answer:1. **Understanding the given problem:** We are given an isosceles trapezoid ( ABCD ) with ( AB ) being one of the legs. The diagonals ( AC ) and ( BD ) intersect at point ( P ). We are required to prove that the circumcenter ( O ) of the trapezoid ( ABCD ) lies on the circumcircle of triangle ( APB ). 2. **Analyzing the angle ( angle APB ):** In an isosceles trapezoid, the diagonals intersect in such a way that: [ angle APB = frac{overarc{AB} + overarc{CD}}{2} ] This result comes from the properties of cyclic quadrilaterals and the fact that the angle subtended by a chord at the circumference is half the sum of the angles subtended by the opposite arc. 3. **Analyzing the angle ( angle AOB ):** Since ( O ) is the circumcenter of trapezoid ( ABCD ): [ angle AOB = 2 cdot angle APB ] But from geometric construction and properties of circles, we know that if ( P ) is the intersection of diagonals: [ angle APB = frac{overarc{AB} + overarc{CD}}{2} ] 4. **Combining these observations:** From the property of the intersection point ( P ) of the diagonals in a trapezoid ( angle APB ) being equal to: [ angle APB ] And the circumcenter ( O ) giving: [ angle AOB = 2 times angle APB = overarc{AB} + overarc{CD} ] 5. **Conclusion:** Since ( angle APB ) and ( angle AOB ) align such that ( O ) would be on the circumcircle of ( triangle APB ), it follows directly by this property: [ O text{ lies on the circumcircle of } triangle APB ] This completes the proof, and we can conclude that: [ boxed{O text{ lies on the circumcircle of triangle } APB} ]

answer:Let's denote the total amount Rahim spent on the books from the first shop as X. He bought 65 books from the first shop and 35 books from the second shop, making a total of 65 + 35 = 100 books. From the second shop, he spent Rs. 2000 for 35 books. The average price he paid per book is Rs. 85, so for 100 books, he would have spent a total of 100 * 85 = Rs. 8500. Now, we know that the total amount spent on both shops is Rs. 8500, and he spent Rs. 2000 on the second shop. So, the amount spent on the first shop (X) can be calculated as follows: X + 2000 = 8500 Subtracting 2000 from both sides, we get: X = 8500 - 2000 X = 6500 Therefore, Rahim spent Rs. boxed{6500} on the books from the first shop.

answer:John earned 18 on Saturday and half of that on Sunday, which is 18 / 2 = 9. So, the total amount he earned over the weekend is 18 + 9 = 27. He also earned 20 the previous weekend, so in total, he has earned 27 + 20 = 47. John needs 60 for the pogo stick, so he still needs 60 - 47 = boxed{13} more to buy the pogo stick.

answer:Solution: (Ⅰ) Since f(x) is a monotonic function on (-infty,+infty), and it is increasing on (-infty,0), Therefore, f(x) is an increasing function on [0,+infty), Thus, a > 1, and f(0)=1+bgeqslant -1, we get bgeqslant -2, Therefore, a > 1, bgeqslant -2. (Ⅱ) Since when x < 0, f(x) < -1, therefore, f(x) has no zero points on (-infty,0), Therefore, when xgeqslant 0, f(x)=2^{x}+b has only one zero point, Since f(x) is increasing on [0,+infty), Therefore, f(0)=1+bleqslant 0, that is, bleqslant -1. Therefore, the range of values for the real number b is bin(-infty,-1]. Thus, for (Ⅰ), the range of values for a and b are boxed{a > 1, bgeqslant -2}, and for (Ⅱ), the range of values for b is boxed{bin(-infty,-1]}.