For $n \in \mathbb N$, let $f(n)$ be the number of positive integers $k \leq n$ that do not contain the digit $9$. Does there exist a positive real number $p$ such that $\frac{f(n)}{n} \geq p$ for all positive integers $n$?

Among the spatial points $ A,B,C,D, $ at most two of are aparted at a distance greater than $ 1. $ Find the the maximum value of the expression: $$ g(A,B,C,D) =AB+BC+ AD+CA+DB+DC. $$

In the plane, there are $n \geqslant 6$ pairwise disjoint disks $D_{1}, D_{2}, \ldots, D_{n}$ with radii $R_{1} \geqslant R_{2} \geqslant \ldots \geqslant R_{n}$. For every $i=1,2, \ldots, n$, a point $P_{i}$ is chosen in disk $D_{i}$. Let $O$ be an arbitrary point in the plane. Prove that \[O P_{1}+O P_{2}+\ldots+O P_{n} \geqslant R_{6}+R_{7}+\ldots+R_{n}.\] (A disk is assumed to contain its boundary.)

Astrophysicists have discovered a minor planet of radius $30$ kilometers whose surface is completely covered in water. A spherical meteor hits this planet and is submerged in the water. This incidence causes an increase of $1$ centimeters to the height of the water on this planet. What is the radius of the meteor in meters?

Prove that the set of all divisors of a positive integer which is not a perfect square can be divided into pairs so that in each pair, one number is divided by another.

There are $1001$ red marbles and $1001$ black marbles in a box. Let $P_s$ be the probability that two marbles drawn at random from the box are the same color, and let $P_d$ be the probability that they are different colors. Find $|P_s-P_d|$. $\textbf{(A) }0\qquad\textbf{(B) }\dfrac1{2002}\qquad\textbf{(C) }\dfrac1{2001}\qquad\textbf{(D) }\dfrac2{2001}\qquad\textbf{(E) }\dfrac1{1000}$

[b]p1.[/b] A light beam shines from the origin into the unit square at an angle of $\theta$ to one of the sides such that $\tan \theta = \frac{13}{17}$ . The light beam is reflected by the sides of the square. How many times does the light beam hit a side of the square before hitting a vertex of the square? [img]https://cdn.artofproblemsolving.com/attachments/5/7/1db0aad33ed9bf82bee3303c7fbbe0b7c2574f.png[/img] [b]p2.[/b] Alex is given points $A_1,A_2,...,A_{150}$ in the plane such that no three are collinear and $A_1$, $A_2$, $...$, $A_{100}$ are the vertices of a convex polygon $P$ containing $A_{101}$, $A_{102}$, $ ...$, $A_{150}$ in its interior. He proceeds to draw edges $A_iA_j$ such that no two edges intersect (except possibly at their endpoints), eventually dividing $P$ up into triangles. How many triangles are there? [img]https://cdn.artofproblemsolving.com/attachments/d/5/12c757077e87809837d16128b018895a8bcc94.png[/img] [b]p3. [/b]The polynomial P(x) has the property that $P(1)$, $P(2)$, $P(3)$, $P(4)$, and $P(5)$ are equal to $1$, $2$, $3$, $4$,$5$ in some order. How many possibilities are there for the polynomial $P$, given that the degree of $P$ is strictly less than $4$? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a$ and $b$ be real numbers from interval $\left[0,\frac{\pi}{2}\right]$. Prove that $$\sin^6 {a}+3\sin^2 {a}\cos^2 {b}+\cos^6 {b}=1$$ if and only if $a=b$

A square $ (n \minus{} 1) \times (n \minus{} 1)$ is divided into $ (n \minus{} 1)^2$ unit squares in the usual manner. Each of the $ n^2$ vertices of these squares is to be coloured red or blue. Find the number of different colourings such that each unit square has exactly two red vertices. (Two colouring schemse are regarded as different if at least one vertex is coloured differently in the two schemes.)

An eighth degree polynomial funtion $ y \equal{} ax^8 \plus{} bx^7 \plus{} cx^6 \plus{} dx^5 \plus{} ex^4 \plus{} fx^3 \plus{} gx^2\plus{}hx\plus{}i\ (a\neq 0)$ touches the line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha ,\ \beta ,\ \gamma ,\ \delta \ (\alpha < \beta < \gamma <\delta).$ Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta ,\gamma ,\ \delta .$

If $A$ is the $3\times 3$ square matrix $\begin{bmatrix} 5 & 3 & 8\\ 2 & 2 & 5\\ 3 & 5 & 1 \end{bmatrix}$ and $B$ is the $4\times 4$ square matrix $\begin{bmatrix} 32 & 2 & 4 & 3 \\ 3 & 4 & 8 & 3 \\ 11 & 3 & 6 & 1 \\ 5 & 5 & 10 & 1 \end{bmatrix} $ find the sum of the determinants of $A$ and $B$.

