Mock PMO1 Problems First Phase, Team Competition Round 1 August 26, 2011 4
1. Find the eighth root of 44 . 2. The points (1, 4), (x, 10) and (5, 12) are collinear. Find x. 3. If 25% of a is b and 40% of b is 16, what is
3 (a 20
+ b)?
4. Find the length of the hypotenuse of a right triangle with legs x and y given that 3x 3˙y = 2187 and (3x )y = 9. 5. On a certain test, the average scores of the boys, girls, and the whole class are 94, 91 and 90, respectively. How many percent of the class are boys? 6. Given that 2x + 3y = 6x − y 6= 0, find the value of
7x − y . x+y
7. Suppose that positive integers a, b,none of them are divisible by 10, satisfy ab = 20000. Find a + b. 8. If (2x − 3y + 3)3 is expanded, what is the sum of the coefficients of its terms? 9. If x +
1 1 = 4, what is the value of x2 + 2 ? x x
10. Compute the number of positive integral factors of 1002001. 11. Find p given that p3 = 29 + 26 (35 ) + 23 (39 ) + 312 . 12. In how many ways can the letters of the word ”olympiad” be rearranged such that the vowels appear in alphabetical order and then consonants also appear in alphabetical order? 13. Calculate the sum of 6 distinct prime factors of 224 − 1. 14. In a right triangle ABC with 6 B = 90◦ , points D, E, F are chosen on sides AB, AC, BC respectively, such that AD = AE and CF = CE. Find 6 DEF . 15. How many integers on the set S = 1, 2, 3 · 2010 are there such that x2 + x3 is a perfect square? 1 1 16. If r, s are the roots of x2 − 6x + 3 = 0 and r + , s + are the roots of ax2 − bx + c = 0, s r find a + b + c. 17. Find the probability that if two dice are thrown, the sum of the numbers on top faces is at least 7. 1
Philippine Mathematical Olympiad
1
18. Assume that a, b, c, d are real numbers such that a + b + c + d = 12 (a + b)(c + d) = 18 (a + c)(b + d) = 27 (a + d)(b + c) = 36
Find the value of a2 + b2 + c2 + d2 . 19. Compute the smallest poitive number of the form p3 + 7p2 that is a perfect square. 20. In how many ways can eight people be divided into 2 groups? 21. If a3 + 12ab2 = 679 and 9a2 b + 12b3 = 978, determine a2 − 4ab + 4b2 . 22. Find the smallest three digit number with exactly 15 divisors. 3 2 23. Quartic polynomial f (x) = x4 +ax √ +bx +cx+d, with real coefficients a, b, c, d, satisfy f (2i) = f (2 + i) = 0, where i = −1. Find a + b + c + d.
24. Find the value of p with 5 < p < 20 such that the roots of x2 − 2(2p − 3)x + 4p2 − 14p + 8 = 0 are integers. 25. Triangle ABC has 6 A = 45◦ . Point P is on side BC such that P B = 2, P C = 6, and O is the circumcenter. Calculate OP 2 . 26. In an isosceles triangle ABC with AB = AC, altitudes BD, CE where D, E lie on sides CA, AB respectively, BD and CE meet at H. If EH = 2 and AD = 4, find DE. 27. Calculate the largest positive integer n such that
5n + 27 is an integer. 2n + 3
28. Square ABCD has point E on BC such that CE = 8 and point F on CD such that F C = 15. A circle, being inscribed in a square, is also tangent to EF . Find the sidelength of the square. 29. Let r be the positive real solution to s r x= If 2r2 = p +
√
3+
q √ 3 + 3 + 3 + x.
q, where p, q are positive integers, find p + q.
30. Circle ω of radius 15 is tangent to x2 + y 2 = 100 at (6, −8). Find the sum of the x-coordinates of the centers of ω. 2