Find the best approximation of $\sqrt{3}$ by a rational number with denominator less than or equal to $15$

Four small spheres each with radius $6$ are each internally tangent to a large sphere with radius $17$. The four small spheres form a ring with each of the four spheres externally tangent to its two neighboring small spheres. A sixth intermediately sized sphere is internally tangent to the large sphere and externally tangent to each of the four small spheres. Its radius is $\frac{m}{n}$ , where m and n are relatively prime positive integers. Find $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/7/2/25955cd6f22bc85f2f3c5ba8cd1ee0821c9d50.png[/img]

The angle bisector of $\angle ABC$ of triangle $ABC$ ($AB>BC$) cuts the circumcircle of that triangle in $K$. The foot of the perpendicular from $K$ to $AB$ is $N$, and $P$ is the midpoint of $BN$. The line through $P$ parallel to $BC$ cuts line $BK$ in $T$. Prove that the line $NT$ passes through the midpoint of $AC$.

Let $ABC$ be such a triangle that $\sphericalangle BAC = 45^{\circ}$ and $ \sphericalangle ACB > 90^{\circ}.$ Show that $BC + (\sqrt{2} - 1)\cdot CA < AB.$

Let a regular polygon with $p$ vertices be given, where $p$ is an odd prime number. At every vertex there is one monkey. An owner of monkeys takes $p$ peanuts, goes along the perimeter of polygon clockwise and delivers to the monkeys by the following rule: Gives the first peanut for the leader, skips the two next vertices and gives the second peanut to the monkey at the next vertex; skip four next vertices gives the second peanut for the monkey at the next vertex ... after giving the $k$-th peanut, he skips the $2 \cdot k$ next vertices and gives $k+1$-th for the monkey at the next vertex. He does so until all $p$ peanuts are delivered. [b]I.[/b] How many monkeys are there which does not receive peanuts? [b]II.[/b] How many edges of polygon are there which satisfying condition: both two monkey at its vertex received peanut(s)?

If $ a\plus{}1\equal{}b\plus{}2\equal{}c\plus{}3\equal{}d\plus{}4\equal{}a\plus{}b\plus{}c\plus{}d\plus{}5$, then $ a\plus{}b\plus{}c\plus{}d$ is $ \text{(A)}\ \minus{}5 \qquad \text{(B)}\ \minus{}10/3 \qquad \text{(C)}\ \minus{}7/3 \qquad \text{(D)}\ 5/3 \qquad \text{(E)}\ 5$

Wendy randomly chooses a positive integer less than or equal to $2020$. The probability that the digits in Wendy's number add up to $10$ is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Two circles $w_{1}$ and $w_{2}$ of different radii touch externally at $L$. A line touches $w_{1}$ at $A$ and $w_{2}$ at $B$ (the points $A$ and $B$ are different from $L$). A point $X$ is chosen in the plane. $Y$ and $Z$ are the second points of intersection of the lines $XA$ and $XB$ with $w_{1}$ and $w_{2}$ respectively. Prove that all $X$ such that $AB||Y Z$ belong to one circle.

For which integers $N$ it is possible to write real numbers into the cells of a square of size $N \times N$ so that among the sums of each pair of adjacent cells there are all integers from $1$ to $2(N-1)N$ (each integer once)? Maxim Didin

For a real number $x$ let $\lfloor x\rfloor$ be the greatest integer less than or equal to $x$ and let $\{x\} = x - \lfloor x\rfloor$. If $a, b, c$ are distinct real numbers, prove that \[\frac{a^3}{(a-b)(a-c)}+\frac{b^3}{(b-a)(b-c)}+\frac{c^3}{(c-a)(c-b)}\] is an integer if and only if $\{a\} + \{b\} + \{c\}$ is an integer.

A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than $ 100$ points. What was the total number of points scored by the two teams in the first half? $ \textbf{(A)}\ 30 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 33 \qquad \textbf{(E)}\ 34$

A collection of 8 cubes consists of one cube with edge-length $k$ for each integer $k,\thinspace 1 \le k \le 8.$ A tower is to be built using all 8 cubes according to the rules: $\bullet$ Any cube may be the bottom cube in the tower. $\bullet$ The cube immediately on top of a cube with edge-length $k$ must have edge-length at most $k+2.$ Let $T$ be the number of different towers than can be constructed. What is the remainder when $T$ is divided by 1000?