Peter's family ordered a 12-slice pizza for dinner. Peter ate one slice and shared another slice equally with his brother Paul. What fraction of the pizza did Peter eat? $\textbf{(A)}\hspace{.05in} \dfrac1{24}\qquad \textbf{(B)}\hspace{.05in}\dfrac1{12} \qquad \textbf{(C)}\hspace{.05in}\dfrac18 \qquad \textbf{(D)}\hspace{.05in}\dfrac16 \qquad \textbf{(E)}\hspace{.05in}\dfrac14 $

A sequence $(a_{n})$ is defined by $a_{1}= 1$ and $a_{n}= a_{1}a_{2}...a_{n-1}+1$ for $n \geq 2.$ Find the smallest real number $M$ such that $\sum_{n=1}^{m}\frac{1}{a_{n}}<M\; \forall m\in\mathbb{N}$.

Let $\mathbb{N}^*$ be the set of positive integers. Define $a_1=2$, and for $n=1, 2, \ldots,$\[ a_{n+1}=\min\{\lambda|\frac{1}{a_1}+\frac{1}{a_2}+\cdots\frac{1}{a_n}+\frac{1}{\lambda}<1,\lambda\in \mathbb{N}^*\}\] Prove that $a_{n+1}=a_n^2-a_n+1$ for $n=1,2,\ldots$.

Positive sequences $\{a_n\},\{b_n\}$ satisfy:$a_1=b_1=1,b_n=a_nb_{n-1}-\frac{1}{4}(n\geq 2)$. Find the minimum value of $4\sqrt{b_1b_2\cdots b_m}+\sum_{k=1}^m\frac{1}{a_1a_2\cdots a_k}$,where $m$ is a given positive integer.

Given an integer $n\ge 2$, define $M_0 (x_0, y_0)$ to be an intersection point of the parabola $y^2=nx-1$ and the line $y=x$. Prove that for any positive integer $m$, there exists an integer $k\ge 2$ such that $(x^m_0, y^m_0)$ is an intersection point of $y^2=mx-1$ and the line $y=x$.

Find the sum of all positive integers $n$ such that $7<n < 100$ and $1573_{n}$ has $6$ factors when written in base $10$. [i]Proposed by Aidan Duncan[/i]

In $\triangle ABC$, we have $AB>AC>BC$. $D,E,F$ are the tangent points of the inscribed circle of $\triangle ABC$ with the line segments $AB,BC,AC$ respectively. The points $L,M,N$ are the midpoints of the line segments $DE,EF,FD$. The straight line $NL$ intersects with ray $AB$ at $P$, straight line $LM$ intersects ray $BC$ at $Q$ and the straight line $NM$ intersects ray $AC$ at $R$. Prove that $PA \cdot QB \cdot RC = PD \cdot QE \cdot RF$.

Let $C$ be a closed plane curve that has a continuously turning tangent and bounds a convex region. If $T$ is a triangle inscribed in $C$ with maximum perimeter, show that the normal to $C$ at each vertex of $T$ bisects the angle of $T$ at that vertex. If a triangle $T$ has the property just described, does it necessarily have maximum perimeter? What is the situation if $C$ is a circle?

Can the product of $2$ consecutive natural numbers equal the product of $2$ consecutive even natural numbers? (natural means positive integers)

$ABCD$ is a quadrilateral having both an inscribed circle (one tangent to all four sides) with center $I,$ and a circumscribed circle with center $O$. Let $S$ be the point of intersection of the diagonals of $ABCD$. Show that if any two of $S, I$ and $O$ coincide, then $ABCD$ is a square (and hence all three coincide).

There are $4$ pairs of men and women, and all $8$ people are arranged in a row so that in each pair the woman is somewhere to the left of the man. How many such arrangements are there?

A tennis competition takes place over four days, the number of participants is $2n$ with $n \ge 5$. Each participant plays exactly once a day (a couple of participants may be more times). Prove that such competition can end with exactly one winner and exactly three players in second place and such that there are no players with four lost games,

The only prime factors of an integer $n$ are 2 and 3. If the sum of the divisors of $n$ (including itself) is $1815$, find $n$.

Let $S$ be the set of all real values of $x$ with $0 < x < \pi/2$ such that $\sin x$, $\cos x$, and $\tan x$ form the side lengths (in some order) of a right triangle. Compute the sum of $\tan^2 x$ over all $x$ in $S$.