For $m = 1, 2, 3, ...$ denote $S(m)$ the sum of the digits of $m$, and let $f(m)=m+S(m)$. Show that for each positive integer $n$, there exists a number that appears exactly $n$ times in the sequence $f(1),f(2),...,f(m),...$

Peter and Basil together thought of ten quadratic trinomials. Then, Basil began calling consecutive natural numbers starting with some natural number. After each called number, Peter chose one of the ten polynomials at random and plugged in the called number. The results were recorded on the board. They eventually form a sequence. After they finished, their sequence was arithmetic. What is the greatest number of numbers that Basil could have called out?

How many non-congruent scalene triangles with perimeter $21$ have integer side lengths that form an arithmetic sequence? (In an arithmetic sequence, successive terms differ by the same amount.) $\text{(A) }0\qquad\text{(B) }1\qquad\text{(C) }3\qquad\text{(D) }4\qquad\text{(E) }6$

Let $ABC$ be a non-degenerate triangle in the euclidean plane. Define a sequence $(C_n)_{n=0}^\infty$ of points as follows: $C_0:=C$, and $C_{n+1}$ is the incenter of the triangle $ABC_n$. Find $\lim_{n\to\infty}C_n$.

Define a sequence $<x_n>$ by $x_1 = 1, x_2 = x, x_{n+2} = xx_{n+1} + nx_n, n \ge 1$. Consider the polynomial $P_n(x) = x_{n-1}x_{n+1} - x_n^2$, for each $n \ge 2$. Prove or disprove that the coefficients of $P_n(x)$ are all non-negative, except for the constant term when $n$ is odd.

Let $N$ be the number of sequences of natural numbers $d_1,d_2,\dots,d_{10}$ such that the following conditions hold: $d_1|d_2$, $\dots$, $d_9|d_{10}$ and $d_{10}|6^{2018}$. Evaluate the remainder when $N$ is divided by $2017$.

Does there exist a strictly increasing sequence $\{a_n\}_{n=1}^\infty$ of natural numbers with the following property: for $\forall$ $c\in \mathbb{Z}$ the sequence $c+a_1,c+a_2,...,c+a_n...$ has finite number of primes? Explain your answer.

[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].

