COMC 2010 Unofficial Solutions
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Transcript of COMC 2010 Unofficial Solutions
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Henry Wise Wood Math Club 11/29/2010
*COMC 2010 Unofficial Solutions
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1. Find
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2. Solve
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3. Three circles centered at O, CD passes through B, A, O. OA=2, OB=4, OC=6, then what is the area of the shaded region?
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4. How many digits in
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5. What point on is closest to ?
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6. On a exam, the average of students who studied was 90%, the average of students who did not study was 40%, and the class average was 85%. What percentage of the class did not study?
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7. ABCD is a rectangle, AB=20, BC=10, WA=KC=12, WB=KD=16, find WK.
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8. Solve
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1a. Find C
A A 50B C 4437 57
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1b. Find n
D D D 30F F E 55F E E 50
50 n 40
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1c. Find P+Q
P Q T R 20Q P T R 20R R R T 33T T T R 19
20 20 19 33
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2a. Parabola intersects line at A and B. Find the coordinates of A and B.
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2b. Find the midpoint M of AB.
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2c. A line parallel to intersects the parabola at and . Prove .
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2d. N is the midpoint of PQ. Explain why MN is vertical.
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3a. O is the center of the circle with diameter AC, radius 1. B is a point on the circle and AB is extended to P with BP=1. Let S be the set of points P. If U is in S and UO is perpendicular to AC, find UO.
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3b. V is in S and VC is perpendicular to AC. Find VC.
𝑃 𝑇2=𝑃𝑀 ⋅𝑃𝑁
Power of a Point Theorem
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3b. V is in S and VC is perpendicular to AC. Find VC.
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3c. Do all points in S lie on one circle?
(AB)(CD) + (AD)(BC) = (AC)(BD)
Ptolemy’s Theorem
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3c. Do all points in S lie on one circle? NO.
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4a. . Determine all x such that .
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4b. Suppose that for some integer . Prove but
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4c. Prove that there are infinitely many rational numbers u, , so that are all distinct and
I don’t know how to do this problem either :(