problem stringlengths 10 5.15k | answer stringlengths 0 1.22k | solution stringlengths 0 11.1k | reward float64 0 1 | length float64 172 8.19k | correct_length float64 -1 8.19k | incorrect_length float64 -1 8.19k |
|---|---|---|---|---|---|---|
Let $A B C$ be an acute triangle with $A$-excircle $\Gamma$. Let the line through $A$ perpendicular to $B C$ intersect $B C$ at $D$ and intersect $\Gamma$ at $E$ and $F$. Suppose that $A D=D E=E F$. If the maximum value of $\sin B$ can be expressed as $\frac{\sqrt{a}+\sqrt{b}}{c}$ for positive integers $a, b$, and $c$,... | 705 | First note that we can assume $A B<A C$. Suppose $\Gamma$ is tangent to $B C$ at $T$. Let $A D=D E=E F=x$. Then, by Power of a Point, we have $D T^{2}=D E \cdot D F=x \cdot 2 x=2 x^{2} \Longrightarrow D T=x \sqrt{2}$. Note that $C T=s-b$, and since the length of the tangent from $A$ to $\Gamma$ is $s$, we have $s^{2}=A... | 0 | 8,192 | -1 | 8,192 |
Given a positive sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, if both $\{a_n\}$ and $\{\sqrt{S_n}\}$ are arithmetic sequences with the same common difference, calculate $S_{100}$. | 2500 | 0.875 | 4,883.4375 | 4,410.785714 | 8,192 | |
A ball invites 2018 couples, each assigned to areas numbered $1, 2, \cdots, 2018$. The organizer specifies that at the $i$-th minute of the ball, the couple in area $s_i$ (if any) moves to area $r_i$, and the couple originally in area $r_i$ (if any) exits the ball. The relationship is given by:
$$
s_i \equiv i \pmod{20... | 1009 | 0 | 8,133.0625 | -1 | 8,133.0625 | |
What is the smallest number, all of whose digits are 1 or 2, and whose digits add up to $10$? | 111111112 | 0 | 6,789.6875 | -1 | 6,789.6875 | |
Given that 5 students are to be distributed into two groups, A and B, with at least one person in each group, and student A cannot be in group A, calculate the number of different distribution schemes. | 15 | 0.875 | 5,827.125 | 5,489.285714 | 8,192 | |
How many three-digit numbers are composed of three distinct digits such that one digit is the average of the other two? | 112 | 0 | 8,192 | -1 | 8,192 | |
Evaluate: $(2^2)^3$. | 64 | 1 | 1,899.1875 | 1,899.1875 | -1 | |
Find the smallest natural number that starts with the digit 5, which, when this 5 is removed from the beginning of its decimal representation and appended to its end, becomes four times smaller. | 512820 | 0.6875 | 5,657 | 4,504.727273 | 8,192 | |
Given the function $f(x)=\log_{a} \frac{2-x}{b+x} (0 < a < 1)$ is an odd function, when $x \in (-2,2a)$, the range of the function $f(x)$ is $(-\infty,1)$, then the sum of the real numbers $a+b=$ __________. | \sqrt{2}+1 | 0 | 5,538.5625 | -1 | 5,538.5625 | |
Eight points are chosen on the circumference of a circle, labelled $P_{1}, P_{2}, \ldots, P_{8}$ in clockwise order. A route is a sequence of at least two points $P_{a_{1}}, P_{a_{2}}, \ldots, P_{a_{n}}$ such that if an ant were to visit these points in their given order, starting at $P_{a_{1}}$ and ending at $P_{a_{n}... | 8744 | Solution 1: How many routes are there if we are restricted to $n$ available points, and we must use all $n$ of them? The answer is $n 2^{n-2}$ : first choose the starting point, then each move after that must visit one of the two neighbors of your expanding region of visited points (doing anything else would prevent yo... | 0 | 8,192 | -1 | 8,192 |
Two workers were assigned to produce a batch of identical parts; after the first worked for \(a\) hours and the second for \(0.6a\) hours, it turned out that they had completed \(\frac{5}{n}\) of the entire job. After working together for another \(0.6a\) hours, they found that they still had \(\frac{1}{n}\) of the bat... | 10 | 0 | 8,171.125 | -1 | 8,171.125 | |
