APMOPS 2010

Find the value of $(1-\frac12)(1-\frac13)(1-\frac14)\cdots(1-\frac1{98})(1-\frac1{99})(1-\frac1{100})$.

Find the value of $\frac12+\frac1{2\times2}+\frac1{2\times2\times2}+\cdots+\frac1{\underbrace{2\times\cdots\times2}_{10\text{ factors}}}$.

Find the total number of ways to arrange 3 identical white balls and 3 identical black balls in a circle on a plane. The two layouts below are considered as one way of arrangement.

Given that one and only one of the following statements is true, which one is true?

Let $n$ be a whole number greater than 1. It leaves a remainder of 1 when divided by any single digit whole number greater than 1. Find the smallest possible value of $n$.

$M$ and $N$ are mid-points of the lines $AD$ and $BC$ respectively. Given that the area of $ABCD$ is $2000\text{ cm}^2$, and the area of the shaded region $ANCM=x\text{ cm}^2$, find the value of $x$.

Source note: all vertex labels are missing from the original image. A consistent proposed labeling and an explanation are supplied in the solution.

Your pocket money had previously been decreased by ________ %. To get back to the same amount of pocket money before the decrease, you need to have an increase of 25%.

A circle of diameter 2 cm rolls along the circumference of a circle of diameter 12 cm, without slipping, until it returns to its starting position. Given that the smaller circle turns $x^\circ$ about its centre, find the value of $x$.

Find the last digit of the number $\underbrace{2\times2\times\cdots\times2}_{859435\text{ factors}}$.

Three bus services operate from the same bus interchange. The first bus service leaves at 24 minute intervals, the second at 30 minute intervals and the third at 36 minute intervals. All three services leave the bus interchange together at 0900. Find the number of minutes that has passed when they next leave the interchange together.

Twenty five boys position themselves in a 5 by 5 formation such that the distances between two adjacent boys in the same row or the same column are equal to 1 m. The two dark circles indicate a pair of boys whose distance is exactly 5 m. Given that there are $n$ pairs whose distance apart are exactly 5 m, find the value of $n$.

When Albert begins walking up slope $AB$ (1 km distance), across level ground $BC$ (12 km distance), and down slope $CD$ (3 km distance), Daniel begins his journey in the opposite direction from $D$ at the same time. Given that the speeds of both travelling up the slope, on level ground and down slope are 2 km/h, 4 km/h and 5 km/h respectively, find the number of hours that has passed when they meet.

Find the value of $1^3+2^3+3^3+4^3+\cdots+20^3+21^3$.

Three identical circles have at most three points of contacts as shown below. Find the least number of identical circles required to have nine points of contact.

A goat in a horizontal ground is tied to one end of a 14 m long rope. The other end of the rope is attached to a ring which is free to slide along a fixed 20 m long horizontal rail. If the maximum possible area that the goat can graze is $x\text{ m}^2$, find the value of $x$. [Ignore the dimension of the ring and take $\pi=22/7$.]

The 13 squares are to be filled with whole numbers. If the sum of any three adjacent numbers is 21, find the value of $x$. The fifth square contains 4, the ninth contains 9, and the thirteenth contains $x$.

A square is divided into three rectangles of widths $w_1,w_2,w_3$, as shown. If $w_1+w_3=w_2$ and the areas of the shaded regions $A,B,C$ are $8\text{ cm}^2,x\text{ cm}^2,2\text{ cm}^2$ respectively, find the value of $x$.

Given that $S=\dfrac1{\frac1{2001}+\frac1{2002}+\cdots+\frac1{2009}+\frac1{2010}}$, find the largest whole number smaller than $S$.

The figure comprises five identical squares. $A,B,C,D$ are vertices of the squares. $AB$ cuts $CD$ at $O$ and angle $AOC=x^\circ$. Find the value of $x$.

Find the smallest whole number that is not a factor of $1\times2\times3\times\cdots\times21\times22\times23$.