Q1 Two positive integers and exist such that is divisible by 5. Prove that is also divisible by 5.
Q2 Find all integers , and such that
Q3 Find all solutions in positive integers to the equation
Q4 Find all non-negative integers such that is a perfect square.
Q5 Let be a positive integer. Prove that there exists a multiple of , the sum of whose digits is equal to .
Q1 Let be any point on the line passing through vertices and of triangle . A circle passing through A is tangent to side of triangle at point and intersects sides and at points and respectively.
Prove that is parallel to if and only if is an internal or external bisector .
Q2 Points , , , and lie in that order on a circle and have the property the lines and are parallel.
Prove that if and only if
Q3 Triangle has centroid . Let be the midpoint of . The line through parallel to intersects at .
Prove that if and only if .
Q4 Let be a non-equilateral triangle with incentre and centroid .
Prove that if and only if is parallel to .
Q5 Two circles of different radius having centres and are externally tangent at . A common tangent (not through ), touches the first circle at and the second one at . The line through which is perpendicular to , and the perpendicular bisector of intersect at .
Prove that .
Q1 Positive real numbers , and satisfy . Prove that
Q2 Find all functions such that
for all .
Q3 Consider the quartic polynomial where , and are positive real numbers.
(a) Prove that if , then the equation cannot have 4 real roots.
(b) Prove that if , then the equation cannot have 4 distinct positive real roots.
Q4 For each pair of real numbers , , let
What is the minimum possible value that can achieve as , are allowed to range over all real numbers?
Q5 Let be any real number. Prove that is an integer if and only if
holds for all positive integers .
(Here denotes the greatest integer not exceeding the real number .)
Q1 Finn likes nine digit positive integers which do not contain the same digit more than once, and which do not contain the digit zero at all.
What is the sum of all the 9 digit numbers Finn likes?
(Express your answer as an actual number, not an uncomputed formula.)
Q2 Let be a positive integer. Krishna is studying the subsets of . Find a simple formula for:
(a) The number of subsets of S which contain an odd number of elements.
(b) The number of subsets of S the sum of whose elements is an odd number.
Q3 Let be a positive integer. A class has students in it, and no two people have the same height. Principal Calvin wishes to arrange these students in two rows for a class photo. He wants two rows of students, one row directly in front of the other, so that each front row student is shorter than the corresponding back row student.
In how many ways can Calvin organise this?
Q4 Satvik attempts all six questions on an exam. Each question is given an integral mark out of 10. It turns out that Satvik never scores more in a later question than in any earlier question.
How many different possible sequences of six marks could Satvik achieve subject to the above question.
Q5 Sharky would like to decorate the surface of a cube of side length 4 with some rectangular strips of paper. Any such decoration must be neat. This means that if each face of the cube is seen as being composed of 16 unit squares in the natural way, and if each rectangular strip is seen as being composed of three unit squares in the natural way, then each unit square of any such rectangular strip must exactly cover a unit square on the surface of the cube. It is permitted to bend such a strip over the edge of the cube. But it is not permitted to have a rectangular strip overlapping another such strip.
(a) What is the maximum number of strips that Sharky can use?
(b) Suppose that Sharky only wants to cover three mutually adjacent faces. What is the maximum number of strips that Sharky can use?
Q1 Find all right angled triangles whose side lengths are positive integers and whose area is equal to its perimeter.
Q2 Let be the circumcentre of acute triangle . Points and are located on sides and , respectively such that triangle and are similar.
Prove that is the orthocentre of triangle .
Q3 Find all function which satisfy
for all .
Q4 Around a circular table are seated people, where is a positive integer. After a break, they again sit around the circular table in a different order.
Prove that there are at least two people such that the number of participants sitting between them before and after the break is the same.
Q5 From the set of natural numbers , four consecutive even integers are removed. The remaining integers have an average value of .
Determine all possible sets of four consecutive even integers whose removal creates this situation.
Q6 Three schools have 200 students each. Every student has at least one friend in each school. (Friendships are mutual of course.) It is known that there exists a set of 300 students among the 600 students such that for any school an any two students which are not in the school , the number of friends in of and are different.
Show that one can find a student in each school such that they are all friends of each other.
Q7 Let be the centre of the square inscribed in an acute triangle with two vertices of the square on side . Thus one of the two remaining vertices of the square on side and the other is on . Points are defined in a similar way for inscribed squares with two vertices on sides and, respectively.
Prove that lines are concurrent.
Q8 Determine all ordered triples of positive rational numbers for which all three of
Q9 A cube of side contains points in its interior, no four of which are coplanar. Amongst the points which include the 999 points as well as the eight vertices of the cube, prove that there exists a tetrahedron of volume no more than 9 units.
Q10 A circle passes though points and of triangle and intersect the lines and in and respectively. The projections of the points and onto are denoted by and , respectively. The projections of the points and onto are denoted by and , respectively.
Prove that the points are concyclic.
Q11 Let be a given positive integer and let , with , be a polynomial with integer coefficients such that
(a) are divisible by all prime factors of ,
(b) and are relatively prime.
Prove that there exists a positive integer , such that is divisible by
Q12 Find a function satisfying
for all as well as and .