2  September

1. What is the half of \(4^8 + 4^8 + 4^8 + 4^8\)?
2. Find the value of \(x\) knowing that \(\alpha = \beta\):

Find \(x\)

3. What is the yellow area:

4. A square with an inscribed right triangle. What fraction is yellow?

5. Find area of regular hexagon:

6. Find sum of digits:

7. Find the area of yellow region:

8. Integrate \[ \int \frac{1}{x(a+bx^n)}dx \]

9. Find \(x\) if \(\sqrt[3]{14 + \sqrt{x}} + \sqrt[3]{14 - \sqrt{x}}=4\).

10. Integrate \[ \int_0^\infty \frac{x}{1 + e^{x \ln 2}}dx \]

11. Find the sum of \[\left(1+\frac{1}{2}\right)\left(1+\frac{1}{3}\right)\left(1+\frac{1}{4}\right)\ldots \left(1+\frac{1}{2006}\right)\left(1+\frac{1}{2007}\right)\]
    [Math Olympiad 2007]

12. Evaluate \[ \sqrt[3]{3}\times \sqrt[3]{72} + \sqrt{3}(\sqrt{96}-\sqrt{12}) - \sqrt{162} \] [Ans: \(3\sqrt{2}\)]

13. Find \(x+y\) if \(2x+y=10\) and \(x^2 + y^2 = 25\).

14. Find \(x\) and \(y\) if \(xy=80\) and \(\log x - 2 \log y = 1\).

15. Find \(\sin x + \sin y\) if \(3 \sin x + 4 \cos y - 3 \sin y = 8\).

16. Find \(x\) and \(y\) if

    \[ \begin{aligned} 2^{x-y} - x - y = 0 \\ 2 - (x+y)^{x-y} = 0 \end{aligned} \]

17. If \(x^3 + \frac{1}{x^3} = 1\), evaluate

    \[ \frac{(x^5 + \frac{1}{x^5})^3 - 1}{x^5 + \frac{1}{x^5}} \]

18. Evaluate \(x^3 + \frac{1}{x^3}\) if \(x=\sqrt{3+\sqrt{8}}\). [Ans: \(10 \sqrt{2}\)]

19. What is the area of shaded region?

20. Four circles each of radius \(x\) and a square are arranged within a circle of radius \(8\) as shown in the following fire. What is the range of \(x\)? [Ans: \(4-2\sqrt{2} \lt x \lt 8(\sqrt{2}-1)\)]

21. Find \(x\) and \(y\) if \(\sqrt{x} + \sqrt{y} = \sqrt{2 + \sqrt{3}}\).
22. If \(x^x = 2\), find \(x^{x^{x+1}} + x^{x^{1+x^{1+x}}}\). [Ans: \(20\)]
23. If \(\log (\frac{x+y}{3}) = \frac{\log x + \log y}{2}\), find \(\frac{x}{y} + \frac{y}{x}\). [Ans: 7]
24. Find \(\sqrt{x}\)

25. Find \(x\) if \([5^{5^x}]^{5^5}=5^5\). [Ans: \(-4\)]
26. Find \(A\) and \(B\): \(\frac{3x+5}{x^2-x-6} = \frac{A}{x+2}+\frac{B}{x-3}\)
27. Find the green area:

28. Find \(a\) and \(b\) if \(a-b=11\) and \(\sqrt{a} + \sqrt{b} = 11\).
29. What is the value of \(x\) in equation \(5^{x+3}- 5^{x+1} + 5^{x-1} = 601\)? [Ans: \(1\)]
30. If \(3^x + 3^y = 10\) and \(3^{x+y} = 5\), evaluate \(3^{x-y} + 3^{y-x}\). [Ans: \(18\)]
31. An equilateral triangle with side \(3\) and a circle with radius \(1\) have the same center. What is the perimeter of the following figure? [Ans: \(6 + \pi\)]

32. Here is a goat attached to one square. Rope length is \(6\). Square has side \(4\). Rope is attached to the center of a side. What is the area of the region the goat has access to? [Ans: \(26 \pi\)]

