True or False?
The infinite sequence \[\big\lfloor 1\times \sqrt{ 2017 } \big\rfloor,\ \big\lfloor 2 \times \sqrt{ 2017 } \big\rfloor,\ \big\lfloor 3 \times \sqrt{ 2017 } \big\rfloor,\ \big\lfloor 4 \times \sqrt{ 2017 } \big\rfloor,\ \ldots\] contains infinitely many perfect squares.
\(\)
Notation: \( \lfloor \cdot \rfloor \) denotes the floor function.
Are you sure you want to view the solution?
Pi Han has written down a perfect square between 1 to 300 inclusive. Calvin wants to know the integer, and is allowed to ask "yes/no" questions to determine it. Pi Han would immediately answer the question, and then Calvin can use that information to ask his next question. As a point of pride, Calvin would not ask questions to which he already knows the answer. Calvin asks the following questions:
At this point, Calvin declares that he knows the answer. However, he turns out to be incorrect because Pi Han gave the wrong answer to each of the 4 questions!
What is the number in the envelope? If you think it cannot be uniquely determined, enter 0 as your answer.
\(\)
Clarification: Before the questions are asked, Calvin knows that Pi Han wrote down a perfect square between 1 and 300 inclusive.
Are you sure you want to view the solution?
Parallelogram \( ABCD \) with \(AB = 10, BC =16, \angle ABC = 60^{\circ} \) has an inscribed ellipse such that the tangent point \( E \) satisfies \( CE = \frac{3}{4} CD \).
Find the length of the semi-major axis of this ellipse.
Are you sure you want to view the solution?
Three points are chosen at random on the circumference of a circle with radius 1. Find the expected value, \(v\), of the area of the triangle formed by the three points.
If \(v = \dfrac{a}{b \pi}\), where \(a\) and \(b\) are coprime positive integers, submit \(a+b\).
Are you sure you want to view the solution?
\[\frac{r^2}{s}+200=\frac{s^2}{r}+200^2\]
In the equation above, \(r\) and \(s\) are both rational numbers and their difference is an integer.
What is the number of solutions to the equation?
Are you sure you want to view the solution?
Problem Loading...
Note Loading...
Set Loading...