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MTAP competitions
General Education
Deparo High School
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MMC 2019 Grade 10 National Team Orals 1
Easy
E 1 If 2 a = 5, what is 4 a+ 2? [ 400 ]
E 2 The legs of a right triangle are 6 and 12 cm long. What is the area of the circle that circumscribes the triangle? [ 45 π cm 2 ]
E 3 Find the coordinates of the vertex of the parabola y = 2 x 2 + 4 x − 3. [(− 1 , − 5 )]
E4 Let g 1 + g 2 + g 3 + ⋯ be an infinite geometric series with a common ratio of
2
3
. If g 3 = 4 , what is the sum
of the series? [ 27 ]
E5 What is the radius of the circle with equation x 2 + y 2 − 6 y = 3? [ 2
√
3 units]
E 6 A line has slope − 2 and passes through the point ( 1 , 4 ). Find an equation of the line. [y = − 2 x + 6 ]
E 7 In an arithmetic sequence, the sum of the first four terms is 2 0 , while that of the first seven is 47. What is the sixth term of the sequence? [ 9 ]
E8 Given two parabolas, what are the possible number of points of intersection of these parabolas? [ 0 , 1 , 2 , 3 , 4 , infinitely many]
E 9 Jose and his younger brother, Pepe, can clean their house together in 3 hours. Working alone, Jose can
clean 4 9 o
f the house in 2 hours. How long will it take Pepe to clean the house alone? [9 hours]
E 10 What is the probability of getting exactly two heads in flipping a fair coin four times? [
3
8
]
Average
A 1 The zeros of the third-degree polynomial P(x) are − 2 , 1 , and 3. If P(− 1 ) = 16 , what is P( 4 )? [ 36 ]
A 2 In a standard deck of playing cards, a full house is a hand of 5 cards consisting of 3 cards of the same rank and a pair of cards of the same rank. How many full houses are possible? [ 3744 ]
A 3 Two vertical poles, one 6 m high and the other 9 m high, are erected 10 m away from each other. A taut wire runs from the top of one pole to an anchor on the ground between the poles and then to the top of the other pole. What is the least possible length of the wire? [ 5
√
13 m]
A4 If 2 , a, b, and 8 are consecutive terms of a harmonic sequence, what is a + b? [
20
3
]
A5 What is the least five-digit number that is divisible by each of the integers from 2 to 1 0? [ 1 0 0 80 ]
Difficult
D 1 The parabola y = − 2 x 2 + 3 x + c intersects the line y = c( 1 − x) at exactly one point. Find the coordinates of this point. [( 0 , − 3 )]
D 2 How many three-digit numbers, greater than 500 , have at least two of their digits the same? [ 139 ]
D 3 Let a 1 , a 2 ,... be an arithmetic sequence. If ap = q and aq = p, where p < q, what is apq in terms of p and q? [p + q − pq]
D4 The points ( 0 , 2 ), ( 0 , − 4 ), ( 3 , 5) lie on a circle. Find the radius of this circle. [ 3
√
5 units]
MTAP competitions
Subject: General Education
School: Deparo High School
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