2014 EMC Online Test: Transgression

2014 EMC Online Math Test: Transgression (18 November 2014, 19:30 PST1 ) General Instructions: 1. This is a test containing 15 questions which cover the topics of algebra, geometry, combinatorics and number theory.

2. Scratch paper, ruler, compass and protractor are permitted. Try not to use calculators in the entire test.2 Computers (WolframAlpha etc. included) are not allowed.

3. (Important) How to submit your answers: we’re glad we make a Google Form as a tool for submission of your answers, which will be attached along with this test sheet. In the form, please write your name first (this is required!), then on each question/problem write your correct answer. All answers are integers from 0 to 999; if your answer is a one-digit number X, please write X and if it is a two-digit answer XY, write XY. (It’s useless to pad zeros before your answer if it is less than three digits, because Google truncates these zeros.) If you didn’t have an answer for a particular question, please leave that question unanswered. The deadline of submission is on 20 November 2014, 23:59 PST. 4. Each correct answer is worth 6 points, each question left unanswered is worth 1 point and each incorrect answer is worth 0 points. There is no penalty for incorrect answers. Scores will be posted at Elite Math Circle Facebook group on 21 November 2014 (time to post the scores are to be announced).

5. The top 5 participants who got the highest scores will be recognized.

6. For comments, clarifications and/or violent reactions, please send your messages to Russelle Guadalupe’s Facebook account www.facebook.com/rhguadalupe2. For those participants who are not friends with him yet, please add him.

Solutions for this online test will be posted at Elite Math Circle Facebook group on or before January 2015.

Good Luck and Happy Problem Solving!!

Problem Proposer: Russelle Guadalupe

1 Philippine

Standard Time course, we don’t have enough time to monitor all of the participants. What we want to say is that all participants must solve all the problems with minimal/no use of calculators.

2 Of

2014 EMC Online Math Test: Transgression Problems 1. Four mangoes are arranged in a row according to their (strictly increasing) weight, with the first as the lightest and the fourth as the heaviest. The sum of the weights of the first mango and the third is 144 grams, the sum of the weights of the second and the fourth is 170 grams and the sum of the weights of the second and the third is 152 grams. Find the sum of the weights of the lightest mango and the heaviest mango, in grams.

2. How many natural numbers n less than 2014 are there such that n4 + 2n3 is a perfect cube?

3. In rectangle ABCD, points E, F, G and H lie on sides AB, BC, CD and DA such that EG ⊥ F H and EG meets F H at P . Suppose that [AEP H] = 60, [HP GD] = 48 and [P F CG] = 32. Find [ABCD]. ([XY ZW ] denotes the area of rectangle XY ZW .)

4. Real numbers a1 , a2 , a3 , · · · , a99 form an arithmetic progression. Suppose that a2 + a5 + a8 + · · · + a98 = 205. Find the value of a1 + a2 + a3 + · · · + a99 .

5. Suppose that you roll a fair die several times until a number divisible by three appears. Find the expected number of rolls you needed to achieve this success.

6. Call a positive integer n zeroless if n doesn’t contain the digit 0. Determine how many zeroless numbers are there whose sum of its digits is exactly 7.

7. Determine the remainder when

2014 2014 2014 2014 1· +2· +3· + · · · + 2014 · 1 2 3 2014 is divided by 1000. √ 8. In triangle ABC, BC = 16 3, ∠B = 30◦ and ∠C = 45◦ . As shown in the figure below, point P is inside ABC and two identical squares P QRS and P T U V , each of sidelength s, are inscribed in ABC such that Q lies on side AC, √ V lies on side AB, R and U lie inside ABC and S and T lie on side BC with ∠SP T = 60◦ . Given that s = a − b c for some positive integers a, b, c with c squarefree3 , find a + b + c.

9. A sequence {an } of real numbers is defined by a1 = 20, a2 = 14 and for n ≥ 3, an is the harmonic mean of an−1 and ∞ X 1 p an−2 . Suppose that the infinite sum = for some relatively prime positive integers p and q. Find p + q.

3k ak q k=1

10. Pete chooses two subsets (possibly empty) A and B of {1, 2, 3, 4, 5}. If the expected value of |A ∪ B| can be written in m the form for some relatively prime positive integers m and n, find mn. (|A ∪ B| denotes the cardinality of A ∪ B, n that is, the number of elements of A ∪ B.) 11. Given that 2017 is prime, determine the remainder when (2013!)3 · (2014!)3 · (2015!)3 · (2016!)3 is divided by 2017.

12. For a positive integer k ≥ 2, let ak , bk , ck be the complex roots of the cubic equation

1 1 1 1 x− x− x− = k−1 k k+1 k and let pk = ak (bk + 1), qk = bk (ck + 1) and rk = ck (ak + 1). Given that

∞ X pk qk rk k=2

k+1

=

m for some relatively prime n

positive integers m and n, find m + n.

3 A natural number is said to be squarefree if it is not divisible by the square of any prime, that is, the exponent of each prime factors in its prime factorization is 1. For instance, 11, 35 and 210 are squarefree, while 12, 100 and 576 are not squarefree.

13. Let r1 , r2 , r3 , r4 , r5 , r6 , r7 be the seven seventh complex roots of unity (that is, the seven complex roots of r7 = 1).

There are relatively prime positive integers m and n such that 1 1 1 1 1 1 m 1 + + + + + + = . r12 + 2 r22 + 2 r32 + 2 r42 + 2 r52 + 2 r62 + 2 r72 + 2 n Find m + n.

14. Determine the sum of all positive integers n ≤ 1000 such that the triple sum

j n X i X X

ijk is a perfect square.

i=1 j=1 k=1

15. Equilateral triangles A1 A2 Z, A2 A3 X and A3 A1 Y are constructed outwards on the sides of an acute triangle A1 A2 A3 , as 23 13 XY 2 + Y Z 2 + ZX 2 shown in the figure below. If 2 = 6 and cos A1 + cos A2 + cos A3 = 9 , then sin A1 + sin A2 + A1 A22 +√ A2 A23 + A A 3 1 √ p r sin A3 can be expressed in the form + , where p, q, r, s are positive integers and p, r are squarefree. Find the q s value of p + q + r + s.