In an island shaped like a regular polygon of $n$ sides, there are airports at each vertex of the island. The island would like to add $k$ new airports into the interior of the island, but it must follow the following rules:\\ $1$. It must be in the interior of the island (none on borders).\\ $2$. No two airports can be at the exact same location.\\ $3$. Every triple of $1$ new and $2$ old airports must form an isoceles triangle.\\ $4$. No three airports can be collinear.\\ Find the maximum value of $k$ for each $n$ [hide=Harder Version]Replace $1$ new and $2$ old with $1$ old and $2$ new.[/hide]

For every dollar Ben spent on bagels, David spent $25$ cents less. Ben paid $\$12.50$ more than David. How much did they spend in the bagel store together? $\textbf{(A)}\ \$37.50 \qquad\textbf{(B)}\ \$50.00\qquad\textbf{(C)}\ \$87.50\qquad\textbf{(D)}\ \$90.00\qquad\textbf{(E)}\ \$92.50$

Let $A$ be a set of $n$ points in the space. From the family of all segments with endpoints in $A$, $q$ segments have been selected and colored yellow. Suppose that all yellow segments are of different length. Prove that there exists a polygonal line composed of $m$ yellow segments, where $m \geq \frac{2q}{n}$, arranged in order of increasing length.

Vertices and Edges of a regular $n$-gon are numbered $1,2,\dots,n$ clockwise such that edge $i$ lies between vertices $i,i+1 \mod n$. Now non-negative integers $(e_1,e_2,\dots,e_n)$ are assigned to corresponding edges and non-negative integers $(k_1,k_2,\dots,k_n)$ are assigned to corresponding vertices such that: $i$) $(e_1,e_2,\dots,e_n)$ is a permutation of $(k_1,k_2,\dots,k_n)$. $ii$) $k_i=|e_{i+1}-e_i|$ indexes$\mod n$. a) Prove that for all $n\geq 3$ such non-zero $n$-tuples exist. b) Determine for each $m$ the smallest positive integer $n$ such that there is an $n$-tuples stisfying the above conditions and also $\{e_1,e_2,\dots,e_n\}$ contains all $0,1,2,\dots m$.

a) Prove that for any positive integer $m > 2$, the equation $$y^3 = x^3_1 + x^3_2 + ... + x^3_m$$ always has a positive integer solution. b) Given a positive integer $n > 1$ and suppose $n \ne 3$. Prove that every rational number $x > 0$ can be expressed as $$x =\frac{a^3_1 + a^3_2 + ... + a^3_n}{b^3_1 + b^3_2 + ... + b^3_n}$$ where $a_i, b_i$ $(i = 1, . . . , n)$ are positive integers.

Given a sequence $<a_1,a_2,a_3,\cdots >$ of real numbers, we define $m_n$ as the arithmetic mean of the numbers $a_1$ to $a_n$ for $n\in\mathbb{Z}^+$. If there is a real number $C$, such that \[ (i-j)m_k+(j-k)m_i+(k-i)m_j=C\] for every triple $(i,j,k)$ of distinct positive integers, prove that the sequence $<a_1,a_2,a_3,\cdots >$ is an arithmetic progression.

Show that the number$$4\sin\frac{\pi}{34}\left(\sin\frac{3\pi}{34}+\sin\frac{7\pi}{34}+\sin\frac{11\pi}{34}+\sin\frac{15\pi}{34}\right)$$ is an integer and determine it.

Let $S=\{(x,y)\mid -1\leq xy\leq 1\}$ be a subset of the real coordinate plane. If the smallest real number that is greater than or equal to the area of any triangle whose interior lies entirely in $S$ is $A$, compute the greatest integer not exceeding $1000A$. [i]Proposed by Yannick Yao[/i]

Determine all positive integers $n$ so that both $20n$ and $5n + 275$ are perfect squares. (A perfect square is a number which can be expressed as $k^2$, where $k$ is an integer.)

a)Let$a,b,c\in\mathbb{R}$ and $a^2+b^2+c^2=1$.Prove that: $|a-b|+|b-c|+|c-a|\le2\sqrt{2}$ b) Let $a_1,a_2,..a_{2019}\in\mathbb{R}$ and $\sum_{i=1}^{2019}a_i^2=1$.Find the maximum of: $S=|a_1-a_2|+|a_2-a_3|+...+|a_{2019}-a_1|$