33. Find \(x\) in \(x! = x^3 - x\).

34. If \(x^2 - 13x + 1 =0\), what is the last digit of \(x^4 + x^{-4}\)? [Ans: \(7\)]

35. Find \(f(0)\) if \(f(x^2 + x+ 1)= x^5 + x + 1\). [Ans: \(0\)]

36. In the range \(-{90}^0 \lt x \lt {90}^0\), how many values of \(x\) are there for which the expression below sums to innity:

    \[ \frac{1}{\tan x} + \frac{1}{\tan^2 x} + \frac{1}{\tan^3 x} + \cdots \]

37. Find yellow area: [Ans: \(11\)]

38. Find \(x\) if \(\log_x (\frac{4x+5}{6-5x}) = -1\).
39. A and B are two centers. \(AB=3.5\) and yellow area \(=21 \pi\). Find the radius of the smaller circle. [Ans: \(1.25\)]

40. If \(\frac{1}{\sin \theta} - \frac{1}{\cos \theta} = \frac{\sqrt{13}}{6}\), find \(\frac{1}{\tan \theta} - \tan \theta\).
41. If \(\sum_{i=1}^n a_i = 2^n - 1\) where \(a_1, a_2, \ldots a_n \in R^+\) and \(n \ge 2\) , find minimum of the following expression \[\frac{a_1}{1} + \frac{a_2}{1 + a_1} + \cdots + \frac{a_n}{1 + a_1 + a_2 + \cdots + a_{n-1}}\]
42. What fraction is yellow? [Ans: \(\frac{1}{4}\)]

43. Find \(AC\)

44. Find \(7^{x+1}\) if \(7^{x-1}=5\).
45. Solve for \(x\): \(81^{\sin^2 x} + 81^{\cos^2 x} = 30\)
46. If \((x-a)(x-b)=1\) and \(a-b+5=0\), then evaluate \((x-a)^3 - \frac{1}{(x-a)^3}\).
47. If \(f(x.y) = f(x).f(y), f(6)=35, f(4)=25\), find \(f(18)\).
48. How many triangles are there in the figure?

49. What is the area shaded in blue?

50. Find \(f(0)\) if \(f(f(x))=x^2 -x + 1\).
51. Solve for \(x\): \((x^2 -5x+5)^{x^2-11x + 30} = 1\)
52. If \(f(x+y)=f(x).f(y), f(3)=64\), find \(f(2) - f(1)\).
53. Suppose that \(m\) and \(n\) are positive integers such that \(75m=n^3\). What is the minimum possible value of \(m+n\)?
54. Assume length of square is \(1\) and we only have tangent touches. Find the radius of small circle. [Ans: \(3 - 2\sqrt{2}\)]

55. What is the smallest positive integer that can be expressed as the sum of nine consecutive integers, the sum of ten consecutive integers and the sum of eleven consecutive integers? [Ans: \(495\)]
56. Find \(\measuredangle BAC\) if \(\measuredangle BAD = \measuredangle ACD\):

57. If \(x^{x^4}=4\), find \(x^{x^8} + x^{x^2}\). [Ans: 258]
58. How many pentagons?

59. Find \(x\) in \(2^{2x}.9^x - 2.6^{3x-1} + 4^{2x-1}.3^{4x-2}=0\). [Ans: 1]
60. Find \(x\) if \(x \sqrt{4!} =12\). [Ans: \(\sqrt{6}\)]
61. Find \(x+y\) if \((\sqrt{x^2 + 1} + x)(\sqrt{y^2 + 1} + y)=1\).
62. Algebra question from MIT-1869/70:

63. Find \(x\): \[4^x + 6^x = 9^x\] [Ans: \(\frac{\log [(-1 \pm \sqrt{5})/2]}{\log (2/3)}\)]
64. Find the ratio

65. Find angle \(x\): [Ans: \(120^0\)]

66. If \(12^m = 18\), find \(2^{\frac{2m-1}{m-2}}\). [Ans: \(\frac{1}{3}\)]
67. Which one is greater? \[222^{333} \text{ or } 333^{222}\] [Ans: \(222^{333}\)]
68. Assume \(x\) and \(y\) are real numbers and \(5^x=10 - 5^{-x}=4^y + 5\). Find \((24)^{1/y}\). [Ans: 16]
69. Evaluate \(\sqrt[4]{(4 + \sqrt[2]{7})^{-1}}\sqrt{1 + \sqrt{7}}\). [Ans: \(\sqrt[4]{2}\)]
70. Find \(xy\) if \(\frac{x+22}{y} + \frac{290}{xy}=\frac{26-y}{x}\). [Ans: -143]
71. Find the value of angle \(F\). [Ans: \(75^0\)]