Three distinct integers, $x$, $y$, and $z$, are randomly chosen from the set $\{1,2,3,4,5,6,7,8,9,10,11,12\}$. What is the probability that $xyz-xy-xz-yz$ is even? | \frac{10}{11} | 0.0625 | 7,633.625 | 7,464 | 7,644.933333 | |
In triangle $BCD$, $\angle CBD=\angle CDB$ because $BC=CD$. If $\angle BCD=80+50+30=160$, find $\angle CBD=\angle CDB$. | 10 | $\angle CBD=\angle CDB=10$ | 0.875 | 2,491.0625 | 1,950.714286 | 6,273.5 |
Given that $21^{-1} \equiv 17 \pmod{53}$, find $32^{-1} \pmod{53}$, as a residue modulo 53. (Give a number between 0 and 52, inclusive.) | 36 | 0 | 6,493.1875 | -1 | 6,493.1875 | |
A special deck of cards contains $49$ cards, each labeled with a number from $1$ to $7$ and colored with one of seven colors. Each number-color combination appears on exactly one card. Sharon will select a set of eight cards from the deck at random. Given that she gets at least one card of each color and at least one c... | 13 | Without loss of generality, assume that the $8$ numbers on Sharon's cards are $1$, $1$, $2$, $3$, $4$, $5$, $6$, and $7$, in that order, and assume the $8$ colors are red, red, and six different arbitrary colors. There are ${8\choose2}-1$ ways of assigning the two red cards to the $8$ numbers; we subtract $1$ because w... | 0 | 8,192 | -1 | 8,192 |
Given point $P$ is a moving point on the ellipse $\frac{x^2}{8} + \frac{y^2}{4} = 1$ in the first quadrant, through point $P$, two tangents $PA$ and $PB$ are drawn to the circle $x^2 + y^2 = 4$, with the points of tangency being $A$ and $B$ respectively. The line $AB$ intersects the $x$-axis and $y$-axis at points $M$ ... | \sqrt{2} | 0 | 5,784.625 | -1 | 5,784.625 | |
In a computer game, a player can choose to play as one of three factions: \( T \), \( Z \), or \( P \). There is an online mode where 8 players are divided into two teams of 4 players each. How many total different matches are possible, considering the sets of factions? The matches are considered different if there is ... | 120 | 0.0625 | 7,965.3125 | 8,159 | 7,952.4 | |
For positive real numbers $a,$ $b,$ $c,$ and $d,$ compute the maximum value of
\[\frac{abcd(a + b + c + d)}{(a + b)^2 (c + d)^2}.\] | \frac{1}{4} | 0 | 8,192 | -1 | 8,192 | |
What is the value of $K$ in the equation $16^3\times8^3=2^K$? | 21 | 1 | 1,211.1875 | 1,211.1875 | -1 | |
What is the area of the region \( \mathcal{R} \) formed by all points \( L(r, t) \) with \( 0 \leq r \leq 10 \) and \( 0 \leq t \leq 10 \) such that the area of triangle \( JKL \) is less than or equal to 10, where \( J(2,7) \) and \( K(5,3) \)? | \frac{313}{4} | The distance between \( J(2,7) \) and \( K(5,3) \) is equal to \( \sqrt{(2-5)^2 + (7-3)^2} = \sqrt{3^2 + 4^2} = 5 \). Therefore, if we consider \( \triangle JKL \) as having base \( JK \) and height \( h \), then we want \( \frac{1}{2} \cdot JK \cdot h \leq 10 \) which means that \( h \leq 10 \cdot \frac{2}{5} = 4 \). ... | 0 | 8,192 | -1 | 8,192 |
Mr. Jones has 6 children. Assuming that the gender of each child is determined independently and with equal likelihood of male and female, what is the probability that Mr. Jones has more sons than daughters or more daughters than sons? | \dfrac{11}{16} | 1 | 2,902.625 | 2,902.625 | -1 | |
Calculate the car's average miles-per-gallon for the entire trip given that the odometer readings are $34,500, 34,800, 35,250$, and the gas tank was filled with $8, 10, 15$ gallons of gasoline. | 22.7 | 0.1875 | 2,044.875 | 5,023.333333 | 1,357.538462 | |
A binary operation $\diamondsuit$ has the properties that $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ and that $a\,\diamondsuit \,a=1$ for all nonzero real numbers $a, b,$ and $c$. (Here $\cdot$ represents multiplication). Find the solution to the equation $2016 \,\diamondsuit\, (6\,\diamon... | \frac{25}{84} | 1 | 2,206.125 | 2,206.125 | -1 | |
