1  August

  1. Evaluate \(\lim_{n \to \infty} \sqrt[n]{n}\). [Ans: \(1\)]
  2. Solve: \(3^{x-2}(3^x -10)=-1\)
  3. Find the yellow area assuming O as the center of the circle.

Find the yellow area

  1. Solve \(x^2 + y^2\) when \[ \begin{aligned} x+y = 2 \\ x^3 + y^3 = 10 \end{aligned} \]
  2. Find the sum: \(1 - 2 + 3 - 4 + 5 - 6 + \ldots + 999\). [Ans: \(500\)]
  3. Find the area of blue shaded region:

Find the blue area

  1. Solve \[\frac{0.001 + 0.011}{0.0111} + \frac{0.03(44.4-23.1)}{0.3 + 0.033} \] [Ans: 3]
  2. Find sum: \(1-4+9-16+\ldots + 99^2 - 100^2\) [Oxford Math Exam 2020]
  3. A triangle has side lengths \(a, a\) and \(b\). It has perimeter \(P\) and area \(A\). Given that \(b\) and \(P\) are integers, and that \(P\) is numerically equal to \(A^2\) i.e. \(P=A^2\), find all possible pairs \((a,b)\). [BMO Nov 2020]
  4. Find \(a+b\) if \(p(x-2) - p(2-x) = ax^2 - bx + 12\) [Ans: 6]
  5. Prove that \(k\) must be a square if \(a^2 + b^2 = k(1+ab)\) for positive integers \(a,b\) and \(k\).
  6. Which one is greater? \((1+1/2)(1+1/4)(1+1/6) \ldots (1+1/2018)\) vs \(50\)? [Ans: 50]
  7. If \(xy=1\), what is the value of \(\frac{2^{(x+y)^2}}{2^{(x-y)^2}}\) ? [Ans: 16]
  8. Solve \[\sqrt{\frac{8^{10} + 4^{10}}{8^4 + 4^{11}}}\]
  9. If \(x=3^2\), then what is the value of \(x^x\)? Ans: \(3^{18}\)
Note

\[e^{\pi} \cong \pi^{e}\]

  1. Two equilateral triangles were drawn using two diagonals on a regular nonagon. What is the angle?

Find the angle

  1. Find the radius of the circle: \(x^2 + y^2 + 4x - 2y + 1=0\)
  2. Square, semicircle and quarter circle. What fraction is yellow?

Yellow fraction

  1. In a room filled with \(7\) people, \(4\) people have exactly \(1\) sibling in the room and \(3\) people have exactly \(2\) siblings in the room. If two individuals are selected from the room at random, what is the probability that those two individuals are NOT siblings? [Ans: \(16/21\)]
  2. Find yellow area

Yellow area

  1. Find the height

Height

  1. If \(N=2^{0.15}\) and \(N^b = 16\), find \(b\).
  2. If \(x=y/4, x^y = 5^{200}\), find \(x\).
  3. If \(3^a=8, 3^b=32\), find \(\frac{a+b}{a-b}\).
  4. If \(m+n=-2\) what is the value of \(m^3 + n^3 - 6mn\) ?
  5. Find the yellow area:

Yellow area

  1. What is the value of \(x\) if \(\frac{(x^2 - 1)!}{x^2-1}=23!\) ? [Ans: \(\pm5\)]
  2. Evaluate: \(\frac{1}{1-2^{-3}} + \frac{1}{1-2^3}\).
  3. Evaluate \(x\) if \(\sqrt{x + \sqrt{2x-1}} + \sqrt{x - \sqrt{2x-1}} = A\).