Prove that for each natural number $k$ there exists a natural number $n(k)$, such that for each $m\geq n(k)$ and each set $M$ of $m$ points in the plane, there can be chosen $k$ triangles, so that each has an angle greater than $120^\circ$.

A uniform disk rotates at a fixed angular velocity on an axis through its center normal to the plane of the disk, and has kinetic energy $E$. If the same disk rotates at the same angular velocity about an axis on the edge of the disk (still normal to the plane of the disk), what is its kinetic energy? (a) $\frac{1}{2}E$ (b) $\frac{3}{2}E$ (c) $2E$ (d) $3E$ (e) $4E$

At CMIMC headquarters, there is a row of $n$ lightbulbs, each of which is connected to a light switch. Daniel the electrician knows that exactly one of the switches doesn't work, and needs to find out which one. Every second, he can do exactly one of 3 things: [list] [*] Flip a switch, changing the lightbulb from off/on or on/off (unless the switch is broken). [*] Check if a given lightbulb is on or off. [*] Measure the total electricity usage of all the lightbulbs, which tells him exactly how many are currently on. [/list] Initially, all the lightbulbs are off. Daniel was given the very difficult task of finding the broken switch in at most $n$ seconds, but fortunately he showed up to work 10 seconds early today. What is the largest possible value $n$ such that he can complete his task on time? [i]Proposed by Adam Bertelli[/i]

Let $O=(0,0)$ be the origin of the $xy$-plane. We say a lattice triangle $ABC$ is [i]marine[/i] if it has centroid $O$ and area $\tfrac{3}{2}$. Let $P$ be any point in the plane which is not a lattice point. Prove that $P$ lies in the interior of some marine triangle if and only if the line segment $\overline{OP}$ does not pass through any lattice points besides $O$. (A [i]lattice point[/i] is a point whose $x$-coordinate and $y$-coordinate are both integers. A [i]lattice triangle[/i] is a triangle whose vertices are lattice points.)

Two congruent rectangles are located as shown in the figure. Find the area of the shaded part. [img]https://cdn.artofproblemsolving.com/attachments/2/e/10b164535ab5b3a3b98ce1a0b84892cd11d76f.png[/img]

Several (at least three) turtles are crawling along the plane, the velocities of which are constant in magnitude and direction (all are equal in magnitude, but pairwise different in direction). Prove that regardless of the initial location, after some time all the turtles will be at the vertices of some convex polygon.

If $x$ is a number satisfying the equation $\sqrt[3]{x+9}-\sqrt[3]{x-9}=3$, then $x^2$ is between: $\textbf{(A)}\ 55\text{ and }65 \qquad \textbf{(B)}\ 65\text{ and }75\qquad \textbf{(C)}\ 75\text{ and }85 \qquad \textbf{(D)}\ 85\text{ and }95 \qquad \textbf{(E)}\ 95\text{ and }105$

For any integer $n > 4$, prove that $2^n > n^2$.

Let $n \geq 2$ be an integer. Find the smallest positive integer $m$ for which the following holds: given $n$ points in the plane, no three on a line, there are $m$ lines such that no line passes through any of the given points, and for all points $X \neq Y$ there is a line with respect to which $X$ and $Y$ lie on opposite sides

Let $\lfloor z \rfloor$ denote the greatest integer less than or equal to $z.$ Compute $$\sum_{j=-1000}^{1000} \left\lfloor \frac{2025}{j+0.5}\right\rfloor.$$

Let $T$ be a set of natural numbers, each of which is greater than 1. A subset $S$ of $T$ is called “good”, if for each $t\in T$ there exists $s\in S$, for which $gcd(t,s)>1$. Prove that the number of "good" subsets of $T$ is odd.

An object is released from rest and falls a distance $h$ during the first second of time. How far will it fall during the next second of time? $ \textbf{(A)}\ h\qquad\textbf{(B)}\ 2h \qquad\textbf{(C)}\ 3h \qquad\textbf{(D)}\ 4h\qquad\textbf{(E)}\ h^2 $

On a given $ 2008 \times 2008$ chessboard, each unit square is colored in a different color. Every unit square is filled with one of the letters C, G, M, O. The resulting board is called [i]harmonic[/i] if every $ 2 \times 2$ subsquare contains all four different letters. How many harmonic boards are there?