LeoIsTheBest

19 Questions
19 Answers

0

773

0

+142

Find all angles theta, 0<=theta<=2pi with the following property:

Find all angles theta, 0<=theta<=2pi with the following property: For all real x, 0<=x<=1: x^2 * cos theta - x(1 - x) + (1 - x)^2 * sin theta > 0

LeoIsTheBest Aug 18, 2019

0

1063

2

+142

In equilateral triangle ABC let points D and E trisect BC. Find sin DAE.

In equilateral triangle ABC let points D and E trisect BC. Find sin DAE.

LeoIsTheBest Aug 18, 2019

0

1031

7

+142

Compute

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LeoIsTheBest Aug 13, 2019

0

1091

6

+142

Stuart has drawn a pair of concentric circles, as shown. He draws chords AB, BC,... of the large circle, each tangent to the small one. If

Stuart has drawn a pair of concentric circles, as shown. He draws chords AB, BC,... of the large circle, each tangent to the small one. If

read more ..

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LeoIsTheBest Aug 12, 2019

0

892

1

+142

Geometry+Algebra

Let A=(10,-10) and O=(0,0). Determine the sum of all x and y-coordinates of all points Q on the line y=-x+6 such that angle OQA = 90.

LeoIsTheBest Aug 4, 2019

0

2025

2

+142

A shop sells 500 smartphones a week for $450 each...

A shop sells 500 smartphones a week for $450 each. A market survey shows that each decrease of $5 on the price will result in the sale of an additional 10 smartphones per week. What price of the smartphone would result in maximum revenue, in dollars? read more ..

LeoIsTheBest Jul 28, 2019

0

1528

1

+142

Jo adds up all the positive integers from 1 to 100...

Jo adds up all the positive integers from 1 to 100. Kate does a similar thing with the first 100 positive integers; however, she first rounds every integer to its nearest multiple of 10 (rounding 5s up) and then adds the 100 values. What is the positive read more ..

LeoIsTheBest Jul 28, 2019

+1

1828

3

+142

For a give-away at the mall, the color of the entry forms...

For a give-away at the mall, the color of the entry forms changes every hour on the hour, and once a color has been used, it is not used again that day. To limit the number of entries, a person may not fill out more than one form of any color. Using the read more ..

LeoIsTheBest Jul 4, 2019

+1

1542

2

+142

What is the median of the distinct positive values...

What is the median of the distinct positive values of all of the fractions less than or equal to 1 with positive integer denominators less than or equal to 5? Express your answer as a common fraction.

LeoIsTheBest Jul 4, 2019

+1

939

1

+142

Let and be nonzero complex numbers such that . Find the sum of all possible values of .

Let and be nonzero complex numbers such that . Find the sum of all possible values of .

LeoIsTheBest Jun 29, 2019

0

3089

1

+142

In the diagram, four circles of radius 1 with centres P, Q, R, and S are tangent to one another and to the sides of triangle ABC, as shown.

In the diagram, four circles of radius 1 with centres P, Q, R, and S are tangent to one another and to the sides of triangle ABC, as shown.

read more ..

LeoIsTheBest Jun 28, 2019

0

2846

3

+142

Help with Geo

A circle with center $O$ has radius $8$ units and circle $P$ has radius $2$ units. The circles are externally tangent to each other at point $Q$. Segment $TS$ is the common external tangent to circle $O$ and circle $P$ at points $T$ and $S$, respectively. read more ..

LeoIsTheBest Jun 22, 2019

+2

859

1

+142

Complex

Let $a$ and $b$ be nonzero complex numbers such that \[|a| = |b| = |a + b|.\]Find the sum of all possible values of $\frac{a}{b}.$

LeoIsTheBest Jun 13, 2019

+2

986

7

+142

Complex Numbers

Let \[\omega\] be a nonreal root of \[z^3 = 1.\] Let \[a_1, a_2, \dots, a_n\] be real numbers such that \[\frac{1}{a_1 + \omega} + \frac{1}{a_2 + \omega} + \dots + \frac{1}{a_n + \omega} = 2 + 5i.\]Compute \[\frac{2a_1 - 1}{a_1^2 - a_1 + 1} + \frac{2a_2 read more ..

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LeoIsTheBest Jun 12, 2019

+1

1609

5

+142

Help On Complex Numbers

The four consecutive digits a, b, c and d are used to form the four-digit numbers abcd and dcba. What is the greatest common divisor of all numbers of the form abcd+dcba?

hectictar ●●●●●

LeoIsTheBest Jun 12, 2019

0

942

5

+142

Complex Numbers

Find all values of the real number so that the four complex roots of form the vertices of a parallelogram in the complex plane. Enter all the values, separated by commas.

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LeoIsTheBest Jun 6, 2019

0

809

2

+142

A square $DEFG$ varies inside equilateral triangle $ABC$ so that $E$ always lies on side $\overline{AB},$ $F$ always lies on side

A square $DEFG$ varies inside equilateral triangle $ABC$ so that $E$ always lies on side $\overline{AB},$ $F$ always lies on side $\overline{BC},$ and $G$ always lies on side $\overline{AC}.$ The point $D$ starts on side $\overline{AB}$ read more ..

LeoIsTheBest Apr 18, 2019

0

1539

4

+142

Help please?

Right $\triangle ABC$ has hypotenuse $AB$. Square $BCDE$ has $BC$ as one of its sides. Suppose that the area of $BCDE$ is a prime number. If $AB$ and $AC$ are each integers less than $20$, how many possibilities are there for their lengths?

LeoIsTheBest Apr 17, 2019

0

866

1

+142

Help on question!!!

At a certain amusement park, there is a bulk discount for tickets. If you buy up to 60 tickets in one order, the price for each ticket is $\$70$. However if you buy more than 60 tickets in a single order, the price of every ticket is reduced by $\$1$ for read more ..

LeoIsTheBest Mar 30, 2019