72. Find the ratio \(AC/BC\): [Ans: \(\frac{5}{2}\)]

73. Evaluate \(\int_0^{\frac{\pi}{2}}f(\theta)d \theta\) if \[ f(\theta) = \sin \theta + \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (\sin \theta + t \cos \theta) f(t) d\theta \] [Ans: \(\pi t f(t) + 1\)]
74. Find the maximum of the expression \[\sin^2 x \cos^4 x\]. [Ans: \(\frac{4}{27}\)]
75. Find \(n\) in \(\sqrt{2 + \sqrt{2-\sqrt{2 + n}}}=n\). [Ans: \(2 \cos \frac{2\pi}{9}\)]
76. Find angle \(x\)

[Ans: \(30^0\)]

77. Find area of shaded region: [Ans: \(2\sqrt{2}\)]

78. Find area of triangle \(ACD\) [Ans: \(1206.22\)]

79. Find \(4^x + 4^{-x}\) if \(2^x + 2^{-x}=5\). [Ans: \(23\)]
80. Find number of possible integers \((a,b)\):

[Ans: \(597\)]

81. Evaluate \[\sqrt{\frac{11^4 + 100^4 + 111^4}{2}}\] [Ans: \(11221\)]
82. What is the smallest value \(\frac{4s^2 + t^2}{5st}\) that can take in the figure below?

[Ans: \(\frac{4}{5}\)]

83. Find the perimeter: [Ans: \(30\)]

84. A number written in base \(a\) is \(123\). The same number written in base \(b\) is \(146\). What is the minimum value of \(a+b\)? [Ans: \(15\)]
85. Solve: [Ans: \(30 + \log_3 \pi\)]

86. Evaluate \[44:4 \times 2^3:8\] [Ans: 11]
87. Find \(x\): [Ans: \(2\)] \[ 2^{\frac{2x-1}{x-1}} + 2^{\frac{3x-2}{x-1}}=24 \]
88. Find the value of \(x\) without calculator: [Ans: \(0\)] \[ 10^2 + 11^2 + 12^2 = 13^2 + 14^2 + x^2 \]
89. Solve: [Ans: \(7\)]

90. Find \(x\) \[ \log \sqrt[3]{x} = \sqrt{\log x} \] [Ans: \(10^9\)]

91. If \(a^2 + 1 = a\), find \(a^3\). [Ans: \(-1\)]

92. If \(x=\frac{1}{2}(1+\sqrt{2021})\), find \((x^3 - 506x -502)^7\). [Ans: \(3^7\)]

93. Find \(x^{x-3}\) if \[ 1- \frac{1}{1-\frac{1}{1-\frac{1}{x}}} = 5 \] [Ans: \(25\)]

94. Find the value of \(x\) and \(y\): \[ \begin{aligned} x-y = 31 \\ \sqrt{x} + \sqrt{y} =31 \end{aligned} \] [Ans: \(x=256, y=225\)]

95. Find \(\frac{3x}{x^2 - 1}\) if \(x + \frac{1}{x}=\sqrt{13}\). [Ans: \(1\)]

96. If \(a:b=2:3\) and \(b:c=4:5\), find \(a^2:b^2:bc\). [Ans: \(16:36:45\)]

97: Find shaded region:

[Ans: \(18\)]

98: Find area of blue triangle:

[Ans: \(12\)]

99. Solve \(10^{x-x^2}=x^x\) [Ans: \(1\)]

100. If \(0 \lt p, 0 \lt q\) and \(p+q \lt 1\), which one is larger: \[ (px+qy)^2 \text{ vs } px^2 + qy^2 \] [Ans: RHS is larger]

101. The large circle has a radius of \(\frac{30}{\sqrt{\pi}}\). Two circles with a diameter of \(\frac{30}{\sqrt{\pi}}\) lie inside the large circle. Two more circles lie inside the large circle so that the five circles touch each other as shown. Find the yellow area.

