All Questions ⁺⁰237754 Questions

0

26

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+977

Geometry

In triangle $ABC,$ $AB = 3,$ $AC = 5,$ $BC = 7,$ and $D$ lies on $\overline{BC}$ such that $\overline{AD}$ bisects $\angle BAC.$ Find $\cos \angle BAD.$

magenta Jan 30, 2025

0

23

1

+977

Geometry

In triangle $ABC,$ $BC = 32,$ $\tan B = \frac{3}{5},$ and $\tan C = \frac{1}{4}.$ Find the area of the triangle.

magenta Jan 30, 2025

0

15

1

+977

Geometry

For what value of $c$ will the circle with equation $x^2 - 10x + y^2 + 10y + c = 0$ have a radius of length 1?

magenta Jan 30, 2025

0

45

0

+977

Geometry

What is the area of the region defined by the equation $x^2+y^2 - 7 = 4y-14x+2x-6y+3$?

magenta Jan 30, 2025

0

44

0

+970

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 $XY = 15$ and $WY = 10,$ find $WX.$

gnistory Jan 30, 2025

0

26

1

+970

Geometry

Let $IJKLMN$ be a hexagon with side lengths $IJ = LM = 3,$ $JK = MN = 3,$ and $KL = NI = 3$. Also, all the interior angles of the hexagon are equal. Find the area of hexagon $IJKLMN$.

gnistory Jan 30, 2025

0

36

1

+970

Geometry

The interior angles of a polygon form an arithmetic sequence. The difference between the largest angle and smallest angle is $56^\circ$. If the polygon has $3$ sides, then find the smallest angle, in degrees.

gnistory Jan 30, 2025

Jan 29, 2025

0

23

1

+970

Geometry

In square ABCD, P is on BC such that BP = 4 and PC = 1, and Q is on line CD such that DQ = 2 and QC = 3. Find sin angle PAQ.

gnistory Jan 29, 2025

0

40

1

+970

Geometry

In quadrilateral $BCED$, sides $\overline{BD}$ and $\overline{CE}$ are extended past $B$ and $C$, respectively, to meet at point $A$. If $BD = 8$, $BC = 3$, $CE = 1$, $AC = 19$ and $AB = 13$, then what is $DE$?

gnistory Jan 29, 2025

0

35

1

+857

Algebra

Five workers have been hired to complete a job. If one additional worker is hired, they could complete the job $4$ days earlier. If the job needs to be completed $20$ days earlier, how many additional workers should be hired?

cooIcooIcooI17 Jan 29, 2025

0

33

1

+857

Algebra

Three runners, Dirk, Edith, and Foley all start at the same time for a $24$ km race, and each of them runs at a constant speed. When Dirk finishes the race, Edith is $10$ km behind, and Foley is $15$ km behind. When Edith finishes the race, how far read more ..

cooIcooIcooI17 Jan 29, 2025

0

52

0

+857

Algebra

For a certain value of $k$, the system
x + y + 3z &= 10, \\
6x + 4y + z &= 18 - 5y - x, \\
kx + 7z &= 8 - 14y
has no solutions. What is this value of $k$?

cooIcooIcooI17 Jan 29, 2025

0

49

0

+857

Algebra

Find the ordered quintuplet (a,b,c,d,e) that satisfies the system of equations
a+2b+3c+4d+5e=47
2a+3b+4c+5d=15
3a+4b+5c=34 read more ..

cooIcooIcooI17 Jan 29, 2025

0

36

1

+810

Algebra

Assume that f(3) = 4. Name a point that must be on the graph of y= (5 + f(2x/3))/11$.

crimefightingvigiI Jan 29, 2025

0

34

1

+810

Algebra

Find all points (x,y) that are 5 units away from the point (2,7) and that lie on the line x + y = 13.

crimefightingvigiI Jan 29, 2025

0

35

1

+810

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.

crimefightingvigiI Jan 29, 2025

0

49

0

+810

Algebra

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

crimefightingvigiI Jan 29, 2025

+1

31

1

+987

Algebra

Find the largest integer k such that the equation
5x^2 - kx + 8 - 20x^2 + 45 = 0
has no real solutions.

booboo44 Jan 29, 2025

+1

45

0

+987

Algebra

Let b be a constant. What is the smallest possible degree of the polynomial f(x) + b \cdot g(x), where f(x) = 2x^5 - 6x^4 - 4x^3 + 12x^2 + 7x - 5 and g(x) = x^6 - 17x^5 + 2x^4 + 6x^3 + 11x^2 - 8x + 1.

booboo44 Jan 29, 2025

+1

42

0

+987

Algebra

Let f be a cubic polynomial such that f(0) = 5, f(2) = 8, f(3)=13, and f(7) = -5. What is the sum of the coefficients of f?

booboo44 Jan 29, 2025

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