aime26-03

Proposed solution

To solve this problem, we need to find the number of integers less than or equal to 100 that can be expressed in the form \( a + b + ab \) for some choice of distinct positive integers \( a \) and \( b \).

First, let's rewrite the expression \( a + b + ab \) in a different form. Notice that:
\[
a + b + ab = (a + 1)(b + 1) - 1
\]
This means that for any distinct positive integers \( a \) and \( b \), the expression \( a + b + ab \) can be written as \( (a + 1)(b + 1) - 1 \).

Next, we need to determine the range of possible values for \( (a + 1)(b + 1) \) such that \( (a + 1)(b + 1) - 1 \leq 100 \). This simplifies to:
\[
(a + 1)(b + 1) \leq 101
\]

Now, we need to find all pairs of integers \( (a + 1, b + 1) \) such that their product is less than or equal to 101. Let \( x = a + 1 \) and \( y = b + 1 \). Then we need to find pairs \( (x, y) \) such that \( xy \leq 101 \) and \( x \) and \( y \) are distinct positive integers. 

We start by listing the factors of 101, which are 1 and 101 (since 101 is a prime number). We then consider all pairs \( (x, y) \) where \( x \leq y \) and \( xy \leq 101 \):

\[
\begin{aligned}
& (1, 1) \quad (1, 2) \quad (1, 3) \quad (1, 4) \quad (1, 5) \quad (1, 6) \quad (1, 7) \quad (1, 8) \quad (1, 9) \quad (1, 10) \quad (1, 11) \quad (1, 12) \quad (1, 13) \quad (1, 14) \quad (1, 15) \quad (1, 16) \quad (1, 17) \quad (1, 18) \quad (1, 19) \quad (1, 20) \quad (1, 21) \quad (1, 22) \quad (1, 23) \quad (1, 24) \quad (1, 25) \quad (1, 26) \quad (1, 27) \quad (1, 28) \quad (1, 29) \quad (1, 30) \quad (1, 31) \quad (1, 32) \quad (1, 33) \quad (1, 34) \quad (1, 35) \quad (1, 36) \quad (1, 37) \quad (1, 38) \quad (1, 39) \quad (1, 40) \quad (1, 41) \quad (1, 42) \quad (1, 43) \quad (1, 44) \quad (1, 45) \quad (1, 46) \quad (1, 47) \quad (1, 48) \quad (1, 49) \quad (1, 50) \quad (1, 51) \quad (1, 52) \quad (1, 53) \quad (1, 54) \quad (1, 55) \quad (1, 56) \quad (1, 57) \quad (1, 58) \quad (1, 59) \quad (1, 60) \quad (1, 61) \quad (1, 62) \quad (1, 63) \quad (1, 64) \quad (1, 65) \quad (1, 66) \quad (1, 67) \quad (1, 68) \quad (1, 69) \quad (1, 70) \quad (1, 71) \quad (1, 72) \quad (1, 73) \quad (1, 74) \quad (1, 75) \quad (1, 76) \quad (1, 77) \quad (1, 78) \quad (1, 79) \quad (1, 80) \quad (1, 81) \quad (1, 82) \quad (1, 83) \quad (1, 84) \quad (1, 85) \quad (1, 86) \quad (1, 87) \quad (1, 88) \quad (1, 89) \quad (1, 90) \quad (1, 91) \quad (1, 92) \quad (1, 93) \quad (1, 94) \quad (1, 95) \quad (1, 96) \quad (1, 97) \quad (1, 98) \quad (1, 99) \quad (1, 100) \\
& (2, 2) \quad (2, 3) \quad (2, 4) \quad (2, 5) \quad (2, 6) \quad (2, 7) \quad (2, 8) \quad (2, 9) \quad (2, 10) \quad (2, 11) \quad (2, 12) \quad (2, 13) \quad (2, 14) \quad (2, 15) \quad (2, 16) \quad (2, 17) \quad (2, 18) \quad (2, 19) \quad (2, 20) \quad (2, 21) \quad (2, 22) \quad (2, 23) \quad (2, 24) \quad (2, 25) \quad (2, 26) \quad (2, 27) \quad (2, 28) \quad (2, 29) \quad (2, 30) \quad (2, 31) \quad (2, 32) \quad (2, 33) \quad (2, 34) \quad (2, 35) \quad (2, 36) \quad (2, 37) \quad (2, 38) \quad (2, 39) \quad (2, 40) \quad (2, 41) \quad (2, 42) \quad (2, 43) \quad (2, 44) \quad (2, 45) \quad (2, 46) \quad (2, 47) \quad (2, 48) \quad (2, 49) \quad (2, 50) \quad (2, 51) \quad (2, 52) \quad (2, 53) \quad (2, 54) \quad (2, 55) \quad (2, 56) \quad (2, 57) \quad (2, 58) \quad (2, 59) \quad (2, 60) \quad (2, 61) \quad (2, 62) \quad (2, 63) \quad (2, 64) \quad (2, 65) \quad (2, 66) \quad (2, 67) \quad (2, 68) \quad (2, 69) \quad (2, 70) \quad (2, 71) \quad (2, 72) \quad (2, 73) \quad (2, 74) \quad (2, 75) \quad (2, 76) \quad (2, 77) \quad (2, 78) \quad (2, 79) \quad (2, 80) \quad (2, 81) \quad (2, 82) \quad (2, 83) \quad (2, 84) \quad (2, 85) \quad (2, 86) \quad (2, 87) \quad (2, 88) \quad (2, 89) \quad (2, 90) \quad (2, 91) \quad (2, 92) \quad (2, 93) \quad (2, 94) \quad (2, 95) \quad (2, 96) \quad (2, 97) \quad (2, 98) \quad (2, 99) \quad (2, 100) \\
& (3, 3) \quad (3, 4) \quad (3, 5) \quad (3, 6) \quad (3, 7) \quad (3, 8) \quad (3, 9) \quad