Infinity Individual Round
Email *
Name *
Quiz Questions
Q.1) The side length of the smallest equilateral triangle in which three discs of radii 2, 3, 4 can be placed without overlap is
Clear selection
Q.2) If  α,β,γ are the roots of x^3-x-1=0, then the value of (1-α)/(1+α)+(1-β)/(1+β)+(1-γ)/(1+γ)  equals
Q.3) Let ∆ABC be an isosceles triangle with AB = AC. Suppose that the angle bisector of ∠B meets AC at D and that BC = BD+AD. Determine ∠A.
Clear selection
Q.4) Find the smallest positive integer K such that every K-element subset of {1, 2, … ,50} contains two distinct elements a, b such that a+b divides ab.
Clear selection
Q.5) In triangle ABC, ∠C = 90 degrees, ∠A = 30 degrees and BC = 1. Find the minimum length of the longest side of a triangle inscribed in triangle ABC (that is, one such that each side of ABC contains a different vertex of the triangle).
Clear selection
Q.6) The minimum value of x^x  for a positive real number x is
Clear selection
Q.7) If x and y are positive real numbers, then which of the following statements is correct?
Clear selection
Q.8) Consider a positive integer n, such that n+1  is prime; then the greatest common divisor of n!+1  and (n+1)!  is ..  
Clear selection
Q.9) Let F be the midpoint of side BC of triangle ABC. Construct isosceles right triangles ABD and ACE externally on sides AB and AC with the right angles at D and E, respectively. What type of triangle is DEF.
Clear selection
Q.10) Let A and B be opposite vertices of a cube of edge length 1. The radius of the sphere with center interior to the cube, tangent to the three faces meeting at A and tangent to the three edges meeting at B is
Clear selection
Q.11) Given an alphabet with three letters a,  b,  c, find the number of words of n letters which contain an even number of a's.
Clear selection
Q.12) Let m and n be positive integers with gcd(m, n) = 1 and m+n is an even number. Then gcd( 5^m+7^m,5^n+7^n) is
Clear selection
Q.13) Let ABCD be a tetrahedron with ∠BAC = ∠ACD and ∠ABD = ∠BDC. Which of the following are true about the length of edges AB and CD.
Clear selection
Q.14) The prime numbers p,q for which the congruence α^(3pq) ≡ α (mod 3pq) holds for all integers α are
Clear selection
Q.15) In the isosceles triangle ABC (AC = BC) point O is the circumcenter, I the incenter, and D lies on BC so that lines OD and BI are perpendicular. Which of the following statements are true?
Clear selection
Q.16) Two piles of coins lie on a table. It is known that the sum of the weights of the coins in the two piles are equal, and for any natural number k, not exceeding the number of coins in either pile, the sum of the weights of the k heaviest coins in the first pile is not more than that of the second pile. For any natural number x, if each coin (in either pile) of weight not less than x is replaced by a coin of weight x, then ….
Clear selection
Q.17) Points E and F are given on side BC of convex quadrilateral ABCD (with E closer than F to B). It is known that ∠BAE = ∠CDF and ∠EAF = ∠FDE. Which of the following is true about ∠FAC & ∠EDB.
Clear selection
Q.18) For certain natural numbers n, there exist relatively prime integers x and y and an integer k > 1 satisfying the equation 3^n=x^k+y^k, then the solution to the equation is
Clear selection
Q.19) Consider a convex polygon of n sides, no two of whose sides are parallel. For each side, if we consider the angle the side subtends at the vertex farthest from the side, then the sum of these angles equals
Clear selection
Q.20) If G is the centroid of the triangle ABC and  AB+GC=AC+GB , then triangle ABC is
Clear selection
Q.21) Let a,b,c be real numbers. Consider the functions  f(x)=ax^2+bx+c   and  g(x)=cx^2+bx+a.  If |f(-1)|≤1,    |f(0)|≤1 and  |f(1)|≤1,  then
Clear selection
Q.22) In a parallelogram ABCD with ∠A<〖90〗^0, the circle with diameter AC meets the lines CB and CD again at E and F, respectively and the tangent to this circle at A meets BD at P. The points P,F and E form a …
Clear selection
Q.23) In a convex quadrilateral ABCD, triangles ABC and ADC have the same area. Let E be the intersection of AC and BD , and let the parallels through E to the lines AD,DC,CB,BA meet AB,BC,CD,DA at K,L,M,N, respectively. The ratio of the areas of the quadrilaterals KLMN and ABCD is
Clear selection
Q.24) A function f defined on the positive integers satisfies f(1)=1996 and f(1)+f(2)+⋯+f(n)=n^2 f(n) (n>1).  Calculate f(1996).
Clear selection
Q.25) ABC is an acute triangle and O its circumcenter. Let Sdenote the circle through A,B,O. The lines CB and CA meet S again at P and Q, respectively. Which of the following statements are true?
Clear selection
Q.26) The average of the numbers nsinn^0 (n=2,4,6,…,180) is
Clear selection
Q.27) The number of permutations of the set {1,2,…,n} such that no three of 1, 2, 3, 4, appear consecutively can be given as
Clear selection
 Q.28) Let P_1,P_2,P_3,P_4 be four points on a circle, and let I_1 be the incenter of the triangle P_2 P_3  P_4, I_2 be the incenter of the triangle P_1 P_3  P_4, I_3 be the incenter of the triangle P_1 P_2  P_4, I_4 be the incenter of the triangle P_1 P_2  P_3. Then I_1,I_2,I_3  and I_4 are the vertices of a ….
Clear selection
Q.29) Let a,b and c be the lengths of the sides of a triangle. If F and G are defined as F=√(a+b-c)+√(b+c-a)+√(c+a-b)    and    G=√a+√b+√c then which of the following inequalities is true?  
Clear selection
Q.30) A cube of side length n (nϵN)  can be divided into 1996 cubes whose side lengths are also natural numbers. The smallest possible value of n is …  
Clear selection
Submit
Clear form
Never submit passwords through Google Forms.
reCAPTCHA
PrivacyTerms
This form was created inside of Aditya Birla Education Trust. Report Abuse
Google Forms