Track / Stage 5 / 161 of 400 #761 of 1964

Problem 761

AIME late

Algebra Difficulty 5.4 Find the answer

1. Let $A=\frac{21 \times 62+22 \times 63+23 \times 64+24 \times 65+25 \times 66}{21 \times 61+22 \times 62+23 \times 63+24 \times 64+25 \times 65} \times 199$ , find the integer part of $\mathrm{A}$

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Official solution

【Answer】202
【Solution】First and last term approximation
$\begin{array}{l} A=\frac{21 \times 61+21+22 \times 62+62+23 \times 63+63+24 \times 64+64+25 \times 65+65}{21 \times 61+22 \times 62+23 \times 63+24 \times 64+25 \times 65} \times 199 \\ =\left(1+\frac{21+62+63+64+65}{21 \times 61+22 \times 62+23 \times 63+24 \times 64+25 \times 65}\right) \times 199 \\ =199+\frac{21+62+63+64+65}{21 \times 61+22 \times 62+23 \times 63+24 \times 64+25 \times 65} \times 199 \end{array}$

Then $A199+\frac{21+22+23+24+25}{(21+22+23+24+25) \times 65} \times 199=199+\frac{199}{65}=202 \frac{4}{65}>202

In summary, the integer part of $\mathrm{A}$ is 202

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.