All Questions ⁺⁰236416 Questions

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+402

Geometry

In the diagram, \overline{BE} bisects \angle ABC, and $\overline{CE}$ bisects $\angle ACD.$ Compute $\angle BAC,$ in degrees.

MEMEG0D Jan 6, 2025

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+402

Geometry

In pentagon ABCDE, \overline{AC} bisects \angle BCE and $\overline{AD}$ bisects $\angle BDE.$ If $\angle CBD = 30^\circ$ and $\angle CED = 60^\circ,$ then find $\angle CAD,$ in degrees.

MEMEG0D Jan 6, 2025

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+433

Geometry

Points $A,$ $B,$ and $C$ are given in the coordinate plane. There exists a point $Q$ and a constant $k$ such that for any point $P$,
PA^2 + PB^2 + PC^2 = 3PQ^2 + k.
If $A = (2,4),$ $B = (-3,1),$ and $C = (1,7)$, then find the constant read more ..

MeIdHunter Jan 6, 2025

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+433

Geometry

In parallelogram EFGH, let M be the point on \overline{EF} such that FM:ME = 1:1, and let N be the point on \overline{EH} such that HN:NE = 1:1. Line segments \overline{FH} and \overline{GM} intersect at P, and line segments \overline{FH} and GN intersect read more ..

MeIdHunter Jan 6, 2025

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+433

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 $2,$ and the area of triangle $RSX$ is $2.$ Find the area of trapezoid read more ..

MeIdHunter Jan 6, 2025

Jan 5, 2025

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+433

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.

MeIdHunter Jan 5, 2025

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+1520

Geometry

Let O be the origin. Points P and Q lie in the first quadrant. The slope of line segment $\overline{OP}$ is $4,$ and the slope of line segment $\overline{OQ}$ is $5.$ If $OP = OQ,$ then compute the slope of line segment $\overline{PQ}.$
Note: read more ..

kittykat Jan 5, 2025

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+1520

Geometry

Let a and b be real numbers, where a < b, and let $A = (a,a^2)$ and $B = (b,b^2)$. The line $\overline{AB}$ (meaning the unique line that contains the point $A$ and the point $B$) has slope $2$. Find $a + b$.

kittykat Jan 5, 2025

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+1520

Geometry

In triangle PQR, M is the midpoint of $\overline{QR}.$ Find $PM.$
PQ = 5, PR = 8, QR = 11

kittykat Jan 5, 2025

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+4

Help with tough diophantine equation

I am asking this because my teacher gave my class a challenge for homework, I still am not able to solve it and I wanted some help. The question is, in the equation , prove that (0,0) is the only solution. I tried using modular arithmetic equations like read more ..

quorter Jan 5, 2025

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+178

Algebra

Find the greatest prime divisor of the value of the arithmetic series
1 + 2 + 3 + \dots + 135 + 136 + 137 + 138 + 139 + 140.

cwimie Jan 5, 2025

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+837

Algebra

Solve the inequality $4t^2 \le -9t + 12 - 14t + 36.$ Write your answer in interval notation.

booboo44 Jan 5, 2025

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+837

Algebra

Solve the inequality $x(x + 6) > 16 - 5x + 33.$ Write your answer in interval notation.

booboo44 Jan 5, 2025

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+1393

need help

For a certain value of k, the system
x + y + 3z = 10 - x + 5y,
4x + 2y + 5z = 7 + 11y + 19z,
kx + z = 3 + 5y
has no solutions. What is this value of k?

learnmgcat Jan 5, 2025

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+16

help plzzzzz

A point $P$ is randomly placed in the interior of the right triangle below. What is the probability that the area of triangle $PBC$ is less than half of the area of triangle $ABC$? Express your answer as a common fraction.

adook101 Jan 5, 2025

Jan 4, 2025

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+845

Geometry

We know \overline {DE}\parallel \overline {BC}. Given $[ADE]=9$ and $[BDE]=13,$ find $[ABC].$

gnistory Jan 4, 2025

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+845

Geometry

In triangle ABC, shown here, \overline{AB} and \overline{AC} have each been divided into five congruent segments. What fraction of triangle $ABC$ is shaded? Express your answer as a common fraction.

gnistory Jan 4, 2025

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+845

Geometry

In the diagram below, ADEF is a square with side length 4. If BC = 2, what is BD?

gnistory Jan 4, 2025

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+18

geometry

Four cubes of volumes $1 \text{ cm}^3$, $8 \text{ cm}^3$, $27 \text{ cm}^3$, and $125 \text{ cm}^3$ are glued together at their faces. What is the number of square centimeters in the smallest possible surface area of the resulting solid figure?

Owinner Jan 4, 2025

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+703

Geometry

In the diagram below, we have \overline{MN}\parallel\overline{PQ}, and $PN = 24.$ Find $OP.$

siIviajendeukie Jan 4, 2025

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+703

Geometry

In rectangle $ABCD,$ $\frac{[AEP]}{[DFP]} = 5.$ Find the ratio of the area of the red region to the area of the yellow region. Enter your answer as a fraction.

siIviajendeukie Jan 4, 2025

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