All Questions ⁺⁰237293 Questions

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help with geo

Let ABCD be a trapezoid with bases AB and CD. Let P be a point on side CD, and let X, Y be the feet of the altitudes from P to AD, BC respectively. Prove that if AD = 5, BC = 7, AB = 6, CD = 12, and CP/PD = 1, then PX = 12*sqrt(6)/5 and PY = read more ..

Pythagorearn Apr 27, 2024

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51

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+2729

Geometry

The tangent to the circumcircle of triangle $WXY$ at $X$ is drawn, and the line through $W$ that is parallel to this tangent intersects $\overline{XY}$ at $Z.$ If $XY = 8$ and $WX = 8,$ find $XY.$

LiIIiam0216 Apr 27, 2024

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52

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+2729

Help math

In triangle $ABC,$ $\angle C = 90^\circ.$ A semicircle is constructed along side $\overline{AC}$ that is tangent to $\overline{BC}$ and $\overline{AB}.$ If the radius of the semicircle is equal to BC, then find AC + BC + AB.

LiIIiam0216 Apr 27, 2024

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44

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+2729

Algebra

Suppose $a(x) = 2x + 5 - 3x^2 + x^3$ and $b(x) = 4 - x^2 + 4x^4$. Find $(a\circ b)(3) - (b\circ a)(3).$

LiIIiam0216 Apr 27, 2024

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15

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+363

1-315, how many 2's

Sam writes down the numbers 1-315
Sam chooses one of the digits written down at random. What is the probability that Sam chooses a 2?

Lilliam0216 Apr 27, 2024

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43

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+2729

Algebra

Let x \mathbin{\spadesuit} y = \frac{x^2}{y} for all x and y such that y\neq 0. Find all values of $a$ such that $a \mathbin{\spadesuit} (a + 1) = 9$. Write your answer as a list separated by commas.\

LiIIiam0216 Apr 27, 2024

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47

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+2729

Counting

Find the number of positive integers that satisfy both the following conditions:
- Each digit is a 1 or a 2 or a 3
- The sum of the digits is 5

LiIIiam0216 Apr 27, 2024

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56

1

+2729

Probability

A member of the junior varsity swim team will be elected as President. Each member is equally likely to be elected. The probability that a sophomore is elected is times the probability that a freshman is elected. Everyone on the junior varsity read more ..

LiIIiam0216 Apr 27, 2024

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25

1

+363

Cubes and probability

The six faces of a cube are painted black. The cube is then cut into 6^3 smaller cubes, all the same size. One of the small cubes is chosen at random, and rolled. What is the probability that when it lands, the face on the top is black?

Lilliam0216 Apr 27, 2024

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58

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+613

Algebra

Find the minimum value of \frac{x^2}{x - 1 + x^3} for x > 1

Pythagorearn Apr 27, 2024

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52

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+613

Algebra

Let f(x) = \left \lfloor \frac{2 - 3x}{x + 3} + x^2 \right \rfloor.
Find f(1) + f(2) + f(3) + ... + f(999) + f(1000).

Pythagorearn Apr 27, 2024

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36

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+613

Floor

Find all values of t such that floor(t) = 3t + 4 - t^2. If you find more than one value, then list the values you find in increasing order, separated by commas.

Pythagorearn Apr 27, 2024

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59

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+613

help with math

The function f(x) is defined for 1 \le x \le 5 as follows:
f(x) = 2x + 8 if 1 \le x \le 2
f(x) = 13 - 5x if 2 < x \le 3
f(x) = 20 - 14x if 3 < x \le 4 read more ..

Pythagorearn Apr 27, 2024

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35

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+1776

Geometry

Altitudes $\overline{AP}$ and $\overline{BQ}$ of an acute triangle $\triangle ABC$ intersect at point $H$. What is the area of triangle $ABC$?

bader Apr 27, 2024

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+1776

Geometry

Point $D$ is the midpoint of median $\overline{AM}$ of triangle $ABC$. Point $E$ is the midpoint of $\overline{AB}$, and point $T$ is the intersection of $\overline{BD}$ and $\overline{ME}$. Find the area of triangle $BET$ if $AB = 20$, $AC = 20$,
<read more ..

bader Apr 27, 2024

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51

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+1776

Geometry

Let $M$, $N$, and $P$ be the midpoints of sides $\overline{TU}$, $\overline{US}$, and $\overline{ST}$ of triangle $STU$, respectively. Let $\overline{UZ}$ be an altitude of the triangle. If $\angle TSU = 60^\circ$ and $\angle STU = 45^\circ$, read more ..

bader Apr 27, 2024

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26

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+613

Geometry

Semicircles are constructed on AB, AC, and BC. A circle is tangent to all three semicircles. Find the radius of the circle.

Pythagorearn Apr 27, 2024

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56

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math problem

In triangle $ABC$, $\angle A = 30^\circ$ and $\angle B = 90^\circ$. Point $X$ is on side $\overline{AC}$ such that line segment $\overline{BX}$ bisects $\angle ABC$. If $BC = 12$, then find the area of triangle $BXA$.

Pythagorearn Apr 27, 2024

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44

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+613

help math

A rectangle is inscribed in a circle of radius $5$ cm. If the area of the rectangle is equal to 24, what is the perimeter of the rectangle? Express your answer as a decimal to the nearest tenth.

Pythagorearn Apr 27, 2024

Apr 26, 2024

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60

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+1776

Geometry

Points $M$, $N$, and $O$ are the midpoints of sides $\overline{KL}$, $\overline{LJ}$, and $\overline{JK}$, respectively, of triangle $JKL$. Points $P$, $Q$, and $R$ are the midpoints of $\overline{NO}$, $\overline{OM}$, and $\overline{MN}$, respectively. read more ..

bader Apr 26, 2024

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27

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Geometry

In triangle $PQR$, let $M$ be the midpoint of $\overline{PQ}$, let $N$ be the midpoint of $\overline{PR}$, and let $O$ be the intersection of $\overline{QN}$ and $\overline{RM}$, as shown. If $\overline{QN}\perp\overline{PR}$, $QN = 15$, and $PR = 20$, read more ..

bader Apr 26, 2024

+1

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+1950

Pythagorean Theorem Proof!

According to Google, there are over 371 proofs for the pythagorean theorem.
My challenge is for you to name as many as could possible name.
I already know a few common ones, so I challenge you guys to read more ..

NotThatSmart Apr 26, 2024

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