All Questions ⁺⁰237783 Questions

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Number Theory

How many numbers between $1000$ and $2000$ leave a remainder of $3$ when divided by $47?$

Jasmo Feb 2, 2025

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27

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+456

Number Theory

For a positive integer n, \phi(n) denotes the number of positive integers less than or equal to $n$ that are relatively prime to $n$. What is $\phi(2835)$?

Jasmo Feb 2, 2025

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11

2

+456

Number Theory

Fill in the blanks with three distinct positive integers.
1/___ + 1/___ + 1/___ = \frac{13}{12}

Jasmo Feb 2, 2025

Feb 1, 2025

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32

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+456

Geometry

The number of diagonals in a certain regular polygon is equal to $2$ times the number of sides. How many sides does this polygon have?

Jasmo Feb 1, 2025

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33

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+456

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 $IJKLMN$.

Jasmo Feb 1, 2025

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43

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Geometry

In the diagram below, each side of convex quadrilateral $ABCD$ is trisected. (For example, $AP = PQ = QB.$) The area of convex quadrilateral $ABCD$ is $180.$ Find the area of the shaded region.

Jasmo Feb 1, 2025

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46

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Geometry

Let $B,$ $A,$ and $D$ be three consecutive vertices of a regular $20$-gon. A regular heptagon is constructed on $\overline{AB},$ with a vertex $C$ next to $A.$ Find $\angle BAD,$ in degrees.

Jasmo Feb 1, 2025

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26

1

+84

Algebra

All members of our painting team paint at the same rate. If $10$ members can paint a $1000$ square foot wall in $10$ minutes, then how long would it take for $5$ members to paint a $1000$ square foot wall, in minutes?

hemiiI8675 Feb 1, 2025

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25

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+84

Algebra

In physics, Ohm's law says that current through a wire, $I$, is directly proportional to voltage, $V$, and inversely proportional to resistance, $R$:
I = V/R
It's also true that resistance is directly proportional to the length of the wire. read more ..

hemiiI8675 Feb 1, 2025

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23

1

+84

Algebra

A plane flies from Penthaven to Jackson and then back to Penthaven. When there is no wind, the round trip takes $3$ hours and $20$ minutes, but when there is a wind blowing from Penthaven to Jackson at $70$ miles per hour, the trip takes $3$ hours read more ..

hemiiI8675 Feb 1, 2025

+1

13

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+7

Remainder Problem Help

When the polynomial is divided by the remainder is . When the polynomial is divided by the remainder is Let be the remainder when is divided by Find read more ..

singlebl Feb 1, 2025

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+823

Counting

Find the number of ways that Magnus can give out $12$ identical stickers to $2$ of his friends, if every friend gets at least one sticker.

crimefightingvigiI Feb 1, 2025

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27

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+823

Counting

Miyu is giving out $8$ identical chocolates to her $5$ friends, including Dhruv. All possible distributions are equally likely. What is the probability that Dhruv gets exactly $7$ chocolates?

crimefightingvigiI Feb 1, 2025

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28

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+823

Counting

How many solutions are there to the equation
u + v + w + x + y + z = 2,
where $u,$ $v,$ $w,$ $x,$ $y,$ and $z$ are nonnegative integers, and $x$ is at most $1?$

crimefightingvigiI Feb 1, 2025

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39

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+456

Geometry

Let P_1 P_2 P_3 \dotsb P_{10} be a regular polygon inscribed in a circle with radius $1.$ Compute
P_1 P_2 + P_2 P_3 + P_3 P_4 + \dots + P_9 P_{10} + P_{10} P_1

Jasmo Feb 1, 2025

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29

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+456

Geometry

Altitudes $\overline{AD}$ and $\overline{BE}$ of acute triangle $ABC$ intersect at point $H$. If $\angle AHB = 144^\circ$ and $\angle BAH = 66^\circ$, then what is $\angle HCA$ in degrees?

Jasmo Feb 1, 2025

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Geometry

In triangle $STU$, let $M$ be the midpoint of $\overline{ST},$ and let $N$ be on $\overline{TU}$ such that $\overline{SN}$ is an altitude of triangle $STU$. If $ST = 8$, $SU = 8$, $TU = 8$, and $\overline{SN}$ and $\overline{UM}$ intersect at $X$,
<read more ..

Jasmo Feb 1, 2025

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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 $\overline{QN}\perp\overline{PR}$, $QN = 1$, and $PR =1$, read more ..

Jasmo Feb 1, 2025

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23

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+31

I'm confused.

Let k be a positive real number. The line x + y = 3k and the circle x^2 + y^2 = k are drawn. Find k so that the line is tangent to the circle.
If someone could send help, it would be much appreciated! :)

Jasino Feb 1, 2025

+1

26

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+1309

Geometry

Find the coordinates of the center of the circle.
The points on the circle are (22,15), (-25,0), (22,-29).

AnswerscorrectIy Feb 1, 2025

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+1309

Geometry

Determine the coordinates of the point $P$ on the line $y=-x+6$ such that $P$ is equidistant from the points $A(10,-12)$ and $O(2,8)$ (that is, so that $PA=PO$). Express your answer as an ordered pair $(a,b)$.

AnswerscorrectIy Feb 1, 2025

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35

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+1309

Geometry

In the diagram, the four points have coordinates $A(0,1)$, $B(1,3)$, $C(5,2)$, and $D(4.5,0)$. What is the area of quadrilateral $ABCD$?

AnswerscorrectIy Feb 1, 2025

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