tmua.me Extreme Set A Paper 2
20 questions · 75 min · Difficulty 81 · by tmua.me
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Questions in this paper
- Question 1Let k≥0. Consider the following statements. S: For every real number x with 0≤ x≤1, there exists a real number y such that |y-x|≤ k(1+x)…
- Question 2A triangle ABC has AB=3, AC=4 and BC=c. Consider the condition c≤3√2 and the statement AP^2≥ BP· PC every point P on the line segment BC…
- Question 3Each of a_0,a_1,…,a_6 is chosen independently, with 1 and -1 equally likely. Define P(x)=x^7+a_6x^6+a_5x^5+…+a_1x+a_0. A student tries to…
- Question 4In the coordinate plane, let O=(0,0), A=(1,0), B=(-frac12,(√3)/(2)), C=(-frac12,-(√3)/(2)). A point P lies on the circle x^2+y^2-4x+2y+1=0…
- Question 5A student wishes to prove that there is no real-valued function f, defined for every real number, such that f(x+f(y))=f(x)+y^2 all real…
- Question 6The integers 1,2,…,27 are divided into nine groups of three, and each group is replaced by its median. The resulting nine numbers are…
- Question 7Eight distinct real numbers have the following property: From every choice of three of the numbers, it is possible to select two whose sum…
- Question 8Each of two bags contains red and blue counters, with at least one counter of each colour in each bag. The first bag contains more red…
- Question 9Five boxes are arranged in a circle. In clockwise order, their initial numbers of counters are 0, 1, 0, 2, 4. In one move, two adjacent…
- Question 10A polynomial P(x) with real coefficients has degree 11, and P(1)≠0. Every real root is simple: whenever P(r)=0, the expression (x-r)^2 is…
- Question 11A right-angled triangle has positive integer side lengths a<b<c and area 60. Let h be the perpendicular distance from the right-angled…
- Question 12The polynomial f(x)=x^2+ax+b satisfies ∫_0^1 f(x) dx=0, ∫_0^1 x f(x) dx=0. What is the total area between the curve y=f(x) and the x-axis…
- Question 13Let a and b be real constants. Consider the condition a^2≤4b and the following statement: For every real angle θ, if sin…
- Question 14A geometric progression has terms a_n=a r^(n-1) (n=1,2,3,…), where a>0 and r>0. It is known that a_2, a_5 and a_9 are integers. Consider…
- Question 15The vertices of a convex quadrilateral are A,B,C,D, in that order. A point P in the plane is called balanced if PA^2+PC^2=PB^2+PD^2…
- Question 16Consider the inequality a^n<1+n(a-1). Which of the following statements are true? 1. There exists a positive integer n for which the…
- Question 17How many composite positive integers N have the following property? For every pair of integers a,b with a>1 and b>1, if ab=N, then a+b is a…
- Question 18For real numbers x and y, define s_n=x^n+y^n for each positive integer n. Consider the condition that s_1, s_2 and s_4 are integers, and…
- Question 19A sequence of real numbers is defined by 0 1, and the statement that x_1 is rational. Which of the following correctly describes the…
- Question 20A function f maps every positive integer to a positive integer and satisfies f(f(n))=n+4 for every positive integer n. How many different…