NMC Sample Problems: Grade 10

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1 NMC Sample Problems: Grade 0. Find the remainder when P (x) = x 00 + x + is divided by x How many positive numbers less than 00 are there that have odd number of positive divisors? 0 9. If what is a b a+b?. 08 a + b a b = 08, A palindrome is a number that reads the same backward or forward. For example, 8 is a palindrome. There is one prime number that is a factor of all palindromes with an even number of digits. What is the prime number? If both roots of the quadratic equation x 99x + k = 0 are prime numbers, find the value of k Simplify a a a, where a is a positive real number. a 9/ a / a / a / 7. If a and b are two distinct numbers such that a 7a = and b 7b =, what is ab? Suppose α and β are nonzero real solutions of the equation x 6x + = 0. Find the sum α + β. 7 7 of 8

2 NMC Sample Problems Grade 0 9. Define a b = ab + a + b for all real numbers a and b. If a pair of positive numbers (x, y) satisfies what is x + y? x y = y x and x y =, How many arrangements of the letters in INF INIT I are there such that the consonants occur in alphabetical order? Compute the perimeter of the triangle whose vertices are (0, 0) and two intercepts of the line x + y = Determine i + i + i + i + + i 08, where i =. + i 0 i i + i. How many consecutive zeros are there at the end of 00! Find sin(cos x) in terms of x. x x x x x. Suppose that f( x) = x x + and f(x) = ax + bx + c. What is a + b + c? What is the remainder when 7 is divided by? 6 7. What is the largest prime factor of? How many real solutions are there to the equation x + x =? 0 of 8

3 NMC Sample Problems Grade 0 9. Suppose that + solves x 6x + a = 0. Find the value of a Two numbers, x and y are selected at random from the interval [0, ]. What is the probability that y x +? /7 /7 / /9 /9. Consider the sequence a n defined by a =, a =, a = +,, a n = + + ( ) n+ n. Find 99 a n. n= equals If a and b are real numbers, then (a b ) is real if and only if a = b or b = 0 a = b or b = 0 a = b and b = 0 a = 0 a = b or b = 0. If y = x + is tangent to the circle centered at the origin in an xy-plane. Find the radius of the circle.. Find the area of the given figure. (Note: Figure not drawn to scale!) How many numbers between and 00 are multiples of or or 7? of 8

4 NMC Sample Problems Grade 0 7. Let a + b = and a + b =. Find a + b. / 8. The largest integer that divides n n for all n is equals: How many terms are there in the expansion of (x + y + z + ) 6? If (x x + 6) (x ) is factored using only real coefficients, then the number of factors (x = x x has two factors) is: 7. Consider the circle below which has center at P with radius. If the acute angle between QR and line L is 0 degrees, and the angle between L and QP is 90 degrees, find the area of the shaded region. (Note: Figure not drawn to scale!) R Q P L π π π 6 π. Let z be a solution of x + x + = 0. What is the value of z 0 + z +? 0 i of 8

5 NMC Sample Problems Grade 0. Let n be the smallest positive integer such that the expansion of ( x + x ) n has a constant term. Determine the constant term Evaluate the sum: cos 0 + cos cos 70 + cos How many positive integers d such that the remainder in dividing 09 by d equals? Grade A oil sells for $78 per unit and grade B oil sells for $ per unit. If a mixture sells for $6 per unit, find the ratio of A to B used in the mixture. 8. Find the sum of If a ball inches in diameter weighs ounces, then what is the weighs of the ball inches in diameter made of the same material? 7 π If x is a solution to 9 x 9 x =, then x satisfies: < x < 0 0 < x < < x < < x <. Let + i be a root of x + ax + bx = 0 where a and b are real numbers. Find a + b? What is the remainder when is divided by? 8. Among the 0 numbered cards, from to 0, we will pick cards randomly. What is the probability that number 7 card is excluded from the selection? 0 of 8

6 NMC Sample Problems Grade 0. The length of an interior diagonal of a rectangular box is 8 cm, and the sum of the lengths of all the edges of the box is 6 cm. Find the surface area of the box. cm cm 6 cm 7 cm 9 cm. In the figure, AB = 0, AC =, AD = DB, angles ACB and ADE are right angles. Find the area of the quadrilateral ADEC. (Note: Figure not drawn to scale!) C E A D B Which one of the following is equal to cos π? The function g(x) = f(+x)+f( x) is symmetric about the vertical line x = a for all functions f defined on the reals. Find the value a A sequence begins with a, a, and for n > is defined by a n = a n a n. Find a 08. a a a a - a a a 9. Which one of the following is equal to The increasing sequence of positive integers a, a, a, has the property that If a 6 = 9, what is a? a n+ = a n+ + a n for all n How many positive factors of are there? of 8

7 NMC Sample Problems Grade 0. In a certain village live 7 families. Each family has one, two, or three cars. There are as many families owning three cars as families with only one. How many cars are there in the village?. For how many number of real numbers x is (x + i) real?. Given that the one roots of the equation x ax + m = 0 is a b, determine m in terms of a and b. (Completely expand the result.). Find the set of all real numbers x for which x + x + x + x + is a rational number. 6. Recall that for real number x, [x] denotes the greatest integer not exceeding x. Find all real number pairs (x, y) satisfying the following equations: x + y [y] = 0. x + [x] + [y] = Compute (9)(0)()() +. 7 of 8

8 KEYS [] [6] [] [6] [] [7] [] [7] [] [8] [] [8] [] [9] [] [9] [] [0] [] [0] [6] [] [6] [] [7] [] [7] [] [8] [] [8] [] [9] [] [9] [] a b [0] [] [] [] [] [6] [7] [8] [0] [] [] [] [] The set of all rational numbers. [6] (9.7,.) [7] 9 [] [9] [] [] [0] []

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