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12
149 Questions
12 Answers
0
1
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+617
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
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AnswerscorrectIy
13.09.2024
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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 $3$
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AnswerscorrectIy
13.09.2024
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1
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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
13.09.2024
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1
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Geometry
In the diagram below, SR = PQ = 20 and RT:TS = 2:5. Find UV.
AnswerscorrectIy
09.09.2024
0
2
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Geometry
In triangle $PQR,$ $M$ is the midpoint of $\overline{QR}.$ Find $PM.$
AnswerscorrectIy
09.09.2024
0
1
1
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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
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AnswerscorrectIy
06.09.2024
0
1
0
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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
06.09.2024
0
2
0
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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
05.09.2024
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1
0
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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
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AnswerscorrectIy
04.09.2024
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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.
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AnswerscorrectIy
04.09.2024
0
1
1
+617
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
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AnswerscorrectIy
31.08.2024
0
2
1
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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 $WXYZ$?
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AnswerscorrectIy
28.08.2024
0
1
1
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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 $AB$.
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AnswerscorrectIy
28.08.2024
0
2
0
+617
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
28.08.2024
0
1
1
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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
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AnswerscorrectIy
28.08.2024
0
1
0
+617
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
23.08.2024
0
1
1
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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.
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AnswerscorrectIy
23.08.2024
0
2
0
+617
Counting
Find the number of 7-digit numbers, where the sum of the digits is divisible by 7.
AnswerscorrectIy
17.08.2024
0
1
0
+617
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
17.08.2024
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