tmua.me Extreme Set B Paper 1
20 questions · 75 min · Difficulty 80 · by tmua.me
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Questions in this paper
- Question 1The quadratic polynomials f and g both have coefficient 1 for x^2. For every real number x, f(g(x))-g(f(x))=4x^2-4x+3. What is the least…
- Question 2Two positive integers a and b, with a<b, satisfy a+b=120. The sum of their highest common factor and their lowest common multiple is 360…
- Question 3Four non-negative real numbers have mean 1. The mean of their squares is dfrac32. What is the greatest possible median of the four numbers?
- Question 4The circles C_1 and C_2 have equations C_1: x^2+y^2=4, C_2: (x-3)^2+y^2=7. They intersect at A and B, with A above B. The tangent to C_1 at…
- Question 5A drone's path relative to the ground follows the four equal sides of a square, travelling east, north, west and south in that order. The…
- Question 6Two distinct points P and Q lie on the curve y=x^3-2x. The tangents to the curve at P and Q are perpendicular, and their point of…
- Question 7The positive real numbers a,b,c satisfy a≤ b≤ c. Both a, b, c frac1a, frac1b, frac1c can be used as the side lengths of triangles of…
- Question 8A real constant c is to be chosen so that three distinct points common to the curves y=2^x y=(x-c)^2 have x-coordinates which, in…
- Question 9For a real constant k, the equation sin^3θ+cos^3θ=k has exactly three distinct solutions θ_1,θ_2,θ_3 with 0≤θ<2π. What is the value of…
- Question 10Initially a list has the single entry dfrac12. In each round, every entry x is replaced simultaneously by the two entries (x)/(3)…
- Question 11Ten cards, six red and four blue, are shuffled uniformly at random into a row. It is given that no two blue cards are next to one another…
- Question 12The lines L_1: y=mx+1, L_2: y=mx-2 each meet the curve y=x^2 at two distinct points. The points on L_1 are A and B, and those on L_2 are C…
- Question 13The curve y=x^4-3x^2+1 meets the x-axis at four distinct points. The gradients of the normals at these points are n_1,n_2,n_3,n_4. What is…
- Question 14For r=0,1,…,30, let a_r be the coefficient of x^r in the expansion of (1+x)^(30). What is the value of a_0+a_3+a_6+…+a_(30) ?
- Question 15The points A and B lie on the curve y=x^2, and the length of the straight line segment AB is 4. What is the least possible y-coordinate of…
- Question 16The terms of a geometric progression are positive. Its first three terms, when rounded to the nearest integer, are 2, 3 and 5, in that…
- Question 17A particle moves along a straight line. At time t seconds, its signed displacement in metres from its starting point is s(t)=t^3-6t^2+bt…
- Question 18Ten counters numbered 1,2,…,10 are divided at random into five pairs, with every division into pairs equally likely. It is known that the…
- Question 19The nine entries in the array ccc 1&a&b c&d&e f&g&225 are distinct positive integers. In each row, the entries from left to right form an…
- Question 20A square has its centre at the origin and all four of its vertices on the curve y=x^3-3x. What is the sum of all the distinct values that…