Does there exist a natural number $N$ which is a power of$2$, such that one can permute its decimal digits to obtain a different power of $2$?

Call two circles in three-dimensional space pairwise tangent at a point $ P$ if they both pass through $ P$ and lines tangent to each circle at $ P$ coincide. Three circles not all lying in a plane are pairwise tangent at three distinct points. Prove that there exists a sphere which passes through the three circles.

Click for solution We use a spatial inversion (just like in 2D) which has one of the 3 pts of tangency as pole. It turns the 2 circles which pass through the pole into lines, which are both tangent to the circle which is the image of the 3'rd circle, so the images of the 3 circles are coplanar (a circle and 2 lines tangent to it), which means that the points were originally on a sphere (which passes through the point chosen as pole). It's a nice problem!

In a deck of $n > 1$ cards, some digits from $1$ to$8$are written on each card. A digit may occur more than once, but at most once on a certain card. On each card at least one digit is written, and no two cards are denoted by the same set of digits. Suppose that for every $k=1,2,\dots,7$ digits, the number of cards that contain at least one of them is even. Find $n$.

A sequence of natural numbers $c_1, c_2,\dots$ is called perfect if every natural number $m$ with $1\le m \le c_1 +\dots+ c_n$ can be represented as $m =\frac{c_1}{a_1}+\frac{c_2}{a_2}+\dots+\frac{c_n}{a_n}$ Given $n$, find the maximum possible value of $c_n$ in a perfect sequence $(c_i)$.

Circles $ C_1$ and $ C_2$ with centers at $ O_1$ and $ O_2$ respectively meet at points $ A$ and $ B$. The radii $ O_1B$ and $ O_2B$ meet $ C_1$ and $ C_2$ at $ F$ and$ E$. The line through $ B$ parallel to $ EF$ intersects $ C_1$ again at $ M$ and $ C_2$ again at $ N$. Prove that $ MN = AE + AF$.

Two triangles $ ABC$and $ A'B'C'$ are positioned in the space such that the length of every side of $ \triangle ABC$ is not less than $ a$, and the length of every side of $ \triangle A'B'C'$ is not less than $ a'$. Prove that one can select a vertex of $ \triangle ABC$ and a vertex of $ \triangle A'B'C'$ so that the distance between the two selected vertices is not less than $ \sqrt {\frac {a^2 + a'^2}{3}}$.

Click for solution let $ O$ be an arbitrary point in the space,and for any point $ X$ let $ x$ denote the vector from $ O$ to $ X$.let $ S = \{A,B,C\}$,$ \sigma_1 = a + b + c$ and $ \sigma_2 = a\cdot b + b\cdot c + c\cdot a$.define $ S',\sigma_1',\sigma_2'$ analogously in terms of $ A',B'$ and $ C'$. given $ (P,P')\in S\times S'$.note that $ PP'^2 = |p|^2 - 2p\cdot p' + |p'|^2$.summing over all 9 possible pairs yields the total $ t$ with: \begin{eqnarray*}t & = & \sum_{P\in S}3|p|^2+\sum_{P'\in S'}3|p'|^2-2\sigma_1\cdot\sigma_1' \\ & = & \sum_{P\in S}3|p|^2 + \sum_{P'\in S'}3|p'|^2 + \left(|\sigma_1 - \sigma_1'|^2 - |\sigma_1|^2 - |\sigma_1'|^2\right) \\ & \geq & \sum_{P\in S}3|p|^2 + \sum_{P'\in S'}3|p'|^2 - |\sigma_1|^2 - |\sigma_1'|^2 \\ & = & \left(\sum_{P\in S}2|p|^2 - 2\sigma_2\right) + \left(\sum_{P'\in S'}2|p'|^2 - 2\sigma_2'\right) \\ & = & |a - b|^2 + |b-c|^2 + |c - a|^2 + |a' - b'|^2 + |b' - c'|^2 + |c' - a'|^2 \\ & = & AB^2 + BC^2 + CA^2 + A'B'^2 + B'C'^2 + C'A'^2 \\ & \geq & 3(a^2 + a'^2)\end{eqnarray*} thus one of the nine distances is greater or equal to: $ \sqrt {\frac t9}\geq\sqrt {\frac {a^2 + a'^2}3}$ as desired.the equality case is never reached.because $ A,B$ and $ C$ do not all lie on a line,the nine values $ PP'^2$ we sum to find $ t$ cannot all be equal to each other.

Two circles intersect at two points $A$ and $B$. A line $\ell$ which passes through the point $A$ meets the two circles again at the points $C$ and $D$, respectively. Let $M$ and $N$ be the midpoints of the arcs $BC$ and $BD$ (which do not contain the point $A$) on the respective circles. Let $K$ be the midpoint of the segment $CD$. Prove that $\measuredangle MKN = 90^{\circ}$.