[b]p1[/b]. Evaluate $1! + 2! + 3! + 4! + 5! $ (where $n!$ is the product of all integers from $1$ to $n$, inclusive). [b]p2.[/b] Harold opens a pack of Bertie Bott's Every Flavor Beans that contains $10$ blueberry, $10$ watermelon, $3$ spinach and $2$ earwax-flavored jelly beans. If he picks a jelly bean at random, then what is the probability that it is not spinach-flavored? [b]p3.[/b] Find the sum of the positive factors of $32$ (including $32$ itself). [b]p4.[/b] Carol stands at a flag pole that is $21$ feet tall. She begins to walk in the direction of the flag's shadow to say hi to her friends. When she has walked $10$ feet, her shadow passes the flag's shadow. Given that Carol is exactly $5$ feet tall, how long in feet is her shadow? [b]p5.[/b] A solid metal sphere of radius $7$ cm is melted and reshaped into four solid metal spheres with radii $1$, $5$, $6$, and $x$ cm. What is the value of $x$? [b]p6.[/b] Let $A = (2,-2)$ and $B = (-3, 3)$. If $(a,0)$ and $(0, b)$ are both equidistant from $A$ and $B$, then what is the value of $a + b$? [b]p7.[/b] For every flip, there is an $x^2$ percent chance of flipping heads, where $x$ is the number of flips that have already been made. What is the probability that my first three flips will all come up tails? [b]p8.[/b] Consider the sequence of letters $Z\,\,W\,\,Y\,\,X\,\,V$. There are two ways to modify the sequence: we can either swap two adjacent letters or reverse the entire sequence. What is the least number of these changes we need to make in order to put the letters in alphabetical order? [b]p9.[/b] A square and a rectangle overlap each other such that the area inside the square but outside the rectangle is equal to the area inside the rectangle but outside the square. If the area of the rectangle is $169$, then find the side length of the square. [b]p10.[/b] If $A = 50\sqrt3$, $B = 60\sqrt2$, and $C = 85$, then order $A$, $B$, and $C$ from least to greatest. [b]p11.[/b] How many ways are there to arrange the letters of the word $RACECAR$? (Identical letters are assumed to be indistinguishable.) [b]p12.[/b] A cube and a regular tetrahedron (which has four faces composed of equilateral triangles) have the same surface area. Let $r$ be the ratio of the edge length of the cube to the edge length of the tetrahedron. Find $r^2$. [b]p13.[/b] Given that $x^2 + x + \frac{1}{x} +\frac{1}{x^2} = 10$, find all possible values of $x +\frac{1}{x}$ . [b]p14.[/b] Astronaut Bob has a rope one unit long. He must attach one end to his spacesuit and one end to his stationary spacecraft, which assumes the shape of a box with dimensions $3\times 2\times 2$. If he can attach and re-attach the rope onto any point on the surface of his spacecraft, then what is the total volume of space outside of the spacecraft that Bob can reach? Assume that Bob's size is negligible. [b]p15.[/b] Triangle $ABC$ has $AB = 4$, $BC = 3$, and $AC = 5$. Point $B$ is reflected across $\overline{AC}$ to point $B'$. The lines that contain $AB'$ and $BC$ are then drawn to intersect at point $D$. Find $AD$. [b]p16.[/b] Consider a rectangle $ABCD$ with side lengths $5$ and $12$. If a circle tangent to all sides of $\vartriangle ABD$ and a circle tangent to all sides of $\vartriangle BCD$ are drawn, then how far apart are the centers of the circles? [b]p17.[/b] An increasing geometric sequence $a_0, a_1, a_2,...$ has a positive common ratio. Also, the value of $a_3 + a_2 - a_1 - a_0$ is equal to half the value of $a_4 - a_0$. What is the value of the common ratio? [b]p18.[/b] In triangle $ABC$, $AB = 9$, $BC = 11$, and $AC = 16$. Points $E$ and $F$ are on $\overline{AB}$ and $\overline{BC}$, respectively, such that $BE = BF = 4$. What is the area of triangle $CEF$? [b]p19.[/b] Xavier, Yuna, and Zach are running around a circular track. The three start at one point and run clockwise, each at a constant speed. After $8$ minutes, Zach passes Xavier for the first time. Xavier first passes Yuna for the first time in $12$ minutes. After how many seconds since the three began running did Zach first pass Yuna? [b]p20.[/b] How many unit fractions are there such that their decimal equivalent has a cycle of $6$ repeating integers? Exclude fractions that repeat in cycles of $1$, $2$, or $3$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Five distinct positive integers form an arithmetic progression. Can their product be equal to $a^{2008}$ for some positive integer $a$ ?

Given the sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1,...,$ find $n$ such that the sum of the first $n$ terms is $2008$ or $2009$.

Let $X_1,X_2,\ldots$ be a sequence of independent random variables distributed exponentially with mean $1$. Suppose $\mathbb N$ is a random variable independent of $X_i$'s that has a Poisson distribution with mean $\lambda>0$. What is the expected value of $X_1+X_2+\ldots+X_N$? $\textbf{(A)}~N^2$ $\textbf{(B)}~\lambda+\lambda^2$ $\textbf{(C)}~\lambda^2$ $\textbf{(D)}~\lambda$

Let $n$ be a positive integer and let $(1+iT)^n=f(T)+ig(T)$ where $i$ is the square root of $-1$, and $f$ and $g$ are polynomials with real coefficients. Show that for any real number $k$ the equation $f(T)+kg(T)=0$ has only real roots.

Show that, for any sequence $a_1,a_2,\ldots$ of real numbers, the two conditions \[ \lim_{n\to\infty}\frac{e^{(ia_1)} + e^{(ia_2)} + \cdots + e^{(ia_n)}}n = \alpha \] and \[ \lim_{n\to\infty}\frac{e^{(ia_1)} + e^{(ia_2)} + \cdots + e^{(ia_{n^2})}}{n^2} = \alpha \] are equivalent.

