All Questions ⁺⁰236007 Questions

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20

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+868

Geometry

In triangle $PQR,$ let $X$ be the intersection of the angle bisector of $\angle P$ with side $\overline{QR}$, and let $Y$ be the foot of the perpendicular from $X$ to side $\overline{PR}$. If $PQ = 8,$ $QR = 5,$ and $PR = 1,$ then compute the length read more ..

eramsby1O1O May 15, 2024

0

25

0

+868

System

The system of equations
{xy}/{x + y} = 1,{xz}/{x + z} = 1, {yz}/{y + z} = 4
has exactly one solution. What is z in this solution?

eramsby1O1O May 15, 2024

-1

32

1

+868

Algebra

Compute
\frac{1}{1 \times 4} + \frac{1}{4 \times 7} + \frac{1}{7 \times 10}

eramsby1O1O May 15, 2024

0

21

0

+868

Algebra

Compute
1 + \frac{3}{6} + \frac{5}{9} + \frac{7}{12} + \dotsb.

eramsby1O1O May 15, 2024

0

16

0

+868

Algebra

Let S be the set of all real numbers of the form
a1/3 + a2/3^2 + a3/3^3 + ....
where a_i \in {0, 1, 2} for all i.
(a) Is the number 1/7 in the set S?
(b) Is the number 1/4 in the set S? read more ..

eramsby1O1O May 15, 2024

0

28

0

+868

Algebra

Let $f(x) = x^2 + bx + 12 - 3x + 10$ for all real numbers $x$. Find the greatest integer value of $b$ such that $-4$ is not in the range of $f(x)$.

eramsby1O1O May 15, 2024

0

19

0

+868

Algebra

Let $f(x)$ be a polynomial with integer coefficients. There exist distinct integers $p,$ $q,$ $r,$ $s,$ $t$ such that
f(p) = f(q) = f(r) = f(s) = 1
and $f(t) > 1.$ What is the smallest possible value of $f(t)?$

eramsby1O1O May 15, 2024

0

35

0

+868

Geometry

The circles $x^2 + y^2 = 4$ and $(x - 5)^2 + (y - 8)^2 = 60$ intersect in two points $A$ and $B.$ Find the distance $AB$.

eramsby1O1O May 15, 2024

0

17

0

+868

Algebra

Let x and y be complex numbers. If $x + y =2$ and $x^3 + y^3 = 5$, then what is $x^2 + y^2$?

eramsby1O1O May 15, 2024

0

23

0

+868

Help with math

Compute the sum
\frac{1}{\sqrt{36} + \sqrt{39}} + \frac{1}{\sqrt{36} + \sqrt{45}} + \frac{1}{\sqrt{39} +\sqrt{96}}

eramsby1O1O May 15, 2024

0

31

0

+1557

Geometry

Triangle $ABC$ has altitudes $\overline{AD},$ $\overline{BE},$ and $\overline{CF}.$ If $AD = 12,$ $BE = 20,$ and $CF$ is a positive integer, then find the largest possible value of $CF.$

parmen May 15, 2024

0

20

0

+1557

Geometry

There are four cities: Capital City, Little Village, Mytown, and Yourtown. The distance from Capital City to Little Village is $5$ miles. The distance from Capital City to Mytown is $2$ miles. The distance from Mytown to Yourtown is $3$ read more ..

parmen May 15, 2024

0

26

0

+1557

geometry

All the sides of a triangle are integers. If the perimeter of the triangle is $3,$ then how many different possible triangles are there? (Assume that the triangle is non-degenerate. Two triangles are considered the same if they are co read more ..

parmen May 15, 2024

0

15

1

+4

easy algebra practice

If x + y = -5 and xy = 3, then what is x2+y2?

easymathpractice May 15, 2024

0

33

0

+772

trig question

Let a be a real number such that 0 < a < \frac{\pi}{2}. Show that \left(\sin(a)\right)^7 + \left(\cos ( a)\right)^7< 2 sin(a)^4 cos(a)^4.

booboo44 May 15, 2024

0

30

0

+1455

Conics

The ellipse x^2 + 4y^2 = 4 and the hyperbola x^2 - my^2 = 1 are tangent. Compute m.

kittykat May 15, 2024

May 14, 2024

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18

0

+1455

Algebra

Let a and b be positive real numbers such that a + b = 1. Find the set of all possible values of a/b.

kittykat May 14, 2024

0

35

0

+1455

Math problem

A right triangle with integer leg lengths is called cool if the number of square units in its area is equal to five times the number of units in the sum of the lengths of its legs. What is the sum of all the different possible areas of cool right read more ..

kittykat May 14, 2024

0

25

0

+1455

Graphs

The graphs of x = y^4 and x + y^2 = 1 + x^2 - y^3 intersect at two points. The distance between these points is of the form sqrt(u + v*sqrt(5)), where u and v are integers. Find the ordered pair (u,v).

kittykat May 14, 2024

0

16

0

+1455

Math problem

A mathematician works for t hours per day and solves p problems per hour, where t and p are positive integers. One day, the mathematician drinks some coffee and discovers that he can now solve 3p + 12 problems per hour. In fact, read more ..

kittykat May 14, 2024

0

25

0

+667

Geometry

In trapezoid $PQRS,$ $\overline{PQ} \parallel \overline{RS}$. Let $X$ be the intersection of diagonals $\overline{PR}$ and $\overline{QS}$. The area of triangle $PQX$ is $4,$ and the area of triangle $PSR$ is $8.$ Find the area of trapezoid read more ..

siIviajendeukie May 14, 2024

-1

14

1

+667

Geometry

Let $WXYZ$ be a trapezoid with bases $\overline{XY}$ and $\overline{WZ}$. In this trapezoid, $\angle ZXW = 120^\circ$, $\angle XWZ = 60^\circ$, and $\angle XYZ = 120^\circ$. Find $\angle YXZ$, in degrees.

siIviajendeukie May 14, 2024

0

28

0

+667

Geometry

The area of a trapezoid is $80$. The length of one base is $16$ units greater than the other base, and one base of the trapezoid is $12$. Find the length of the median of the trapezoid.

siIviajendeukie May 14, 2024

0

14

0

+667

Geometry

Let P_1 P_2 P_3 \dotsb P_{10} be a regular polygon inscribed in a circle with radius $1.$ Compute
P_1 P_2 + P_2 P_3 + P_3 P_4 + \dots + P_9 P_{10} + P_{10} P_1

siIviajendeukie May 14, 2024

-1

21

0

+1839

Algebra

Let x, y, and z be positive real numbers. Find the largest possible value of
sqrt((3x + 4y)/(6x + 5y + 4z)) + sqrt((y + 4z)/(2x + 3y + 8z)) + sqrt((5z + 7x)/(4x + 3y + 5z))

ABJeIIy May 14, 2024

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