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2012 Spring CHMMC Mixer Round - Caltech Harvey Mudd Mathematics Competition

Source:

February 29, 2024
CHMMCalgebrageometrycombinatoricsnumber theory

Problem Statement

Part 1
You might think this round is broken after solving some of these problems, but everything is intentional.
1.1. The number nn can be represented uniquely as the sum of 66 distinct positive integers. Find nn.
1.2. Let ABCABC be a triangle with AB=BCAB = BC. The altitude from AA intersects line BCBC at DD. Suppose BD=5BD = 5 and AC2=1188AC^2 = 1188. Find ABAB.
1.3. A lemonade stand analyzes its earning and operations. For the previous month it had a \45dollarbudgettodividebetweenproductionandadvertising.Ifitspent45 dollar budget to divide between production and advertising. If it spent kdollarsonproduction,itcouldmake dollars on production, it could make 2k - 12glassesoflemonade.Ifitspent glasses of lemonade. If it spent kdollarsonadvertising,itcouldselleachglassatanaveragepriceof dollars on advertising, it could sell each glass at an average price of 15 + 5kcents.Theamountitmadeinsalesforthepreviousmonthwas cents. The amount it made in sales for the previous month was \40.5040.50. Assuming the stand spent its entire budget on production and advertising, what was the absolute di erence between the amount spent on production and the amount spent on advertising?
1.4. Let AA be the number of di erent ways to tile a 1×n1 \times n rectangle with tiles of size 1×11 \times 1, 1×31 \times 3, and 1×61 \times 6. Let B be the number of different ways to tile a 1×n1 \times n rectangle with tiles of size 1×21 \times 2 and 1×51 \times 5, where there are 2 different colors available for the 1×21 \times 2 tiles. Given that A=BA = B, find nn. (Two tilings that are rotations or reflections of each other are considered distinct.)
1.5. An integer n≥0n \ge 0 is such that nn when represented in base 22 is written the same way as 2n2n is in base 55. Find nn.
1.6. Let xx be a positive integer such that 33, log⁡6(12x) \log_6(12x), log⁡6(18x)\log_6(18x) form an arithmetic progression in some order. Find xx.
Part 2
Oops, it looks like there were some intentional printing errors and some of the numbers from these problems got removed. Any ■\blacksquare that you see was originally some positive integer, but now its value is no longer readable. Still, if things behave like they did for Part 1, maybe you can piece the answers together.
2.1. The number nn can be represented uniquely as the sum of ■\blacksquare distinct positive integers. Find nn.
2.2. Let ABCABC be a triangle with AB=BCAB = BC. The altitude from AA intersects line BCBC at DD. Suppose BD=■BD = \blacksquare and AC2=1536AC^2 = 1536. Find ABAB.
2.3. A lemonade stand analyzes its earning and operations. For the previous month it had a $50\$50 dollar budget to divide between production and advertising. If it spent k dollars on production, it could make 2k−22k - 2 glasses of lemonade. If it spent kk dollars on advertising, it could sell each glass at an average price of 25+5k25 + 5k cents. The amount it made in sales for the previous month was $■\$\blacksquare. Assuming the stand spent its entire budget on production and advertising, what was the absolute di erence between the amount spent on production and the amount spent on advertising?
2.4. Let AA be the number of different ways to tile a 1×n1 \times n rectangle with tiles of size 1×■1 \times \blacksquare, 1×■1 \times \blacksquare, and 1×■1 \times \blacksquare. Let BB be the number of different ways to tile a 1×n1\times n rectangle with tiles of size 1×■1 \times \blacksquare and 1×■1 \times \blacksquare, where there are ■\blacksquare different colors available for the 1×■1 \times \blacksquare tiles. Given that A=BA = B, find nn. (Two tilings that are rotations or reflections of each other are considered distinct.)
2.5. An integer n≥■n \ge \blacksquare is such that nn when represented in base 99 is written the same way as 2n2n is in base ■\blacksquare. Find nn.
2.6. Let xx be a positive integer such that 11, log⁡96(6x)\log_{96}(6x), log⁡96(■x)\log_{96}(\blacksquare x) form an arithmetic progression in some order. Find xx.

PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
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