AnswerscorrectIy

149 Questions
12 Answers

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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 lire plus ..

AnswerscorrectIy 13 sept. 2024

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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 $3$ lire plus ..

AnswerscorrectIy 13 sept. 2024

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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 13 sept. 2024

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Geometry

In the diagram below, SR = PQ = 20 and RT:TS = 2:5. Find UV.

AnswerscorrectIy 9 sept. 2024

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Geometry

In triangle $PQR,$ $M$ is the midpoint of $\overline{QR}.$ Find $PM.$

AnswerscorrectIy 9 sept. 2024

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Algebra

Find the area of the region enclosed by the graph of x^2+y^2=2x-6y+6+4x-4y+12

AnswerscorrectIy 6 sept. 2024

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Algebra

The graph of $y = ax^2 + bx + c$ has an axis of symmetry of $x = 2.$ If $(4,7)$ lies on the graph, find another point on the graph.

AnswerscorrectIy 6 sept. 2024

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Algebra

The parabola y = ax^2 + bx + c is graphed below. Find a \cdot b \cdot c. (The grid lines are one unit apart.)

AnswerscorrectIy 5 sept. 2024

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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 $4,$ and the area of triangle $PSR$ is $8.$ Find the area of trapezoid lire plus ..

AnswerscorrectIy 4 sept. 2024

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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.

AnswerscorrectIy 4 sept. 2024

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Number Theory

When fractions are expressed in different bases, they can be terminating or repeating. For example, when $\frac{1}{5}$ is expressed in base $3,$ the result is
\[0.\overline{0121}_3 = 0.01210121 \dots,\]
which is repeating.
When lire plus ..

AnswerscorrectIy 31 août 2024

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Geometry

Points $A$ and $B$ are on side $\overline{YZ}$ of rectangle $WXYZ$ such that $\overline{WA}$ and $\overline{WB}$ trisect $\angle ZWX$. If $WX = 2$ and $XY = 3$, then what is the area of rectangle $WXYZ$?

AnswerscorrectIy 28 août 2024

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Geometry

In rectangle $WXYZ$, $A$ is on side $\overline{WX}$, $B$ is on side $\overline{YZ}$, and $C$ is on side $\overline{XY}$. If $AX = 15$, $BY = 20$, $\angle ACB= 60^\circ$, and $CY = 3 \cdot CX$, then find $AB$.

AnswerscorrectIy 28 août 2024

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Algebra

For certain values of $k$ and $m,$ the system
a + 2b = -3 - 7a + b,
4a + 2b = k - 5a - mb
has infinitely many solutions $(a,b).$ What are $k$ and $m?$

AnswerscorrectIy 28 août 2024

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Algebra

Find the ordered pair (s, t) that satisfies the system
(s/2) + 5t =3 - 7t + 8s
3t - 6s = 9 - 2t

AnswerscorrectIy 28 août 2024

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Algebra

Let $x$, $y$, and $z$ be nonzero real numbers. Find all possible values of
\frac{x}{|x|} + \frac{y}{|y|} + \frac{z}{|z|} + \frac{x + y + z}{|x + y + z|}

AnswerscorrectIy 23 août 2024

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Algebra

Find all $c$ such that $|c + 5| - 3|c + 7| = 10 + 2|c - 8|.$ Enter all the solutions, separated by commas.

AnswerscorrectIy 23 août 2024

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Counting

Find the number of 7-digit numbers, where the sum of the digits is divisible by 7.

AnswerscorrectIy 17 août 2024

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Counting

In how many ways can the numbers 1, 2, 3, 4, 5, 6 be arranged in a row, so that the product of any two adjacent numbers is at least 4?

AnswerscorrectIy 17 août 2024