SMO(J) Mock Paper Duration: One hour May 18, 2012 1. Given Given a tra trapeziu pezium m ABCD with AB C D, AB > CD and ∠A + ∠B = 90 . Points M and N are midpoints of AB and C D respectively. Prove that M N = AB − C D . 2 ◦
2. Around Around a circle, circle, 5 ones and 4 zeros zeros are arranged arranged in any order. order. Then, Then, between between any two equal digits you write 0 and between any two different digits you write 1. Finall Finally y, the origina originall digits are wiped out. Can there there be an instance instance where all 9 digits will become 0? a,b,c, prove that 3. For nonnegative nonnegative reals a,b,c,
1
c a b + + ≥ 2. a b+c c
2 1. Given Given a tra trapeziu pezium m ABCD with AB C D, AB > CD and ∠A + ∠B = 90 . Points M and N are midpoints of AB and C D respectively. Prove that M N = AB − C D . 2 ◦
Extend AD and BC to meet at point E . We note note tha thatt E , M M ,, N are collinear because the segment EM must intersect the midpoint of C D since AB C D. Also Also,, ∠AEB = 90 . Hence, Hence, we must must have have EM = AM and EN = C N . N . Taking the difference of the two equations, we obtain Solution.
◦
M N = AM − C M =
AB − C D . 2
2. Around Around a circle, circle, 5 ones and 4 zeros zeros are arranged arranged in any order. order. Then, Then, between between any two equal digits you write 0 and between any two different digits you write 1. Finall Finally y, the origina originall digits are wiped out. Can there there be an instance instance where all 9 digits will become 0? Suppose Suppose it is possible. possible. Before Before the final final state state is achie achieved ved,, all digits digits must ust be 1s. 1s. Befor Beforee all all digi digits ts are are 1s, 1s, all all adjac adjacen entt digi digits ts must must be differ differen ent. t. However this is not possible with an odd number of digits.
Solution.
3. For nonnegative nonnegative reals a,b,c, a,b,c, prove that Solution.
c a b + + ≥ 2. a b+c c
Using AM-GM inequality, we have: c a b c a b+c + + = + + −1 a b+c c a b+c c ≥3−1 =2
Equality is achieved when a = c, b = 0.
