Math IOQM

Q1. "In a room of n people (where n >= 2), prove that there are always at least two people who have shaken hands with the exact same number of people." Q2. "In a tournament with n players (where n >=2 ), prove that at any given moment during the tournament, there are always at least two players who have completed the exact same number of games." Q3. Twenty pairwise distinct positive integers are all < 70. Prove that among their pairwise differences there are four equal numbers. Q4. Fifty-one small insects are placed inside a square of side 1. Prove that at any moment there are at least three insects which can be covered by a single disk of radius 1/7. Solution1: Possible number of handshakes: 0,1,2 ... n-1 But 0 and n-1 are not possible together, one of them has to be removed. So now there are n people and (n-1) possible values. By pigeonhole principle, at least 2 people need to have same number of handshakes. Solution 2: Same as above. Solution 3: 1,2...68...

Q1. A worker suffers a 20% cut in wages. He regains his original pay by obtaining a rise of: Q2. If m men can do a job in d days, then the number of days in which m + r men can do the job is: Q3. A boy walks from his home to school at 6 km per hour (kmph). He walks back at 2 kmph. His average speed, in kmph, is. Q4. A car travels from P to Q at 30 kilometres per hour (kmph) and returns from Q to P at 40 kmph by the same route. Its average speed, in kmph, is nearest to Q5. 5. A man invests Rs. 10,000 for a year. Of this Rs. 4,000 is invested at the interest rate of 5% per year, Rs. 3,500 at 4% per year and the rest at α% per year. His total interest for the year is Rs. 500. Then α equals Q6. Let (x_1, x_2, \ldots, x_{100}) be positive integers such that (x_i + x_{i+1} = k) for all (i), where (k) is a constant. If (x_{10} = 1), then the value of (x_1) is 7. If (a0 = 1), (a1 = 1) and (an = a_{n-1}a_{n-2} + 1) for (n > 1), then a465 and a466 are odd or even? 15.  A1. 25% A2. md/(m+r...

Distance of (x0,y0) from ax + by + c = 0 formula is: d = (ax0 + by0 + c)/sqrt(a^2 + b^2) Proof: Let the perpendicular from (x0,y0) fall on the line at (x1,y1). ax1 + by1 + c = 0 Slope of the perpendicular line = (y1 - y0)/(x1- x0) = b/a Let y1 - y0 = kb x1 - x0 = ka d^2 = (x1 - x0)^2 + (y1 - y0)^2 = k^2(a^2 + b^2) Now ax1 + by1 + c = 0 => a(ka + x0) + b(kb + y0) + c = 0 => k = -(ax0 + by0 + c)/(a^2 + b^2) =>  d^2 = (ax0 + by0 + c)^2/(a^2 + b^2) H.P. Distance between 2 parallel lines ax + by + c1 = 0  and  ax + by + c2 = 0 formula is: d = |c1 - c2|/sqrt(a^2 + b^2) Proof: Let (x0,y0) lie on the first line =>  ax0 + by0 + c1 = 0 Distance of (x0,y0) from the second line is: (ax0 + by0 + c2)/sqrt(a^2 b+2) = |c1 - c2|/sqrt(a^2 + b^2) = answer.

Q1. Let X be the midpoint of the side AB of △ABC. Let Y be the midpoint of CX. Let BY cut AC at Z. Prove that AZ=2ZC. Q2. ABC is a equilateral triangle with vertex A fixed and B moving in a given straight line. Find the locus of C. Q3. Let  A be one of the two points of intersection of two circles with centres  X and  Y. The tangents at  A to these two circles meet the circles again at  B,C. Let the point  P be located so that  PXAY is a parallelogram. Show that  P is the circumcentre of triangle  ABC. Q4: Let ABC be a triangle and h be the altitude from A to BC. Prove: (b+c)^2 >= a^2 + 4h^2 Q5: The interior of a quadrilateral is bounded by the graphs of  ( x + a y ) 2 = 4 a 2 ( x + a y ) 2 = 4 a 2  and  ( a x − y ) 2 = a 2 ( a x − y ) 2 = a 2 , where  a a  is a positive real number. What is the area of this region in terms of  a a , valid for all  a > 0 a > 0  ? Q6. Three equally spaced p...

Q1) Solve the cubic equation (9x^3 - 27x^2 + 26x - 8 = 0), given that one of the root of this equation is double the other. Q2) If the product of two roots of the equation (4x^4 - 24x^3 + 31x^2 + 6x - 8 = 0) is 1, find all the roots. Q3 Obtain a polynomial of lowest degree with integral coefficient, whose one of the zeros is sqrt{5} + sqrt{2}. Solution 1: Assume a,2a are the roots. Put the values in the given equation and equate them. You will get a cubic in 'a' with one root as 0. a can't be 0 since that would mean -8 = 0 Other roots will be 2/3,13/21 Putting a = 2/3 in the original equation works out nicely and 13/21 doesn't quite fit it. So roots will be 2/3,4/3,1 = answer. Solution 2: Solution 3: Simplest way to solve such question is to start with x = given root = sqrt(5) + sqrt(2) Square both sides: x^2 = 7 + 2sqrt(10) x^2 - 7 = 2.sqrt(10) Again square to eliminate the sqrt on the right side and now you will have a degree 4 polynomial with all integer coefficients...

Q1). Andy and Bethy are at same point. Andy leaves at 1:30 toward north at a steady 8 miles/hr speed. Bethy leaves at 2:30, toward east at a steady 12 miles/hr speed. At what time they will be exactly the same distance away from their starting point? Q2). A box contains 10 pounds of a nut mix i.e., 50% peanuts, 20% cashews, 30% almonds. A 2nd nut mix containing 20% peanuts, 40% cashews, 40% almonds is added to the box resulting in a new nut mix i.e., 40% peanuts. How many pounds of cashews are now in the box? Q3). How many isosceles triangles are there with positive area whose side lengths are all positive integers and whose longest side has length 2025? Q4) 1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, 2, 1, 2, 3, 4, 5, 4, 3, 2, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 2, ... What is the 2025th term? Q5) Suppose a and b are real numbers. When the polynomial (x^3 + x^2 + ax + b) is divided by (x - 1), the remainder is 4, and when divided by (x - 2), the remainder is 6. b - a = ? Q6) The sequence (1, x, y, ...