剑桥模拟试题剑桥国际高中入学

剑桥模拟试题剑桥国际高

中入学

The latest revision on November 22, 2020

Model Test for Cambridge International School

Solve the following questions comprehensively.

1.Three unit circles are arranged so that each touches the other two.

Find the radii of the two circles which touch all three.

2.Find all real numbers x such that x + 1 = |x + 3| - |x - 1|.

3.(1) Given x = (1 + 1/n)n, y = (1 + 1/n)n+1, show that x y = y x.

(2) Show that 12 - 22 + 32 - 42 + ... + (-1)n+1n2 = (-1)n+1(1 + 2 + ...

+ n).

4.All coefficients of the polynomial p(x) are non-negative and none

exceed p(0). If p(x) has degree n, show that the coefficient of x n+1 in p(x)2 is at most p(1)2/2.

5.What is the maximum possible value for the sum of the absolute

values of the differences between each pair of n non-negative real numbers which do not exceed 1

6.AB is a diameter of a circle. X is a point on the circle other

than the midpoint of the arc AB. BX meets the tangent at A at P, and AX meets the tangent at B at Q. Show that the line PQ, the tangent at X and the line AB are concurrent.

7.Four points on a circle divide it into four arcs. The four

midpoints form a quadrilateral. Show that its diagonals are

perpendicular.

8.Find the smallest positive integer b for which 7 + 7b + 7b2 is a

fourth power.

9.Show that there are no positive integers m, n such that 4m(m+1) =

n(n+1).

10.ABCD is a convex quadrilateral with area 1. The lines AD, BC

meet at X. The midpoints of the diagonals AC and BD are Y and Z.

Find the area of the triangle XYZ.

11. A square has tens digit 7. What is the units digit

12.Find all ordered triples (x, y, z) of real numbers which

satisfy the following system of equations:xy = z - x - yxz = y - x - zyz = x - y - z