AMO 2027 Grades 8-9 SET 5 - Self-study

In the figure below, $PSR$, $RTQ$ and $PAQ$ are three semicircles of diameters 10cm, 3cm and 7cm respectively. Given that the area of the shaded region is $k\pi$. Find $k$.

Each figure in the sequence is composed entirely of $1\times1$ shaded squares

In the 28th figure, given that the ratio (in simplest form) of the area of the inner shaded rectangle to the area of the outermost rectangle is $m/n$, what is the value of $mn$?

In the figure, $\angle A$, $\angle B$, and $\angle C$, are right angles. If $\angle AEB=40^\circ$ and $\angle BED=\angle BDE$, then $\angle CDE=$

What is the smallest positive integer $n$ such that $n!$ is a multiple of $100,000$?

If numbers are arranged in 3 rows A, B and C according to the following table, which row will contain the number 1000 ?

| | |
|---|---|
| A | $1,6,7,12,13,18,19,\ldots$ |
| B | $2,5,8,11,14,17,20,\ldots$ |
| C | $3,4,9,10,15,16,21,\ldots$ |

In the Fibonacci sequence, $T_1=T_2=1$ and $T_n=T_{n-1}+T_{n-2}$ for all positive integers $n\ge3$. Each of the first 2015 Fibonacci terms is written on an identical card. All the 2015 cards are then placed inside a box. Philip randomly picks up a card from the box. What is the probability that he will pick an odd number?

Daniel had some money. He spent \(\$695\) on an iPhone. He then spent $2/5$ of the remaining money on a pair of shoes. The amount of money spent was \(\$65\) less than the amount of money he had left. How much did Daniel have at first?

Three men, Albert, Ben and Charles, and three women, Denise, Evelyn and Fiona, met at a restaurant.
- Each of the men is married to exactly one of the women, and vice versa.
- Albert’s wife and Denise’s husband do not know each other.
- Evelyn’s husband and Ben’s wife do not know each other.
- Ben knows everybody.

Who is Fiona’s husband?

A tank has a leak which would empty the tank in 20 minutes. There is an inlet pipe which admits 5 liters a minute into the tank. Due to this, the tank is emptied in 30 minutes. How many liters does the cistern hold?

How many subsets of $\{1,2,3,4,\ldots,10\}$ are there so that, there will exactly any one element of $\{1,2\}$?

Suppose $p\ast q=2p+2q-pq$, where $p$ and $q$ are positive integers. If $8\ast x=4$, then what is the value of $x$?

The average age of Terry, Celine and Ken is 48 years. The average age of Ken and Terry is 53 years. How old is Celine?

Determine the of non–negative integer $M$ that satisfies exactly three of the four statements below.

(1) $M$ is prime.

(2) $M+3$ is prime.

(3) $1<\sqrt M<8$.

(4) $M+5$ has an odd number of factors.

In the following number sequence,
$$\ldots,245,234,223,212,201,\ldots,$$
234 is the 55th term.

a) Find the first term.

b) Find the 70th term.

A square is inscribed in a circle with diameter 2. Four smaller circles are then constructed with their diameters on each of the sides of the square. Determine the area of the shaded region.