All Questions ⁺⁰235720 Questions

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Geometry

In triangle ABC, angle ACB = 90 degrees. Let H be the foot of the altitude from C to side AB.
If BC = 1 and AH = 2, find CH.

gnistory Sep 28, 2024

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8

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+834

Geometry

Let ABC be a triangle, and let its angle bisectors be AD, BE, and CF which intersect at I. If DI=3, BD=4 and BI=6 then compute the area of triangle BID.

gnistory Sep 28, 2024

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help

Let $ABC$ be a triangle. Let $X$ be a point on $\overline{AC}$ such that $\angle BXA = 90^\circ.$ If $\angle ABC = 90^\circ,$ $AX = 15,$ and $CX = 6,$ then what is $BX?$

Jasino Sep 28, 2024

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3

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+1439

Counting

Prove that given any set of $17$ integers, there exist nine of them whose sum is divisible by $2.$

kittykat Sep 28, 2024

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6

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+838

Algebra

Find the smallest positive real number such that
\lfloor x^2 \rfloor - x \lfloor x \rfloor = 1.

magenta Sep 28, 2024

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2

+279

Algebra

Find one ordered pair $(x,y)$ of real numbers such that $x + y = 10$ and $x^3 + y^3 = 162 + x^2 + y^2.$

bIueb3rry Sep 28, 2024

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+279

Algebra

Find the minimum value of 9^x - 2 \cdot 3^x + 1 over all real numbers x.

bIueb3rry Sep 28, 2024

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9

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+279

Algebra

I own a large truck, and my neighbor owns four small trucks that are all identical. My truck can carry a load of at least 600 pounds more than each of her trucks, but no more than 1/2 of the total load her four trucks combined can carry. Based on these read more ..

bIueb3rry Sep 28, 2024

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6

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+279

Algebra

The product $(3-\sqrt{5})(4+\sqrt{5})(1+6\sqrt{5})$ can be expressed in the form $a+b\sqrt{5}$, where $a$ and $b$ are integers. Find $a+b$.

bIueb3rry Sep 28, 2024

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5

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+176

Algebra

Let $x$ and $y$ be nonnegative real numbers. If $x + y = 25$, then find the minimum value of $6x + 3y.$

AUnVerifedTaxPayer Sep 28, 2024

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4

2

+176

Algebra

For certain values of k and m, the system
3a + 2b = 2
a + 2b = k - 8a - mb
has infinitely many solutions (a,b). What are k and m? read more ..

AUnVerifedTaxPayer Sep 28, 2024

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8

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+176

Algebra

The function $f(x)$ is defined only on domain $[-1,2]$, and is defined on this domain by the formula
$$f(x) = 2x^2-8x+1.$$
What is the range of $f(x)$? Express your answer as an interval or as a union of intervals.

AUnVerifedTaxPayer Sep 28, 2024

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4

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+176

Algebra

Let
$$f(x) = \frac{1}{1+\frac{2}{1+\frac 3x}}.$$
There are three real numbers $x$ that are not in the domain of $f(x)$. What is the sum of those three numbers?

AUnVerifedTaxPayer Sep 28, 2024

Sep 27, 2024

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9

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+544

Algebra

My street hockey team plays three games each week. My team lost all $9$ games in the first three weeks. Then, my team won one game and lost two games in the fourth week, bringing our record to $1$ win and $11$ losses. Each week after that, my team read more ..

cooIcooIcooI17 Sep 27, 2024

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5

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+544

Algebra

When the same constant is added to the numbers $a,$ $b,$ and $c,$ a three-term geometric sequence arises. If $a = 60,$ $b = 100,$ and $c = 140,$ what is the common ratio of the resulting sequence?

cooIcooIcooI17 Sep 27, 2024

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5

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+544

Algebra

Let a_1, a_2, a_3, \dots, a_8, a_9, a_{10} be an arithmetic sequence. If $a_1 + a_3 = 5$ and $a_2 + a_4 = 6$, then find $a_1$.

cooIcooIcooI17 Sep 27, 2024

Sep 26, 2024

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+552

Algebra

In a store window, there was a box of berries having a total weight of 200 kg. The berries were 98% water, by weight. After two days in the sun, the water content of the berries was only 95%, by weight. What was the total weight of the berries read more ..

Pythagorearn Sep 26, 2024

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+552

Algebra

All members of our painting team paint at the same rate. If $10$ members can paint a $1000$ square foot wall in $10$ minutes, then how long would it take for $5$ members to paint a $1000$ square foot wall, in minutes?

Pythagorearn Sep 26, 2024

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9

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+552

Algebra

The graph of the equation
4x^2 - 12x + 4y^2 + 16y + 17 = -4x + 2y + 5
is a circle. Find the radius of the circle.

Pythagorearn Sep 26, 2024

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5

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+544

Counting

A standard 8 times 8 chessboard has 64 unit squares.
Eight rooks are randomly placed on different squares of a chessboard. A rook is said to attack all of the squares in its row and its column.
Compute the probability read more ..

cooIcooIcooI17 Sep 26, 2024

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8

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+544

Counting

Twelve points are given in the plane, so that no three points lie on the same line.
Five distinct segments joining pairs of these points are chosen at random, with all \binom{12}{2} = 66 such segments equally likely to be chosen. What read more ..

cooIcooIcooI17 Sep 26, 2024

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