Question bank APMOPS 2008

What is the difference between the sum of the first 2008 even numbers and the sum of the first 2008 odd numbers?

The side lengths of a triangle are consecutive whole numbers, and its perimeter is greater than 2008 cm. If the least possible perimeter is $x$ cm, find $x$.

Find $2008\times20072007-2007\times20082007$.

A rectangular sheet has length 8 cm. When it is folded exactly in half, the ratio of its length to its width remains unchanged. If the square of the original width is $x$ cm$^2$, find $x$.

The numbers 1 to 10 are placed in the circles so that the four numbers on each of the three lines have sum 21. The three lines share the central circle, labelled n. Find n.

Find $(56789+67895+78956+89567+95678)\div5$.

A circle touches the sides of a large square and passes through the vertices of a small square. The small square has area 9 cm$^2$ and the large square has area $x$ cm$^2$. Find $x$.

One hundred numbers are placed along the circumference of a circle. The sum of any five adjacent numbers is always 40. Find the difference between the largest and smallest numbers. Source note: the question explicitly says 100, but the supplied working uses 888 numbers. The literal question has no unique numerical answer; it requires review.

In triangle $PQR$, $PQ=6$ cm, $PR=4$ cm and $QR=6$ cm. If $PQ$ and $PR$ are tripled while $QR$ is unchanged, which statement is correct?

Find the last five digits of $1+22+333+4444+55555+666666+7777777+88888888+999999999$.

An $80^\circ$ arc of circle A has the same length as a $60^\circ$ arc of circle B. Their areas are in ratio $a:b$. Find the smallest value of $a+b$.

A circle with circumference 1 m rolls around the outside of an equilateral triangle of perimeter 3 m without slipping. How many turns does it make in one complete circuit?

Rectangle $ABCD$ has area 32 cm$^2$. Triangle $ADF$ has area 2 cm$^2$ and triangle $ABE$ has area 8 cm$^2$, with $F$ on $DC$ and $E$ on $BC$. The shaded triangle $AEF$ has area $x$ cm$^2$. Find $x$.

Containers A and B each contain 9 white, 9 black and 9 red marbles. Ten marbles are moved from A to B. How many marbles must be returned from B to A to guarantee that A contains at least 8 marbles of each colour?

$9^{10}$ is a 10-digit number. A is the sum of its digits; B is the sum of the digits of A; C is the sum of the digits of B. Find C.

A car travels from A to B at constant speed $V$ km/h. Increasing the speed by 20% makes it arrive 1 hour earlier. If it travels the first 120 km at $V$ and then increases its speed by 25%, it arrives 48 minutes earlier. If the distance is $x$ km, find $x$.

All faces of a rectangular block are painted green, then it is cut into unit cubes of volume 1 cm$^3$. Seven cubes have no painted faces. How many cubes have exactly two painted faces?

$ABC$ is equilateral, and $OD,OE,OF$ are perpendicular to $AB,BC,CA$, respectively, with O inside the triangle. If $OD+OE+OF=28$ cm and the triangle height is $x$ cm, find $x$.

The diagram shows triangle $PQR$. Three lines parallel to its sides pass through O. The source states that the areas of the three shaded “triangles” are 32, 48 and 96 cm$^2$; the drawn shaded regions are in fact parallelograms. Using those shaded regions, find the area $x$ cm$^2$ of triangle $PQR$.

A straight line can cross at most 3 unit squares of a $2\times2$ grid and at most 5 of a $3\times3$ grid, as shown. What is the largest number of unit squares a straight line can cross in a $2008\times2008$ grid?