7

Part of 2019 AIME Problems

Problems(2)

high quality system

Source: 2019 AIME I #7

There are positive integers

x

y

that satisfy the system of equations \begin{align*} \log_{10} x + 2 \log_{10} (\gcd(x,y)) &= 60 \\ \log_{10} y + 2 \log_{10} (\text{lcm}(x,y)) &= 570. \end{align*} Let

m

be the number of (not necessarily distinct) prime factors in the prime factorization of

x

n

be the number of (not necessarily distinct) prime factors in the prime factorization of

y

3m+2n

AMCAIMEAIME I2019 AIME I2019 AMCnumber theory

Similar Triangles

Source: 2019 AIME ii #7

ABC

has side lengths

AB=120

BC=220

AC=180

\ell_{A}

\ell_{B}

\ell_{C}

are drawn parallel to

\overline{BC}

\overline{AC}

\overline{AB}

, respectively, such that the intersection of

\ell_{A}

\ell_{B}

\ell_{C}

with the interior of

\triangle ABC

are segments of length

55

45

15

, respectively. Find the perimeter of the triangle whose sides lie on

\ell_{A}

\ell_{B}

\ell_{C}

geometrysimilar trianglesAMCAIMEAIME II2019 AIME II2019 AMC