1. Several months ago, Deven bought two plants. The two plants had the same height at the time when they were purchased. Ever since then, Plant A grew $10$ centimeters every month. Plant B grew $1$ centimeter in the first month after its purchase, $2$ centimeters in the following month, and generally grew $n$ centimeters in the $n^{\text{th}}$ month after its purchase. Deven finds out today that the two plants have the same height again. How many months ago did he buy the two plants?

2. What is the 50th letter in this pattern: $ABCAABBCCAAABBBCCC\ldots$?

3. Compute$$\frac{999999999\times 999999999}{1 + 2 + 3 + \cdots +8+9+8+\cdots + 3 + 2 + 1} .$$

4. When $36$ consecutive integers are added, the result is $-18$. What is the greatest product that can be obtained by multiplying two of these $36$ integers?

5. What is the smallest positive integer that can be added to the sum of consecutive integers\[1 + 2 + 3 + \dots + 327 \]so that the resulting total is divisible by 5?

Guest Jan 10, 2021

#1**0 **

1. After x months, plant a grows 10x centimeters, and after x months, plant b grows x(x+1)/2 because of the formula for triangluar numbers. this means those two are equal, so 10x=x(x+1)/2 or 20x=x(x+1), which means 20=x+1 so x=19.

hihihi Jan 10, 2021

#4**0 **

Hello Guest! You may not have realized that this person copied this answer off of someone else's work. Here is the real answer for number 1: https://web2.0calc.com/questions/mathcounts-question-d

DewdropDancer
Jan 11, 2021

#3**0 **

1) 19

1+2+3.../10 = whole number. We can first check 9, where (1+9)+(2+8)... = 45 so that doesn't work. The next number that does work is 19.

2) A

3+6+9+12+15 = 45, 50-45 = 5.Since 15 has groups of 5, 18 has groups of 3, and A falls in as the 5th letter

3) 12345678987654321 (actually)

I did some experiments with smaller number (99*99, 999*999), and you get a pattern of 12321 and 1234321, so with 9, 9's on each side, we get... that number.

4) 306

0 is also an integer, and since we have an even number of terms, we can start from 17, and count back to -18 (where 17+(-17) cancels out). We get that -18 * -17 = 306.

5) 2

we can take out the 1+2 temporarily to get 3+4+5... +327 = (3+327)+(4+326)... (and 327-3 is even so they match up).

Putting back in the 1+2, and adding 2, we get 5.

Guest Jan 10, 2021