All Questions ⁺⁰235714 Questions

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Algebra

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 ..

gnistory Sep 14, 2024

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+27

Probability Question

A fair coin is flipped until either three heads in a row or four tails in a row occur. What is the expected number of times the coin is flipped? Express your answer as a common fraction.

Halplz Sep 14, 2024

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+331

Algebra

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).

Stanry Sep 14, 2024

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+331

Algebra

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 ..

Stanry Sep 14, 2024

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+331

Algebra

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

Stanry Sep 14, 2024

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+36

Geometry help

The two circles below are externally tangent. A common external tangent intersects line PQ at R. Find QR
Circle radius-> 12 (p)and 8(q)

surfz Sep 14, 2024

Sep 13, 2024

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+544

Geometry

In triangle $ABC$, point $D$ is on side $\overline{AC}$ such that line segment $\overline{BD}$ bisects $\angle ABC$. If $\angle A = 45^\circ$, $\angle C = 45^\circ$, and $AC = 12$, then find the area of triangle $ABD$.

cooIcooIcooI17 Sep 13, 2024

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+829

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 ..

AnswerscorrectIy Sep 13, 2024

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+829

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 ..

AnswerscorrectIy Sep 13, 2024

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+829

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 $CF.$

AnswerscorrectIy Sep 13, 2024

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+253

Counting

Six children are each offered a single scoop of any of $3$ flavors of ice cream from the Combinatorial Creamery. In how many ways can each child choose a flavor for their scoop of ice cream so that some flavor of ice cream is selected by exactly five c read more ..

jaekg Sep 13, 2024

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+157

Confusing Probability Question...

Ryan has 3 red lava lamps and 3 blue lava lamps. He arranges them in a row on a shelf randomly, then turns 3 random lamps on. What is the probability that the leftmost lamp on the shelf is red, and the leftmost lamp which is turned on is also red?
Thanks read more ..

BlackjackEd Sep 13, 2024

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Please Help! - Probability Question

A fair coin is tossed repeatedly until either heads comes up three times in a row or tails comes up three times in a row. What is the probability that the coin will be tossed more than 10 times? Express your answer as a common fraction.
Any read more ..

NotLatePY Sep 13, 2024

Sep 12, 2024

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12

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+1222

Algebra

Solve the inequality (x - 2)(x + 6) \le (x - 2)(x + 5). Write your answer in interval notation.

learnmgcat Sep 12, 2024

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+1222

Algebra

Solve the inequality x(x + 6) > 16 - x + 14 + x^2. Write your answer in interval notation.

learnmgcat Sep 12, 2024

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+1222

Algebra

Solve the inequality 4t^2 \le -9t + 12 - 4t + 23 + (2t - 1)(2t + 1). Write your answer in interval notation.

learnmgcat Sep 12, 2024

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+1222

Algebra

Solve the inequality
\frac{3-z}{z+1} \ge 2(z + 4).
Write your answer in interval notation.

learnmgcat Sep 12, 2024

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+834

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 $AB$.

RedDragonl Sep 12, 2024

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+1443

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 $f(t)?$

blackpanther Sep 12, 2024

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9

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+1443

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 $f(x)$.

blackpanther Sep 12, 2024

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3

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+552

Counting

Catherine rolls a standard 6-sided die six times. If the product of her rolls is 200, then how many different sequences of rolls could there have been? (The order of the rolls matters.)

Pythagorearn Sep 12, 2024

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