tmua.me Tau Paper 1
20 questions · 75 min · Difficulty 91 · by tmua.me
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Questions in this paper
- Question 1Pairwise distinct positive real numbers x,y,z satisfy frac1x+frac1y+frac1z=1, xyz=32. Let E=(x-y)^2+(y-z)^2+(z-x)^2. Which one of the…
- Question 2A sequence of positive real numbers satisfies a_(n+2)=frac1+a_(n+1)a_n (n≥1). For which real values of t are there exactly four ordered…
- Question 3A rectangle has area 1. Four circles of positive radii lie inside it, one at each vertex, with each circle tangent to the two sides meeting…
- Question 4For how many real values of a does the equation |log_2(x^2-ax+frac52)| =|log_2 x| have exactly three distinct real solutions which, in…
- Question 5Four distinct integers are chosen uniformly at random from \1,2,…,12\. It is given that their sum is divisible by 3. Let P be the monic…
- Question 6For a real number k, the equation 64cos^4x-100cos^2x-9=64kcos x has exactly six distinct solutions x_1,…,x_6 in 0≤ x<2π. As k varies over…
- Question 7For real numbers a and b, let P(x)=x^3-3x+a, Q(x)=x^2-bx-1. What is the greatest positive number C with the following property? For every…
- Question 8A function f maps the integers to the integers and satisfies f(f(f(n)))=n+18, f(n+12)=f(n)+12 for every integer n. In addition…
- Question 9Call a line admissible if it intersects y=x^3-3x at three distinct points and the two finite regions between the line and the curve have…
- Question 10Let a be real and f(x)=x^2-ax. Define f_1(x)=f(x), f_(n+1)(x)=fbigl(f_n(x)bigr) (n≥1). Suppose that f_2(x)=0 has four distinct real roots…
- Question 11The numbers X_1,…,X_(12) consist of six 1s and six -1s, with every choice of the six positive positions equally likely. Set X_(13)=X_1…
- Question 12For a>1 and real x, define F_a(x)=(sin x)/(a+sin x)+(cos x)/(a+cos x), H_a(x)=F_a(x)F_a(x+π). For which values of a is the minimum of…
- Question 13For a positive real number c, let C be the curve x+y+xy=c(x^2+y^2). For every t with 0≤ t≤1, the line x=t meets C at a point with 0≤ y≤1…
- Question 14For real numbers a,b, let P(x)=(x^2+1)^3+ax(x^2+1)^2+bx^3. Suppose that P(1)=0 and P(x)≥0 for every real x. Also suppose that P(x)=P(-x)…
- Question 15Let S be the set of all points N with the following property: three distinct normals to y=x^2 pass through N, and their three corresponding…
- Question 16Eight cards numbered 1,2,…,8 are placed uniformly at random around a circle. Let T be the sum of the absolute differences between the…
- Question 17A polynomial p has degree 6 and leading coefficient 1. For every real t, ∫_(t-1)^(t+1)p(x) dx≥0, with equality if and only if t=-2 or t=2…
- Question 18For every choice of a<0<b, let P=(a,a^2) and Q=(b,b^2). The tangents to y=x^2 at P and Q meet at R, and C_(a,b) is the circle through…
- Question 19A monic polynomial of degree 12 has twelve distinct real roots, none equal to 0 or 1. Replacing every root r by 1-r leaves the set of roots…
- Question 20Seven coins are tossed independently. Their seven probabilities of heads are all distinct and lie strictly between 0 and 1. For each…