MathDB

12

Part of 2012 AIME Problems

Problems(2)

Steiner's Theorem Returns

Source: 2012 AIME I Problem 12

3/16/2012
Let △ABC\triangle ABC be a right triangle with right angle at CC. Let DD and EE be points on AB‾\overline{AB} with DD between AA and EE such that CD‾\overline{CD} and CE‾\overline{CE} trisect ∠C\angle C. If DEBE=815\frac{DE}{BE} = \frac{8}{15}, then tan⁡B\tan B can be written as mpn\frac{m\sqrt{p}}{n}, where mm and nn are relatively prime positive integers, and pp is a positive integer not divisible by the square of any prime. Find m+n+pm+n+p.
trigonometryratioLaTeXAMCAIMEnumber theoryrelatively prime
P-Safe Numbers

Source: 2012 AIME II Problem 12

3/29/2012
For a positive integer pp, define the positive integer nn to be pp-safe if nn differs in absolute value by more than 22 from all multiples of pp. For example, the set of 1010-safe numbers is {3,4,5,6,7,13,14,15,16,17,23,…}\{3, 4, 5, 6, 7, 13, 14, 15, 16, 17,23, \ldots \}. Find the number of positive integers less than or equal to 10,00010,000 which are simultaneously 77-safe, 1111-safe, and 1313-safe.·
modular arithmeticabsolute valueAMC