings. Find the number of these colorings with the property that no four vertices colored the same color are the four vertices of a rectangle.