Found problems: 85335
A wobbly number is a positive integer whose $digits$ in base $10$ are alternatively non-zero and zero the units digit being non-zero. Determine all positive integers which do not divide any wobbly number.
Let $n$ be a posititve integer and let $M$ the set of all all integer cordinates $(a,b,c)$ such that $0 \le a,b,c \le n$. A frog needs to go from the point $(0,0,0)$ to the point $(n,n,n)$ with the following rules:
$\cdot$ The frog can jump only in points of $M$
$\cdot$ The frog can't jump more than $1$ time over the same point.
$\cdot$ In each jump the frog can go from $(x,y,z)$ to $(x+1,y,z)$, $(x,y+1,z)$, $(x,y,z+1)$ or $(x,y,z-1)$
In how many ways the Frog can make his target?
$a, b, c$ are three distinct integers and $n$ is a positive integer. Show that $$\frac{a^n}{(a - b)(a - c)}+\frac{ b^n}{(b - a)(b - c)} +\frac{ c^n}{(c - a)(c - b)}$$ is an integer.
Prove that, for any natural numbers $k,m,n$: $[k,m] \cdot [m,n] \cdot [n,k] \ge [k,m,n]^2$
Let set $A=\{1,2,\ldots,n\} ,$ and $X,Y$ be two subsets (not necessarily distinct) of $A.$ Define that $\textup{max} X$ and $\textup{min} Y$ represent the greatest element of $X$ and the least element of $Y,$ respectively. Determine the number of two-tuples $(X,Y)$ which satisfies $\textup{max} X>\textup{min} Y.$
Find all natural numbers that can be expressed in a unique way as a sum of five or less perfect squares.
A convex polygon $ A_1,A_2,\cdots ,A_n$ is inscribed in a circle with center $ O$ and radius $ R$ so that $ O$ lies inside the polygon. Let the inradii of the triangles $ A_1A_2A_3, A_1A_3A_4, \cdots , A_1A_{n \minus{} 1}A_n$ be denoted by $ r_1,r_2,\cdots ,r_{n \minus{} 2}$. Prove that $ r_1 \plus{} r_2 \plus{} ... \plus{} r_{n \minus{} 2}\leq R(n\cos \frac {\pi}{n} \minus{} n \plus{} 2)$.
Find
$$ \inf_{\substack{ n\ge 1 \\ a_1,\ldots ,a_n >0 \\ a_1+\cdots +a_n <\pi }} \left( \sum_{j=1}^n a_j\cos \left( a_1+a_2+\cdots +a_j \right)\right) . $$
Show that in each set of ten consecutive integers there is one that is coprime with each of the other integers. (For example, in the set $ \{ 114,115,...,123 \}$ there are two such numbers: $ 119$ and $ 121.)$
ff nonnegative real numbers$ x_1,x_2,...,x_n$ satisfy $x_1 +...+x_n\le \frac12$, prove that
$$(1-x_1)(1-x_2)...(1-x_n) \ge \frac12$$
The straight river is one and a half kilometers wide and has a current of $8$ kilometers per hour. A boat capable of traveling $10$ kilometers per hour in still water, sets out across the water. How many minutes will it take the boat to reach a point directly across from where it started?
In the circle $\omega$, we draw a chord $BC$, which is not a diameter. Point $A$ moves in a circle $\omega$. $H$ is the orthocenter triangle $ABC$. Prove that for any location of point $A$, a circle constructed on $AH$ as on diameter, touches two fixed circles $\omega_1$ and $\omega_2$.
(Dmitry Prokopenko)
A quadrilateral $ABCD$ is inscribed in a circle $k$ where $AB$ $>$ $CD$,and $AB$ is not paralel to $CD$.Point $M$ is the intersection of diagonals $AC$ and $BD$, and the perpendicular from $M$ to $AB$ intersects the segment $AB$ at a point $E$.If $EM$ bisects the angle $CED$ prove that $AB$ is diameter of $k$.
Proposed by Emil Stoyanov,Bulgaria
The number of intengral points $(x,y)$ that fit $(|x|-1)^2+(|y|-1)^2<2$ is
$\text{(A)}16\qquad\text{(B)}17\qquad\text{(C)}18\qquad\text{(D)}25$
The zeroes of a fourth degree polynomial $f(x)$ form an arithmetic progression. Prove that the three zeroes of the polynomial $f'(x)$ also form an arithmetic progression.
