Found problems: 1
LMT Guts Rounds, 2013
[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].