There is a $4 \times 4$ array of integers $A$, all initially equal to $0$. An operation may be performed on the array for any row or column such that every number in that row or column has $1$ added to it, and then is replaced with its remainder modulo $3$. Given a random $4 \times 4$ array of integers between $0$ and $2$ not identical to $A$, the probability that it can be reached through a series of operations on $A$ is $\frac{p}{q},$ where $p,q$ are relatively prime positive integers. Find $p$.

In a triangle $ABC$, let $D$ and $E$ be the feet of the angle bisectors of angles $A$ and $B$, respectively. A rhombus is inscribed into the quadrilateral $AEDB$ (all vertices of the rhombus lie on different sides of $AEDB$). Let $\varphi$ be the non-obtuse angle of the rhombus. Prove that $\varphi \le \max \{ \angle BAC, \angle ABC \}$.

For $n$ a positive integer, denote by $P(n)$ the product of all positive integers divisors of $n$. Find the smallest $n$ for which \[ P(P(P(n))) > 10^{12} \]

Consider a paper punch that can be centered at any point of the plane and that, when operated, removes from the plane precisely those points whose distance from the center is irrational. How many punches are needed to remove every point?

A circular disk is partitioned into $ 2n$ equal sectors by $ n$ straight lines through its center. Then, these $ 2n$ sectors are colored in such a way that exactly $ n$ of the sectors are colored in blue, and the other $ n$ sectors are colored in red. We number the red sectors with numbers from $ 1$ to $ n$ in counter-clockwise direction (starting at some of these red sectors), and then we number the blue sectors with numbers from $ 1$ to $ n$ in clockwise direction (starting at some of these blue sectors). Prove that one can find a half-disk which contains sectors numbered with all the numbers from $ 1$ to $ n$ (in some order). (In other words, prove that one can find $ n$ consecutive sectors which are numbered by all numbers $ 1$, $ 2$, ..., $ n$ in some order.) [hide="Problem 8 from CWMO 2007"]$ n$ white and $ n$ black balls are placed at random on the circumference of a circle.Starting from a certain white ball,number all white balls in a clockwise direction by $ 1,2,\dots,n$. Likewise number all black balls by $ 1,2,\dots,n$ in anti-clockwise direction starting from a certain black ball.Prove that there exists a chain of $ n$ balls whose collection of numbering forms the set $ \{1,2,3\dots,n\}$.[/hide]

We are given the set $$M = \{-2^{2022}, -2^{2021}, . . . , -2^{2}, -2, -1, 1, 2, 2^2, . . . , 2^{2021}, 2^{2022}\}.$$ Let $T$ be a subset of $M$, such that neighbouring numbers have the same difference when the elements are ordered by size. (a) Determine the maximum number of elements that such a set $T$ can contain. (b) Determine all sets $T$ with the maximum number of elements. [i](Walther Janous)[/i]

Show that there exists a positive integer $N$ such that for all integers $a>N$, there exists a contiguous substring of the decimal expansion of $a$, which is divisible by $2011$. Note. A contiguous substring of an integer $a$ is an integer with a decimal expansion equivalent to a sequence of consecutive digits taken from the decimal expansion of $a$.

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] $\textbf{(A) }\frac15\qquad\textbf{(B) }\frac14\qquad\textbf{(C) }2-\sqrt3\qquad\textbf{(D) }\sqrt3-\sqrt2\qquad\textbf{(E) }\sqrt2-1$