Let $\mathbb{N}_0$ denote the set of all nonnegative integers. Determine all functions $f:\mathbb{N}_0\to\mathbb{N}_0$ with the following two properties: [list] [*] $0\le f(x)\le x^2$ for all $x\in\mathbb{N}_0$ [*] $x-y$ divides $f(x)-f(y)$ for all $x,y\in\mathbb{N}_0$ with $x>y$[/list]

We have some identical paper squares which are black on one side of the sheet and white on the other side. We can join nine squares together to make a $3$ by $3$ sheet of squares by placing each of the nine squares either white side up or black side up. Two of these $3$ by $3$ sheets are distinguishable if neither can be made to look like the other by rotating the sheet or by turning it over. How many distinguishable $3$ by $3$ squares can we form?

The circle inscribed in the triangle $ABC$ is tangent to side $AC$ at point $B_1$, and to side $BC$ at point $A_1$. On the side $AB$ there is a point $K$ such that $AK = KB_1, BK = KA_1$. Prove that $ \angle ACB\ge 60$

[u]Round 9[/u] [b]p25.[/b] Define a hilly number to be a number with distinct digits such that when its digits are read from left to right, they strictly increase, then strictly decrease. For example, $483$ and $1230$ are both hilly numbers, but $123$ and $1212$ are not. How many $5$-digit hilly numbers are there? [b]p26.[/b] Triangle ABC has $AB = 4$ and $AC = 6$. Let the intersection of the angle bisector of $\angle BAC$ and $\overline{BC}$ be $D$ and the foot of the perpendicular from C to the angle bisector of $\angle BAC$ be $E$. What is the value of $AD/AE$? [b]p27.[/b] Given that $(7+ 4\sqrt3)^x+ (7-4\sqrt3)^x = 10$, find all possible values of $(7+ 4\sqrt3)^x-(7-4\sqrt3)^x$. [u]Round 10[/u] Note: In this set, the answers for each problem rely on answers to the other problems. [b]p28.[/b] Let X be the answer to question $29$. If $5A + 5B = 5X - 8$ and $A^2 + AB - 2B^2 = 0$, find the sum of all possible values of $A$. [b]p29.[/b] Let $W$ be the answer to question $28$. In isosceles trapezoid $ABCD$ with $\overline{AB} \parallel \overline{CD}$, line segments $ \overline{AC}$ and $ \overline{BD}$ split each other in the ratio $2 : 1$. Given that the length of $BC$ is $W$, what is the greatest possible length of $\overline{AB}$ for which there is only one trapezoid $ABCD$ satisfying the given conditions? [b]p30.[/b] Let $W$ be the answer to question $28$ and $X$ be the answer to question $29$. For what value of $Z$ is $ |Z - X| + |Z - W| - |W + X - Z|$ at a minimum? [u]Round 11[/u] [b]p31.[/b] Peijin wants to draw the horizon of Yellowstone Park, but he forgot what it looked like. He remembers that the horizon was a string of $10$ segments, each one either increasing with slope $1$, remaining flat, or decreasing with slope $1$. Given that the horizon never dipped more than $1$ unit below or rose more than $1$ unit above the starting point and that it returned to the starting elevation, how many possible pictures can Peijin draw? [b]p32.[/b] DNA sequences are long strings of $A, T, C$, and $G$, called base pairs. (e.g. AATGCA is a DNA sequence of 6 base pairs). A DNA sequence is called stunningly nondescript if it contains each of A, T, C, G, in some order, in 4 consecutive base pairs somewhere in the sequence. Find the number of stunningly nondescript DNA sequences of 6 base pairs (the example above is to be included in this count). [b]p33.[/b] Given variables s, t that satisfy $(3 + 2s + 3t)^2 + (7 - 2t)^2 + (5 - 2s - t)^2 = 83$, find the minimum possible value of $(-5 + 2s + 3t) ^2 + (3 - 2t)^2 + (2 - 2s - t)^2$. [u]Round 12[/u] [b]p34.[/b] Let $f(n)$ be the number of powers of 2 with n digits. For how many values of n from $1$ to $2013$ inclusive does $f(n) = 3$? If your answer is N and the actual answer is $C$, then the score you will receive on this problem is $max\{15 - \frac{|N-C|}{26039} , 0\}$, rounded to the nearest integer. [b]p35.[/b] How many total characters are there in the source files for the LMT $2013$ problems? If your answer is $N$ and the actual answer is $C$, then the score you receive on this problem is $max\{15 - \frac{|N - C|}{1337}, 0\}$, rounded to the nearest integer. [b]p36.[/b] Write down two distinct integers between $0$ and $300$, inclusive. Let $S$ be the collection of everyone’s guesses. Let x be the smallest nonnegative difference between one of your guesses and another guess in $S$ (possibly your other guess). Your team will be awarded $min(15, x)$ points. PS. You should use hide for answers.Rounds 1-4 are [url=https://artofproblemsolving.com/community/c3h3134546p28406927]here [/url] and 6-8 [url=https://artofproblemsolving.com/community/c3h3136014p28427163]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ C$ be a simple arc with monotone curvature such that $ C$ is congruent to its evolute. Show that under appropriate differentiability conditions, $ C$ is a part of a cycloid or a logarithmic spiral with polar equation $ r\equal{}ae^{\vartheta}$. [i]J. Szenthe[/i]

