tmua.me Tau Paper 2
20 questions · 75 min · Difficulty 94 · by tmua.me
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Questions in this paper
- Question 1For -1≤ x≤2 and real y, define H(x,y)=(x^2-2)^2+(x-y)^2. For a real number k, consider the statements P:& For every x∈[-1,2] there is a…
- Question 2For k≥0, let M(k) be the largest number of distinct real solutions that the equation x^4+a x^2+b x+1=0 can have as a,b range over all real…
- Question 3Fifteen lamps are arranged in a circle and labelled 0,1,…,14. All are initially off. Pressing switch i changes the states of lamps i, i+1…
- Question 4A line is called a *double bisector* of a triangle if it divides the triangle into two polygons which have both equal areas and equal…
- Question 5For real k, let N(k) be the number of ordered triples (x,y,z) of pairwise distinct positive real numbers, none equal to 1, for which xyz=1…
- Question 6For a real number a, define F_a(x)=(x^3+ax)/(x^2+1) (x>0). Call a admissible if both of the following hold: - F_a is not one-to-one on the…
- Question 7For each choice of integers r,s,t,u satisfying 0≤ r,s,t,u≤12, put N=2^r3^s5^t7^u. Let D(N) be the largest positive integer with the…
- Question 8A circle meets the parabola y=x^2 at four distinct points A,B,C,D, in increasing order of their x-coordinates a<b<c<d. Suppose…
- Question 9Let P(x)=x^3-3x. For t∈\1,2,3\ and a non-negative integer d, consider the following claim about every real polynomial G of degree at most…
- Question 10An integer n is chosen uniformly at random from 1≤ n<2^(18). It is given that exactly 64 coefficients in the expansion of (1+x)^n are odd…
- Question 11A real number c is called admissible if the following equivalence is true for every monic polynomial p of degree 4 whose derivative has…
- Question 12An equilateral triangle ABC has circumcentre O and circumradius 1. For r>0, let R_r be the set of points P on the circle OP=r for which the…
- Question 13Let T be the set of all possible values of f(2)-λ, where λ is a non-zero real number and f:mathbb R→mathbb R satisfies f…
- Question 14For each real number h, let L_h be the set of distinct straight lines which pass through (h,-1) and are tangent to y=x^4-2x^2. A line…
- Question 15How many monic polynomials p of degree 6 with integer coefficients satisfy all of the following conditions? - The coefficient of x^5 is 0…
- Question 16Let R(x)=(x^3-3x)/(3x^2-1). A real number a is called admissible if the sequence x_1=a, x_(n+1)=R(x_n) is defined and positive for every…
- Question 17Ten coins are tossed together. For every choice of three of the coins, each of the eight possible patterns of heads and tails for those…
- Question 18Let P be the set of real polynomials p of degree at most 6 such that p(x)≥0 (-1≤ x≤1), ∫_(-1)^1p(x) mathrm dx=1. Write a_6(p) for the…
- Question 19A set S consists of six distinct real numbers. For every r∈ S, suppose that Σ_s∈ S s≠ rfrac1r-s=r^3. What is the value of Σ_(r∈ S)r^2?
- Question 20For a real number x, lfloor xrfloor is the greatest integer not exceeding x. Let 0<α<1 and -1≤ t≤0. For each positive integer q, consider…