Subcontests
(24)IOQM 2022-23 P-22
A binary sequence is a sequence in which each term is equal to 0 or 1. A binary sequence is called friendly if each term is adjacent to at least on term that is equal to 1. For example , the sequence 0,1,1,0,0,1,1,1 is friendly. Let Fn denote the number of friendly binary sequences with n terms. Find the smallest positive integer n≥2 such that Fn>100 IOQM 2022-23 P-20
For an integer n≥3 and a permutation σ=(p1,p2,⋯,pn) of {1,2,⋯,n}, we say pl is a landmark point if 2≤l≤n−1 and (pl−1−pl)(pl+1−pl)>0. For example , for n=7,\\
the permutation (2,7,6,4,5,1,3) has four landmark points: p2=7, p4=4, p5=5 and p6=1. For a given n≥3 , let L(n) denote the number of permutations of {1,2,⋯,n} with exactly one landmark point. Find the maximum n≥3 for which L(n) is a perfect square. IOQM 2022-23 P-16
Let a,b,c be reals satisfying\\
3ab+2=6b,3bc+2=5c,3ca+2=4a.\\
\\
Let Q denote the set of all rational numbers. Given that the product abc can take two values sr∈Q and ut∈Q , in lowest form, find r+s+t+u. IOQM 2022-23 P-15
Let x,y be real numbers such that xy=1. Let T and t be the largest and smallest values of the expression \\
(x+y)2+(x−y)−2(x+y)2−(x−y)−2\\.
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If T+t can be expressed in the form nm where m,n are nonzero integers with GCD(m,n)=1, find the value of m+n. IOQM 2022-23 P-14
Let x,y,z be complex numbers such that\\
y+zx+z+xy+x+yz=9\\
y+zx2+z+xy2+x+yz2=64\\
y+zx3+z+xy3+x+yz3=488\\
\\
If yzx+zxy+xyz=nm where m,n are positive integers with GCD(m,n)=1, find m+n. IOQM 2022-23 P-12
Given △ABC with ∠B=60∘ and ∠C=30∘, let P,Q,R be points on the sides BA,AC,CB respectively such that BPQR is an isosceles trapezium with PQ∥BR and BP=QR.\\
Find the maximum possible value of [BPQR]2[ABC] where [S] denotes the area of any polygon S. IOQM 2022-23 P-8
Suppose the prime numbers p and q satisfy q2+3p=197p2+q.Write qp as l+nm, where l,m,n are positive integers , m<n and GCD(m,n)=1. Find the maximum value of l+m+n. IOQM 2022-23 P-7
Find the number of ordered pairs (a,b) such that a,b∈{10,11,⋯,29,30} and \\
GCD(a,b)+LCM(a,b)=a+b.