[u]Round 1[/u] [b]p1.[/b] Let $n$ be a two-digit positive integer. What is the maximum possible sum of the prime factors of $n^2 - 25$ ? [b]p2.[/b] Angela has ten numbers $a_1, a_2, a_3, ... , a_{10}$. She wants them to be a permutation of the numbers $\{1, 2, 3, ... , 10\}$ such that for each $1 \le i \le 5$, $a_i \le 2i$, and for each $6 \le i \le 10$, $a_i \le - 10$. How many ways can Angela choose $a_1$ through $a_{10}$? [b]p3.[/b] Find the number of three-by-three grids such that $\bullet$ the sum of the entries in each row, column, and diagonal passing through the center square is the same, and $\bullet$ the entries in the nine squares are the integers between $1$ and $9$ inclusive, each integer appearing in exactly one square. [u]Round 2 [/u] [b]p4.[/b] Suppose that $P(x)$ is a quadratic polynomial such that the sum and product of its two roots are equal to each other. There is a real number $a$ that $P(1)$ can never be equal to. Find $a$. [b]p5.[/b] Find the number of ordered pairs $(x, y)$ of positive integers such that $\frac{1}{x} +\frac{1}{y} =\frac{1}{k}$ and k is a factor of $60$. [b]p6.[/b] Let $ABC$ be a triangle with $AB = 5$, $AC = 4$, and $BC = 3$. With $B = B_0$ and $C = C_0$, define the infinite sequences of points $\{B_i\}$ and $\{C_i\}$ as follows: for all $i \ge 1$, let $B_i$ be the foot of the perpendicular from $C_{i-1}$ to $AB$, and let $C_i$ be the foot of the perpendicular from $B_i$ to $AC$. Find $C_0C_1(AC_0 + AC_1 + AC_2 + AC_3 + ...)$. [u]Round 3 [/u] [b]p7.[/b] If $\ell_1, \ell_2, ... ,\ell_{10}$ are distinct lines in the plane and $p_1, ... , p_{100}$ are distinct points in the plane, then what is the maximum possible number of ordered pairs $(\ell_i, p_j )$ such that $p_j$ lies on $\ell_i$? [b]p8.[/b] Before Andres goes to school each day, he has to put on a shirt, a jacket, pants, socks, and shoes. He can put these clothes on in any order obeying the following restrictions: socks come before shoes, and the shirt comes before the jacket. How many distinct orders are there for Andres to put his clothes on? [b]p9. [/b]There are ten towns, numbered $1$ through $10$, and each pair of towns is connected by a road. Define a backwards move to be taking a road from some town $a$ to another town $b$ such that $a > b$, and define a forwards move to be taking a road from some town $a$ to another town $b$ such that $a < b$. How many distinct paths can Ali take from town $1$ to town $10$ under the conditions that $\bullet$ she takes exactly one backwards move and the rest of her moves are forward moves, and $\bullet$ the only time she visits town $10$ is at the very end? One possible path is $1 \to 3 \to 8 \to 6 \to 7 \to 8 \to 10$. [u]Round 4[/u] [b]p10.[/b] How many prime numbers $p$ less than $100$ have the properties that $p^5 - 1$ is divisible by $6$ and $p^6 - 1$ is divisible by $5$? [b]p11.[/b] Call a four-digit integer $\overline{d_1d_2d_3d_4}$ [i]primed [/i] if 1) $d_1$, $d_2$, $d_3$, and $d_4$ are all prime numbers, and 2) the two-digit numbers $\overline{d_1d_2}$ and $\overline{d_3d_4}$ are both prime numbers. Find the sum of all primed integers. [b]p12.[/b] Suppose that $ABC$ is an isosceles triangle with $AB = AC$, and suppose that $D$ and $E$ lie on $\overline{AB}$ and $\overline{AC}$, respectively, with $\overline{DE} \parallel \overline{BC}$. Let $r$ be the length of the inradius of triangle $ADE$. Suppose that it is possible to construct two circles of radius $r$, each tangent to one another and internally tangent to three sides of the trapezoid $BDEC$. If $\frac{BC}{r} = a + \sqrt{b}$ forpositive integers $a$ and $b$ with $b$ squarefree, then find $a + b$. PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2800986p24675177]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].