What is the sum of all positive integer values of $n$ that satisfy the equation \[ \cos \Bigl( \frac{\pi}{n} \Bigr) \cos \Bigl( \frac{2\pi}{n} \Bigr) \cos \Bigl( \frac{4\pi}{n} \Bigr) \cos \Bigl( \frac{8\pi}{n} \Bigr) \cos \Bigl( \frac{16\pi}{n} \Bigr) = \frac{1}{32} \, ? \]

If the number $3n +1$, where n is integer, is multiple of $7$, find the possible remainders of the following divisions: (a) of $n$ with divisor $7$, (b) of $n^{m}$ with divisor $7$, for all values of the positive integer $m, m >1$.

A square of area $2$ is inscribed in a square of area $3$, creating four congruent triangles, as shown below. What is the ratio of the shorter leg to the longer leg in the shaded right triangle? [asy] size(200); defaultpen(linewidth(0.6pt)+fontsize(10pt)); real y = sqrt(3); pair A,B,C,D,E,F,G,H; A = (0,0); B = (0,y); C = (y,y); D = (y,0); E = ((y + 1)/2,y); F = (y, (y - 1)/2); G = ((y - 1)/2, 0); H = (0,(y + 1)/2); fill(H--B--E--cycle, gray); draw(A--B--C--D--cycle); draw(E--F--G--H--cycle); [/asy] $\textbf{(A) }\frac15\qquad\textbf{(B) }\frac14\qquad\textbf{(C) }2-\sqrt3\qquad\textbf{(D) }\sqrt3-\sqrt2\qquad\textbf{(E) }\sqrt2-1$

A square of side 1 is decomposed into 9 equal squares of sides 1/3 and the one in the center is painted in black. The remaining eight squares are analogously divided into nine squares each and the square in the center is painted in black. Prove that after 1000 steps the total area of black region exceeds 0.999[/b]

The midpoints of four sides of a cyclic pentagon were marked, after this the pentagon was erased. Restore it.

Determine the sum of absolute values for the complex roots of $ 20 x^8 \plus{} 7i x^7 \minus{}7ix \plus{} 20.$

If $a, b, c$ are real numbers such that $a+b+c=6$ and $ab+bc+ca = 9$, find the sum of all possible values of the expression $\lfloor a \rfloor + \lfloor b \rfloor + \lfloor c \rfloor$.

Palmer correctly computes the product of the first $1,001$ prime numbers. Which of the following is NOT a factor of Palmer's product? $\text{(A) }2,002\qquad\text{(B) }3,003\qquad\text{(C) }5,005\qquad\text{(D) }6,006\qquad\text{(E) }7,007$

There are coins worth $1, 2, \ldots , b$ rubles, blue bills with worth $a{}$ rubles and red bills worth $a + b$ rubles. Ilya wants to exchange a certain amount into coins and blue bills, and use no more than $a-1$ coins. Pasha wants to exchange the same amount in coins and red bills, but use no more than $a{}$ coins. Prove that they have equally many ways of doing so.

Find all complex polynomial $ P(x)$ such that for any three integers $ a,b,c$ satisfying $ a \plus{} b \plus{} c\not \equal{} 0, \frac{P(a) \plus{} P(b) \plus{} P(c)}{a \plus{} b \plus{} c}$ is an integer.