[Ans: \(250\)]

102. Find \(A, B\):

[Ans: \(A=6, B=4\)]

103. Find \(a^3 + 12a\) if \(a=\sqrt[3]{16} - \sqrt[3]{4}\). [Ans: \(12\)]

104. Find \(x^4 + y^4\) if \(x+y=4\) and \(x^3 + y^3=16\). [Ans: \(32\)]

105. Which one is larger? \[ 1999^{1999} \text{vs} 2000^{1998} \] [Ans: \(1999^{1999}\)]

106. Solve:

[Ans: See here]

107. Find the value of triangle:

[Ans: \(1\)]

108. Find \(A-B+C\) if \[ \begin{aligned} A + B = 5\\ A:C =7 \\ A - C =18 \end{aligned} \] [Ans: \(40\)]

109. Evaluate the expression: \[ \frac{2^{20}+4^{10}+4^{11}+8^8}{2^{21}+8^7+4^{12}} \] [Ans: \(1.1\)]

110. Find the area of green sector:

[Ans: \((\pi + 3 - 3\sqrt{3})/12\) = \(0.07878\)]

111. Find the side of the square:

[Ans: \(\frac{2}{5}\)]

112. Find the side of the square:

[Ans: \(\frac{20}{23}\)]

113. The curve below is \(y=\sin x\). Find the area:

[Ans: \(\int_0^\pi \sin x\,dx+\frac{7\pi}{6}\times\frac{1}{2}-\int_0^\frac{\pi}{6} \sin x\,dx\) or \(\frac{7\pi}{6}\times\frac{3}{2}-\int_0^\frac{7\pi}{6} 1-\sin x\,dx\)]

114. Solve \((x^2 -7x + 11)^{(x^2 - 11x + 30)} = 1\). [Ans: \(2, 5, 6\)]
115. If the length of rectangle is \(1\) unit. What is the perimeter of hexagon?

116. Find the values of triangle and square.

[Ans: \(x=1, y=0\)]

117. Solve:

[Ans: \(\pm 1\)]

118. Solve for \(x\):

[Ans: \(\frac{\pi}{4} + n \pi\)]

119. Find \(?\):

[Ans: \(6\)]

120. Find the area of shaded region

[Ans: \(0.3958\)]

121. If \(AC \times BC = KKK\) find \(A + B + C + K\). [Ans: \(21\)]

122. Find \(A\), \(B\) and \(C\) if \[ \begin{aligned} A + B = 13 \\ A / C = 6 \\ A - C = 15 \end{aligned} \] [Ans: \(A=18, B=-5, C=3\)]

123. If \(\frac{1}{x} + \frac{1}{3} = \frac{1}{x+3}\), find \(x^6 - 8\). [Ans: \(721\)]

124. Find \(x\) and \(y\)

[Ans: \(x=4, y=0\)]

125. Find the side of regular decagon:

126. Find \(x+y\) if \[ \begin{aligned} x^2 + y^2 = 7 \\ x^3 + y^3 = 10 \end{aligned} \] [Ans: \(x+y \in {-5, 1, 4}\)]

127. Find the remainder of the expression \(\frac{7^{700}}{100}\). [Ans: \(1\)]

128. Find \(\frac{a}{b}\)

[Ans: \(\frac{3}{2}\)]

129. Solve for \(x\) \[ 3^{x-2} + 5^{x-2}=34 \] [Ans: \(4\)]

130. Solve

[Ans: \(\frac{a^2}{1-2a}\)]

131. Find the area

[Ans: \(27\)]

132. ABCDE has an area of \(26\) cm². ABD and CBE are straight lines. Find the area of the shaded triangle BDE.

[Ans: \(4\)]

133. If \(x^4 + x^3 + x^2 + x+ 1=0\), find \(x^{10}+ x^{20}\). [Ans: \(2\)]
134. Solve \(\sqrt[3]{x+28} - \sqrt[3]{x-28} = 2\). [Ans: \(\pm 36\)]
135. How many?

[Ans: \(249\)]

136. Find the probability

[Ans: \(\frac{2}{9}\)]

137. Find \(x\)

[Ans: \(\sqrt[6]{3}\)]

138. Find angle \(x\) if lines d1 and d2 are parallel.

[Ans: \(90-y\)]

139. Find minimum of \(x+y\):

[Ans: \(30\)]

140. Find the ratio of sides of the polygons: