2004 xIMC keystage2-EMIC individual

Five trucks carry loads: A and B each $3$ tonnes, C and D each $4.5$ tonnes. E carries $1$ tonne more than the average load of all five trucks. How many tonnes does E carry?

Let $A=200320032003\times2004200420042004$ and $B=200420042004\times2003200320032003$. Find $A-B$.

Five boxes contain $110,105,100,115,130$ marbles respectively; each box contains only red or only green marbles. Remove one box so that the remaining green count is three times the remaining red count. How many marbles are in the removed box?

Find the smallest natural number which, multiplied by $123$, produces a number ending in $2004$.

Peter has a two-pan balance, a $200$ g weight and a $1000$ g weight. From a packet containing $2000$ g of sugar he wants to take $600$ g. What is the minimum number of moves?

Each side of a fish needs $6$ minutes in a pan which holds at most $4$ fish at once. Find the minimum time to fry $5$ fish on both sides.

John and Carlson alternately take $1,2,3,4,\ldots$ candies from a bag. When too few remain for the next full turn, the boy whose turn it is takes all the rest. John receives $1012$ candies in total. How many candies were initially in the bag?

Five positive numbers have first plus fifth equal to $13$. The second, third and fourth are respectively $1/3,1/4,1/5$ of the total sum. Find the largest number.

In one class, $80\%$ participate in basketball, $85\%$ football, $74\%$ baseball and $68\%$ volleyball. Find the smallest possible percentage participating in all four.

Among $100,101,\ldots,999$, how many numbers have digits in strictly decreasing order? Examples: $986,852,741$ qualify; $342,551,622$ do not.

On the seven positions A–G in the original figure, both balls start at A. Each move takes the black ball one position clockwise and the white ball two positions counterclockwise. After how many moves do they meet again?

Compute $1^2-2^2+3^2-4^2+\cdots-2002^2+2003^2-2004^2+2005^2$.

Exactly one of A–E wrote something nasty on a blackboard. A says “It was B and C”; B says “Neither E nor I did it”; C says “A and B are both lying”; D says “Either A or B is telling the truth”; E says “D is not telling the truth”. The teacher knows three never lie while the other two may lie. Who wrote it?

$PQRS$ is a rectangle. T lies on diagonal $PR$ with $ST\perp PR$, $PT=9$, $TR=16$, $PS=a$, $ST=b$, $SR=c$. Find $a+b+c$.

In the original figure, A, E, D are collinear, $\angle ECD=32^\circ$, $\angle CDE=36^\circ$, and $CA=CE$. Find $x=\angle ACE$.