Let $A$ and $B$ be arbitrary finite sets and let $f: A\longrightarrow B$ and $g: B\longrightarrow A$ be functions such that $g$ is not onto. Prove that there is a subset $S$ of $A$ such that $\frac{A}{S}=g(\frac{B}{f(S)})$.

Suppose $f : \mathbb{N} \longrightarrow \mathbb{N}$ is a function that satisfies $f(1) = 1$ and $f(n + 1) =\{\begin{array}{cc} f(n)+2&\mbox{if}\ n=f(f(n)-n+1),\\f(n)+1& \mbox{Otherwise}\end {array}$ $(a)$ Prove that $f(f(n)-n+1)$ is either $n$ or $n+1$. $(b)$ Determine$f$.

Let us denote $\prod = \{(x, y) | y > 0\}$. We call a semicircle in $\prod$ with center on the $x-\text{axis}$ a semi-line. Two intersecting semi-lines determine four semi-angles. A bisector of a semi-angle is a semi-line that bisects the semi-angle. Prove that in every semi-triangle (determined by three semi-lines) the bisectors are concurrent.

Find all f:N $\longrightarrow$ N that: a) $f(m)=1 \Longleftrightarrow m=1 $ b) $d=gcd(m,n) f(m\cdot n)= \frac{f(m)\cdot f(n)}{f(d)} $ c) $ f^{2000}(m)=f(m) $

Let $n$ points be given on a circle, and let $nk + 1$ chords between these points be drawn, where $2k+1 < n$. Show that it is possible to select $k+1$ of the chords so that no two of them intersect.

In a tennis tournament where $ n$ players $ A_1,A_2,\dots,A_n$ take part, any two players play at most one match, and $ k \leq \frac {n(n - 1)}{2}$ $ 2$ matches are played. The winner of a match gets $ 1$ point while the loser gets $ 0$. Prove that a sequence $ d_1,d_2,\dots,d_n$ of nonnegative integers can be the sequence of scores of the players ($ d_i$ being the score of$ A_i$) if and only if $ (i)\ \ d_1 + d_2 + \dots + d_n = k$, and $ (ii)\ \text{for any} X\subset\{A_1,\dots,A_n\}$, the number of matches between the players in $ X$ is at most $ \sum_{A_j\in X}d_j$

Isosceles triangles $A_3A_1O_2$ and $A_1A_2O_3$ are constructed on the sides of a triangle $A_1A_2A_3$ as the bases, outside the triangle. Let $O_1$ be a point outside $\Delta A_1A_2A_3$ such that $\angle O_1A_3A_2 =\frac 12\angle A_1O_3A_2$ and $\angle O_1A_2A_3 =\frac 12\angle A_1O_2A_3$. Prove that $A_1O_1\perp O_2O_3$, and if $T$ is the projection of $O_1$ onto $A_2A_3$, then $\frac{A_1O_1}{O_2O_3} = 2\frac{O_1T}{A_2A_3}$.

A circle$\Gamma$ with radius $R$ and center $\omega$, and a line $d$ are drawn on a plane, such that the distance of $\omega$ from $d$ is greater than $R$. Two points $M$ and $N$ vary on $d$ so that the circle with diameter $MN$ is tangent to $\Gamma$. Prove that there is a point $P$ in the plane from which all the segments $MN$ are visible at a constant angle.

Let $n$ be a positive integer. Suppose $S$ is a set of ordered $n-\mbox{tuples}$ of nonnegative integers such that, whenever $(a_1,\dots,an)\in S$ and $b_i$ are nonnegative integers with$b_i\le a_i$, the $n-\text{tuple}$ $(b_1,\dots,b_n)$ is also in $S$. If $h_m$ is the number of elements of $S$ with the sum of components equal to$m$, prove that $h_m$ is a polynomial in $m$ for all sufficiently large$m$.

Suppose that $a, b, c$ are real numbers such that for all positive numbers $x_1,x_2,\dots,x_n$ we have $(\frac{1}{n}\sum_{i=1}^nx_i)^a(\frac{1}{n}\sum_{i=1}^nx_i^2)^b(\frac{1}{n}\sum_{i=1}^nx_i^3)^c\ge 1$ Prove that vector $(a, b, c)$ is a nonnegative linear combination of vectors $(-2,1,0)$ and $(-1,2,-1)$.

Prove that for every natural number $ n$ there exists a polynomial $ p(x)$ with integer coefficients such that$ p(1),p(2),...,p(n)$ are distinct powers of $ 2$ .