Subcontests
(30)IOQM 2023-24 P-27
A quadruple (a,b,c,d) of distinct integers is said to be balanced if a+c=b+d. Let S be any set of quadruples (a,b,c,d) where 1⩽a<b<d<c⩽20 and where the cardinality of S is 4411. Find the least number of balanced quadruples in S. IOQM 2023-24 P-24
A trapezium in the plane is a quadrilateral in which a pair of opposite sides are parallel. A trapezium is said to be non-degenerate if it has positive area. Find the number of mutually non-congruent, non-degenerate trapeziums whose sides are four distinct integers from the set {5,6,7,8,9,10} IOQM 2023-24 P-21
For n∈N, consider non-negative valued functions f on {1,2,⋯,n} satisfying f(i)⩾f(j) for i>j and ∑i=1n(i+f(i))=2023. Choose n such that ∑i=1nf(i) is at least. How many such functions exist in that case? IOQM 2023-24 P-29
A positive integer n>1 is called beautiful if n can be written in one and only one way as n=a1+a2+⋯+ak=a1⋅a2⋯ak for some positive integers a1,a2,…,ak, where k>1 and a1≥a2≥⋯≥ak. (For example 6 is beautiful since 6=3⋅2⋅1=3+2+1, and this is unique. But 8 is not beautiful since 8=4+2+1+1=4⋅2⋅1⋅1 as well as 8=2+2+2+1+1=2⋅2⋅2⋅1⋅1, so uniqueness is lost.) Find the largest beautiful number less than 100. IOQM 2023-24 P-17
Consider the set
S={(a,b,c,d,e):0<a<b<c<d<e<100}
where a,b,c,d,e are integers. If D is the average value of the fourth element of such a tuple in the set, taken over all the elements of S, find the largest integer less than or equal to D. IOQM 2023-24 P-14
Let ABC be a triangle in the xy plane, where B is at the origin (0,0). Let BC be produced to D such that BC:CD=1:1,CA be produced to E such that CA:AE=1:2 and AB be produced to F such that AB:BF=1:3. Let G(32,24) be the centroid of the triangle ABC and K be the centroid of the triangle DEF. Find the length GK. IOQM 2023-24 P-13
The ex-radii of a triangle are 1021,12 and 14. If the sides of the triangle are the roots of the cubic x3−px2+qx−r=0, where p,q,r are integers , find the nearest integer to p+q+r. IOQM 2023-24 P26
In the land of Binary , the unit of currency is called Ben and currency notes are available in denominations 1,2,22,23,.. Bens. The rules of the Government of Binary stipulate that one can not use more than two notes of any one denomination in any transaction. For example, one can give change for 2 Bens in two ways : 2 one Ben notes or 1 two Ben note. For 5 Ben one can given 1 one Ben and 1 four Ben note or 1 Ben note and 2 two Ben notes. Using 5 one Ben notes or 3 one Ben notes and 1 two Ben notes for a 5 Ben transaction is prohibited. Find the number of ways in which one can give a change 100 Bens following the rules of the Government. IOQM 2023-24 P-1
Let n be a positive integer such that 1≤n≤1000. Let Mn be the number of integers in the set Xn={4n+1,4n+2,…,4n+1000}. Let
a=max{Mn:1≤n≤1000}, and b=min{Mn:1≤n≤1000}.
Find a−b.