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2024/25 AHSMC Part 1 Problem 14

Problem 14 of 16

How many positive integers n≤2024n \le 2024 have the property that log⁡3(n)\log_3(n) can be written as the sum of an odd positive integer and some number 0≤x<10 \le x < 1?

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Solution

We want 2k−1≤log⁡3n<2k2k-1 \le \log_3 n < 2k for some positive integer kk, i.e. 32k−1≤n<32k3^{2k-1} \le n < 3^{2k}. Since n≤2024n \le 2024 we have k=1,2,3k = 1,2,3. The count is (32−3)+(34−33)+(36−35)=6(1+32+34)=546. (3^2-3) + (3^4-3^3) + (3^6-3^5) = 6(1+3^2+3^4) = 546. The answer is (C).

University of Alberta