MathDB

6

Part of 2019 AIME Problems

Problems(2)

Quadrilateral Diagonals

Source: 2019 AIME I #6

3/14/2019
In convex quadrilateral KLMNKLMN side MN‾\overline{MN} is perpendicular to diagonal KM‾\overline{KM}, side KL‾\overline{KL} is perpendicular to diagonal LN‾\overline{LN}, MN=65MN = 65, and KL=28KL = 28. The line through LL perpendicular to side KN‾\overline{KN} intersects diagonal KM‾\overline{KM} at OO with KO=8KO = 8. Find MOMO.
AMCAIMEAIME I2019 AIME I2019 AMCgeometryquadrilateral
Martian logs

Source: 2019 AIME II #6

3/22/2019
In a Martian civilization, all logarithms whose bases are not specified are assumed to be base bb, for some fixed b≥2b \geq 2. A Martian student writes down \begin{align*}3 \log(\sqrt{x}\log x) &= 56\\\log_{\log (x)}(x) &= 54 \end{align*} and finds that this system of equations has a single real number solution x>1x > 1. Find bb.
AMCAIMElogarithmsalgebrasystem of equations2019 AIME II2019 AMC