Subcontests
(40)Find r
In triangle ABC lines CE and AD are drawn so that DBCD=13 and EBAE=23. Let r=PECP where P is the intersection point of CE and AD. Then r equals:[asy]
size(8cm);
pair A = (0, 0), B = (9, 0), C = (3, 6);
pair D = (7.5, 1.5), E = (6.5, 0);
pair P = intersectionpoints(A--D, C--E)[0];
draw(A--B--C--cycle);
draw(A--D);
draw(C--E);
label("A", A, SW);
label("B", B, SE);
label("C", C, N);
label("D", D, NE);
label("E", E, S);
label("P", P, S);
//Credit to MSTang for the asymptote
[/asy]<spanclass=′latex−bold′>(A)</span> 3<spanclass=′latex−bold′>(B)</span> 23<spanclass=′latex−bold′>(C)</span> 4<spanclass=′latex−bold′>(D)</span> 5<spanclass=′latex−bold′>(E)</span> 25 Find BE
Point F is taken on the extension of side AD of parallelogram ABCD. BF intersects diagonal AC at E and side DC at G. If EF=32 and GF=24, then BE equals:
[asy]
size(7cm);
pair A = (0, 0), B = (7, 0), C = (10, 5), D = (3, 5), F = (5.7, 9.5);
pair G = intersectionpoints(B--F, D--C)[0];
pair E = intersectionpoints(A--C, B--F)[0];
draw(A--D--C--B--cycle);
draw(A--C);
draw(D--F--B);
label("A", A, SW);
label("B", B, SE);
label("C", C, NE);
label("D", D, NW);
label("F", F, N);
label("G", G, NE);
label("E", E, SE);
//Credit to MSTang for the asymptote
[/asy]<spanclass=′latex−bold′>(A)</span> 4<spanclass=′latex−bold′>(B)</span> 8<spanclass=′latex−bold′>(C)</span> 10<spanclass=′latex−bold′>(D)</span> 12<spanclass=′latex−bold′>(E)</span> 16 Points P
Given points P1,P2,⋯,P7 on a straight line, in the order stated (not necessarily evenly spaced). Let P be an arbitrarily selected point on the line and let s be the sum of the undirected lengths
PP1,PP2,⋯,PP7. Then s is smallest if and only if the point P is:<spanclass=′latex−bold′>(A)</span> midway between P1 and P7<spanclass=′latex−bold′>(B)</span> midway between P2 and P6<spanclass=′latex−bold′>(C)</span> midway between P3 and P5<spanclass=′latex−bold′>(D)</span> at P4<spanclass=′latex−bold′>(E)</span> at P1 Coefficients b and c
Consider equations of the form x2+bx+c=0. How many such equations have real roots and have coefficients b and c selected from the set of integers {1,2,3,4,5,6}?<spanclass=′latex−bold′>(A)</span> 20<spanclass=′latex−bold′>(B)</span> 19<spanclass=′latex−bold′>(C)</span> 18<spanclass=′latex−bold′>(D)</span> 17<spanclass=′latex−bold′>(E)</span> 16 Find BE
Point F is taken in side AD of square ABCD. At C a perpendicular is drawn to CF, meeting AB extended at E. The area of ABCD is 256 square inches and the area of triangle CEF is 200 square inches. Then the number of inches in BE is:[asy]
size(6cm);
pair A = (0, 0), B = (1, 0), C = (1, 1), D = (0, 1), E = (1.3, 0), F = (0, 0.7);
draw(A--B--C--D--cycle);
draw(F--C--E--B);
label("A", A, SW);
label("B", B, S);
label("C", C, N);
label("D", D, NW);
label("E", E, SE);
label("F", F, W);
//Credit to MSTang for the asymptote
[/asy]<spanclass=′latex−bold′>(A)</span> 12<spanclass=′latex−bold′>(B)</span> 14<spanclass=′latex−bold′>(C)</span> 15<spanclass=′latex−bold′>(D)</span> 16<spanclass=′latex−bold′>(E)</span> 20 Similar Triangles
Chord EF is the perpendicular bisector of chord BC, intersecting it in M. Between B and M point U is taken, and EU extended meets the circle in A. Then, for any selection of U, as described, triangle EUM is similar to triangle:[asy]
pair B = (-0.866, -0.5);
pair C = (0.866, -0.5);
pair E = (0, -1);
pair F = (0, 1);
pair M = midpoint(B--C);
pair A = (-0.99, -0.141);
pair U = intersectionpoints(A--E, B--C)[0];
draw(B--C);
draw(F--E--A);
draw(unitcircle);
label("B", B, SW);
label("C", C, SE);
label("A", A, W);
label("E", E, S);
label("U", U, NE);
label("M", M, NE);
label("F", F, N);
//Credit to MSTang for the asymptote
[/asy]<spanclass=′latex−bold′>(A)</span> EFA<spanclass=′latex−bold′>(B)</span> EFC<spanclass=′latex−bold′>(C)</span> ABM<spanclass=′latex−bold′>(D)</span> ABU<spanclass=′latex−bold′>(E)</span> FMC Exponents a,b,c,d
If 2a+2b=3c+3d, the number of integers a,b,c,d which can possibly be negative, is, at most:<spanclass=′latex−bold′>(A)</span> 4<spanclass=′latex−bold′>(B)</span> 3<spanclass=′latex−bold′>(C)</span> 2<spanclass=′latex−bold′>(D)</span> 1<spanclass=′latex−bold′>(E)</span> 0 Find the missing vertex
Three vertices of parallelogram PQRS are P(−3,−2), Q(1,−5), R(9,1) with P and R diagonally opposite. The sum of the coordinates of vertex S is:<spanclass=′latex−bold′>(A)</span> 13<spanclass=′latex−bold′>(B)</span> 12<spanclass=′latex−bold′>(C)</span> 11<spanclass=′latex−bold′>(D)</span> 10<spanclass=′latex−bold′>(E)</span> 9 Points not on the graph
Which one of the following points is not on the graph of y=x+1x?<spanclass=′latex−bold′>(A)</span> (0,0)<spanclass=′latex−bold′>(B)</span> (−21,−1)<spanclass=′latex−bold′>(C)</span> (21,31)<spanclass=′latex−bold′>(D)</span> (−1,1)<spanclass=′latex−bold′>(E)</span> (−2,2)