IMSO 2011 Mathematics - International Essay

There are 53 Tuesdays in the year 2019. What day is Pi day (14 March) in the year 2020?

Let $a=0.363636\ldots$ and $b=0.515$. Find the digit in the 2011th decimal place of $ab$.

There are six business establishments on both sides of a certain street, named A to F, as shown in the figure. The shaded store is shop A. The other locations are described as follows:
(a) The store on the right of shop A is a bookstore.
(b) In front of the bookstore is a flower shop.
(c) A bakery is beside the flower shop.
(d) Shop E is in front of shop D.
(e) The hotel is a neighbour of shop E.
(f) Shop E and the stationery store are on the same side of the street.
What kind of establishment is shop A?

Source warning: the supplied schematic shows six shop icons but lacks the required A–F labels and identification of shaded shop A. The printed answer sheet awards a 3-point bonus for this item.

There are 111 apples to be distributed to 9 children. Each child receives a different number. The child receiving the fewest is kid #1 and the child receiving the most is kid #9. What is the least number of apples kid #9 can receive?

Find all six-digit numbers $\overline{x2011y}$ divisible by 36, where $x$ and $y$ are digits.

A palindrome reads the same forwards and backwards. For example, 246642, 131 and 5005 are palindromes. Find the smallest even palindrome greater than 56789 that is divisible by 7.

Ben and Josh together must paint 3 houses and 20 fences. Ben takes 5 hours per house and 3 hours per fence. Josh takes 2 hours per house and 1 hour per fence. What is the minimum time, in hours, for them to finish all the painting?

A farmer asks his son to select five watermelons to sell at the weekly market. They are sold by weight, so they must be weighed before the trip. By mistake, the son weighs them in pairs instead of individually. The ten pair weights, in pounds, are 20, 22, 23, 24, 25, 26, 27, 28, 30 and 31. How much does each watermelon weigh?

Andy will fill either tank A or tank B, aiming for the highest water level after 20 minutes. Tank A measures $60\text{ cm}\times10\text{ cm}\times30\text{ cm}$ and tank B measures $20\text{ cm}\times20\text{ cm}\times30\text{ cm}$. Tank B leaks at $150\text{ cm}^3$ per minute. Which tank should he fill using a pipe delivering $600\text{ cm}^3$ per minute?

How many ways can six consecutive positive integers be selected from 1 to 999 so that their product ends in exactly four zeros?

276 pupils are seated in a school hall. There are 22 rows of seats, each with 15 seats. At least how many rows have an equal number of pupils?

In triangle $ABC$, $X$ and $Y$ are the midpoints of $AB$ and $BC$, respectively. Let $AY$ and $CX$ intersect at $G$. If $AG=12$, $CG=16$ and $XY=10$, find the area of triangle $GXY$.

The figure shows a square and 12 congruent semicircles. Points $C,D$ are endpoints, while $A,B$ are centres of semicircles. If $AB=2011$, find the total area of all shaded regions.