Math IOQM

 Q1. Let ABCD be a convex quadrilateral with ∠BAD=2∠BCD and AB=AD. Let P be a point such that ABCP is a parallelogram. Prove that CP=DP. S1. Draw the ||gram first. That will make it easier to draw the rest of the diagram. Let O be center of circumcircle of BCD BOD is 2.theta. OB = OD = r Given AB = AD = x. => AO is perpendicular bisector of BD. Now we can show BOD and BAD are congruent triangles. BD common. Both are isosceles so AO is angle bisector of 2.theta for both. Angle ABD = ADB = OBD = ODB = 90 - theta By ASA congruent. So corresponding sides are equal. => r = x. BODA is Rhombus. Now CP is also r cause ABCP is a ||gram. Since BODA is Rhombus AB || OD. Also ABCP is ||gram so AB || CP => AB || OD || CP Also AB = CP = r = OD OD = CP and OD || CP => ODPC is a ||gram. => DP = OC = r H.P. that CP = DP. Q2. The ratio of the median AM of a triangle ABC to the side BC equals sqrt(3):2 The points on the sides of ABC dividing these sides into 3 equal parts are marked. P...

PHP1: n holes, n+1 pigeons, at least 1 hole with > 1 pigeons. PHP2: n holes, nk+1 pigeons, at least 1 hole with > k pigeons. Q1. Pr. th. If there are 5 points inside a unit square, then the distance between at least any two points is less than 1/sqrt(2). A1. Divide in 4 smaller squares => at least 2 will be inside one square. Max distance inside that square is 1/sqrt(2). Q2. There are 14 sock pairs in a basket. How many you need to pick to be sure that you have 1 pair at least. A2. 15. PHP3: If mean of n +ive integers is k then at least 1 of them is >= k and at least 1 of them is <= k. Proof: If a1 = a2 = ... an = k then H.P. If a1 <= a2 <= a3 ... <= an then if a_i > k => a_j < k H.P. Q3. Suppose numbers from 1 through 20 are placed in some order around a circle. Prove that i) The sum of some three consecutive numbers must be at least 32. ii) The sum of some four consecutive numbers must be at least 42. Let the triplets be (a1,a2,a3) (a2,a3,a4) ... (a20...

Q. The six faces of a cube are coloured using all three colours red, blue and green. Two colourings are considered the same if one can be obtained from the other by rotating the cube. How many distinct colourings are there? S21. Since all 3 colors need to be used, there are 3 cases: 4,1,1 3,2,1 2,2,2 Case 1: 4,1,1 3 ways to choose the color with 4 faces. Now the remaining 2 faces with 2 different colors have 2 options, opposite or adjacent. So 3*2 = 6 ways. Red Blue Yellow Yellow Yellow Yellow Drag the cube. The red and blue faces are adjacent. The other four faces have the same colour. ↶ Left ↑ Up Reset ↓ Down Right ↷ Case 2: 3,2,1 Choose first color in 3 ways, second in 2 and third in 1 way: 3*2*1 = 6 total. For each of them 3 sub cases as mentioned below...

Q13. How many sets of three edges of a cube can be chosen so that no two chosen edges share a vertex? S13. 12 ways to choose the first edge. Each of the 2 vertices of this edge had 2 more edges to it. So they are excluded from future selections. 7 edges and 6 vertices remain. Out of the 6 vertices, 2 have 3 edges each and 4 have 2 edges each. Total ways to select 2 edges from 7 = 7C2 = 21. From this we have to subtract those edge pairs which share a vertex. Consider a vertex with degree 3. It gives us 3C2 such pairs. And there are 2 such vertices. Total 6 such edge pairs. Similarly vertices with degree 2 give us 2C2*4 = 4 such pairs. 21 - 6 - 4 = 11 valid edge pairs. So total: 12*11 = 132 valid pairs. But in the first step we could have any of the 3 edges as the first edge. Later we just counted pairs. So 132/3 = 44 = answer. Now let's modify the original problem. What if we need to count any pair of edges from a cube which don't share a vertex? Each vertex has degree 3. So 8*3...

Q1. How many distinguishable arrangements are there of 1 brown tile, 1 purple tile, 2 green tiles, and 3 yellow tiles in a row from left to right? (Tiles of the same color are indistinguishable.)  S1. 7!/3!2! Q2. Seven 6-sided dice are rolled. The probability that the sum of the numbers on the top faces is 10 can be written as n/6^7 where n is a positive integer. What is n? S2. x1+x2..x7 = 10 Each of them is minimum 1. For stars and bars: x1+x2...x7 = 3 9C6 ways out of 6^7 ways. 84/6^7 n = 84 Q3. Suzanne went to the bank and withdrew 800 dollars. The teller gave her this amount using 20 dollars bills, 50 dollars bills, and 100 dollars bills, with at least one of each denomination. How many different collections of bills could Suzanne have received? S3. 20x + 50y + 100z = 800 2x + 5y + 10z = 80 5y will be even, y = 2k 2x + 10k + 10z = 80 x + 5k + 5z = 40 x = 5(8 - k - z) x >= 1 => 8-k-z > 0 => k+z <8 => k+z <= 7 For k+z <= 7 with k>=1 and z>=1 how many ...

1. How many ways to distribute 10 different balls into 3 different bags such that none of the bags is empty? S1. Distinct objects/distinct buckets => PIE(Principle of Inclusion Exclusion) Step 1: Total cases: 3^10 since each ball can go into any of the bags. Step 2: Subtract those cases where one bag is empty: 3C1 to choose one bag. And now each ball has 2 options: 2^10. So total 3*2^10 cases need to be subtracted to account for one empty bucket. In step 2 consider what happens in detail: When we made one bucket empty, 10 balls have to go into 2 buckets: 2^10. This also includes the cases when all the balls go to one bucket. So considering empty bucket1 also considers the cases where bucket 1 and 2 are both empty. Similarly when we consider bucket2 as empty we are also considering the case where both bucket 1 and 2 are empty. Essentially we are double subtracting the cases where 2 buckets are empty. So we need to add them back. 3C2 to choose 2 buckets to remain empty. Now each ball ...