Let $ ABC$ be an equilateral triangle with side length equal to $ N \in \mathbb{N}.$ Consider the set $ S$ of all points $ M$ inside the triangle $ ABC$ satisfying
\[ \overrightarrow{AM} \equal{} \frac{1}{N} \cdot \left(n \cdot \overrightarrow{AB} \plus{} m \cdot \overrightarrow{AC} \right)\]
with $ m, n$ integers, $ 0 \leq n \leq N,$ $ 0 \leq m \leq N$ and $ n \plus{} m \leq N.$
Every point of S is colored in one of the three colors blue, white, red such that
[b](i) [/b]no point of $ S \cap [AB]$ is coloured blue
[b](ii)[/b] no point of $ S \cap [AC]$ is coloured white
[b](iii)[/b] no point of $ S \cap [BC]$ is coloured red
Prove that there exists an equilateral triangle the following properties:
[b](1)[/b] the three vertices of the triangle are points of $ S$ and coloured blue, white and red, respectively.
[b](2)[/b] the length of the sides of the triangle is equal to 1.
[i]Variant:[/i] Same problem but with a regular tetrahedron and four different colors used.
Suppose sequence $\{a_i\} = a_1, a_2, a_3, ....$ satisfies $a_{n+1} = \frac{1}{a_n+1}$ for all positive integers $n$. Define $b_k$ for positive integers $k \ge 2$ to be the minimum real number such that the product $a_1 \cdot a_2 \cdot ...\cdot a_k$ does not exceed $b_k$ for any positive integer choice of $a_1$. Find $\frac{1}{b_2}+\frac{1}{b_3}+\frac{1}{b_4}+...+\frac{1}{b_{10}}.$
.
Find all integers $ n\ge 2$ with the following property:
There exists three distinct primes $p,q,r$ such that
whenever $ a_1,a_2,a_3,\cdots,a_n$ are $ n$ distinct positive integers with the property that at least one of $ p,q,r$ divides $ a_j - a_k \ \forall 1\le j\le k\le n$,
one of $ p,q,r$ divides all of these differences.
Let $M$ be the midpoint of side $BC$ of a triangle $ABC$. Let the median $AM$ intersect the incircle of $ABC$ at $K$ and $L,K$ being nearer to $A$
than $L$. If $AK = KL = LM$, prove that the sides of triangle $ABC$ are in the ratio $5 : 10 : 13$ in some order.
Let $D$ be a point on side $[BC]$ of $\triangle ABC$ such that $|AB|=3, |CD|=1$ and $|AC|=|BD|=\sqrt{5}$. If the $B$-altitude of $\triangle ABC$ meets $AD$ at $E$, what is $|CE|$?
$
\textbf{(A)}\ \dfrac{2}{\sqrt{5}}
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ \dfrac{2}{\sqrt{3}}
\qquad\textbf{(D)}\ \dfrac{\sqrt{5}}{2}
\qquad\textbf{(E)}\ \dfrac{3}{2}
$
For some $99$-digit number $k$, there exist two different $100$-digit numbers $n$ such that the sum of all natural numbers from $1$ to $n$ ends in the same $100$ digits as the number $kn$, but is not equal to it. Prove that $k-3$ is divisible by $5$.
There are $100$ white points on a circle. Asya and Borya play the following game: they alternate, starting with Asya, coloring a white point in green or blue. Asya wants to obtain as much as possible pairs of adjacent points of distinct colors, while Borya wants these pairs to be as less as possible. What is the maximal number of such pairs Asya can guarantee to obtain, no matter how Borya plays.
There are two equal circles of radius $1$ placed inside the triangle $ABC$ with side $BC = 6$. The circles are tangent to each other, one is inscribed in angle $B$, the other one is inscribed in angle $C$.
(a) Prove that the centroid $M$ of the triangle $ABC$ does not lie inside any of the given circles.
(b) Prove that if $M$ lies on one of the circles, then the triangle $ABC$ is isosceles.
Segment $AB$ is the diameter of a circle. Points $C$ and $D$ lie on the circle. The rays $AC$ and $AD$ intersect the tangent to the circle at point $B$ at points $P$ and $Q$, respectively. Show that points $C, D, P$ and $Q$ lie on a circle.
Inside the triangle $DD_1D_3$ the cevian $DD_2$ is constructed. Perpendiculars from $D_1, D_2$ and $D_3$ to lines $DD_1, DD_2$ and $DD_3$, respectively, intersect in points $A,B$ and $C$ such that $AB\perp DD_1, AC\perp DD_2, BC\perp DD_3$. Prove that $\frac{AC}{DD_2}=\frac{AB}{DD_1}+\frac{BC}{DD_3}$.