Subcontests
(35)Pension of a retiring employee
A retiring employee receives and annual pension proportional to the square root of the number of years of his service. Had he served a years more, his pension would have been p dollars greater, whereas, had he served b years more b=a, his pension would have been q dollars greater than the original annual pension. Find his annual pension in terms of a,b,p, and q.<spanclass=′latex−bold′>(A)</span>2(a−b)p2−q2<spanclass=′latex−bold′>(B)</span>2ab(p−q)2<spanclass=′latex−bold′>(C)</span>2(ap−bq)ap2−bq2<spanclass=′latex−bold′>(D)</span>2(bp−aq)aq2−bp2<spanclass=′latex−bold′>(E)</span>(a−b)(p−q) Integers dividing large numbers
The greatest integer that will divide 13,511, 13,903, and 14,589 and leave the same remainder is<spanclass=′latex−bold′>(A)</span>28<spanclass=′latex−bold′>(B)</span>49<spanclass=′latex−bold′>(C)</span>98<spanclass=′latex−bold′>(D)</span>an odd multiple of 7 greater than 49<spanclass=′latex−bold′>(E)</span>an even multiple of 7 greater than 98 Find the sum of the digits
Find the sum of the digits of all numerals in the sequence 1,2,3,4,⋯,10000.<spanclass=′latex−bold′>(A)</span>180,001<spanclass=′latex−bold′>(B)</span>154,756<spanclass=′latex−bold′>(C)</span>45,001<spanclass=′latex−bold′>(D)</span>154,755<spanclass=′latex−bold′>(E)</span>270,001 Find AB in the diagram
In the accompanying figure, segments AB and CD are parallel, the measure of angle D is twice the measure of angle B, and the measures of segments AB and CD are a and b respectively. Then the measure of AB is equal to<spanclass=′latex−bold′>(A)</span>21a+2b<spanclass=′latex−bold′>(B)</span>23b+43a<spanclass=′latex−bold′>(C)</span>2a−b<spanclass=′latex−bold′>(D)</span>4b−21a<spanclass=′latex−bold′>(E)</span>a+b[asy]
size(175);
defaultpen(linewidth(0.8));
real r=50, a=4,b=2.5,c=6.25;
pair A=origin,B=c*dir(r),D=(a,0),C=shift(b*dir(r))*D;
draw(A--B--C--D--cycle);
label("A",A,SW);
label("B",B,N);
label("C",C,E);
label("D",D,S);
label("a",D/2,N);
label("b",(C+D)/2,NW);
//Credit to djmathman for the diagram[/asy] Binary Operations
Given the binary operation ∗ defined by a∗b=ab for all positive numbers a and b. The for all positive a,b,c,n, we have<spanclass=′latex−bold′>(A)</span>a∗b=b∗a<spanclass=′latex−bold′>(B)</span>a∗(b∗c)=(a∗b)∗c<spanclass=′latex−bold′>(C)</span>(a∗bn)=(a∗n)∗b<spanclass=′latex−bold′>(D)</span>(a∗b)n=a∗(bn)<spanclass=′latex−bold′>(E)</span>None of these