Let \( N \) be the smallest positive integer such that \( \frac{N}{15} \) is a perfect square, \( \frac{N}{10} \) is a perfect cube, and \( \frac{N}{6} \) is a perfect fifth power. Find the number of positive divisors of \( \frac{N}{30} \). | 8400 | 0.375 | 6,549.8125 | 5,349.333333 | 7,270.1 | |
Given that the positive integers \( a, b, c \) satisfy \( 2017 \geqslant 10a \geqslant 100b \geqslant 1000c \), find the number of possible triples \( (a, b, c) \). | 574 | 0.3125 | 7,511.75 | 6,135 | 8,137.545455 | |
Calculate:<br/>$(1)-9+5-\left(-12\right)+\left(-3\right)$;<br/>Calculate:<br/>$(2)-(+1.5)-(-4\frac{1}{4})+3.75-(-8\frac{1}{2})$;<br/>$(3)$Read the following solution process and answer the question:<br/>Calculate:$\left(-15\right)\div (-\frac{1}{2}×\frac{25}{3}$）$÷\frac{1}{6}$<br/>Solution: Original expression $=\left(... | \frac{108}{5} | 0.625 | 3,092.625 | 2,866.6 | 3,469.333333 | |
Let $ABCD$ be a convex quadrilateral, and let $G_A,$ $G_B,$ $G_C,$ $G_D$ denote the centroids of triangles $BCD,$ $ACD,$ $ABD,$ and $ABC,$ respectively. Find $\frac{[G_A G_B G_C G_D]}{[ABCD]}.$
[asy]
unitsize(0.6 cm);
pair A, B, C, D;
pair[] G;
A = (0,0);
B = (7,1);
C = (5,-5);
D = (1,-3);
G[1] = (B + C + D)/3;
G[2... | \frac{1}{9} | 0.4375 | 7,307.375 | 6,170 | 8,192 | |
Two teachers, A and B, and four students stand in a row. (Explain the process, list the expressions, and calculate the results, expressing the results in numbers)<br/>$(1)$ The two teachers cannot be adjacent. How many ways are there to arrange them?<br/>$(2)$ A is to the left of B. How many ways are there to arrange t... | 12 | 0.0625 | 7,911.0625 | 6,705 | 7,991.466667 | |
How many six-digit multiples of 27 have only 3, 6, or 9 as their digits? | 51 | Divide by 3. We now want to count the number of six-digit multiples of 9 that only have 1, 2, or 3 as their digits. Due to the divisibility rule for 9, we only need to consider when the digit sum is a multiple of 9. Note that $3 \cdot 6=18$ is the maximum digit sum. If the sum is 18, the only case is 333333. Otherwise,... | 0.0625 | 8,192 | 8,192 | 8,192 |
Let the original number be expressed as $x$. When the decimal point of $x$ is moved one place to the right, the resulting number can be expressed as $100x$. According to the given information, we have the equation $100x = x + 34.65$. | 3.85 | 0.4375 | 2,973.75 | 1,950.285714 | 3,769.777778 | |
How many positive divisors do 9240 and 13860 have in common? | 24 | 0 | 2,636.625 | -1 | 2,636.625 | |
In the diagram, \(P Q R S T\) is a pentagon with \(P Q=8\), \(Q R=2\), \(R S=13\), \(S T=13\), and \(T P=8\). Also, \(\angle T P Q=\angle P Q R=90^\circ\). What is the area of pentagon \(P Q R S T\) ? | 100 | 0.5 | 7,472.75 | 6,753.5 | 8,192 | |
Find the smallest constant $ C$ such that for all real $ x,y$
\[ 1\plus{}(x\plus{}y)^2 \leq C \cdot (1\plus{}x^2) \cdot (1\plus{}y^2)\]
holds. | 4/3 | 0.125 | 8,103.1875 | 7,481.5 | 8,192 | |
In triangle \( ABC \), it is given that \( AC = 5\sqrt{2} \), \( BC = 5 \), and \( \angle BAC = 30^\circ \). What is the largest possible size in degrees of \( \angle ABC \)? | 135 | 0.75 | 5,835.5 | 5,199.25 | 7,744.25 | |
A bag contains four pieces of paper, each labeled with one of the digits $1$, $2$, $3$ or $4$, with no repeats. Three of these pieces are drawn, one at a time without replacement, to construct a three-digit number. What is the probability that the three-digit number is a multiple of $3$? | \frac{1}{2} | 1. **Total number of ways to draw three digits from four**:
The bag contains four distinct digits: $1, 2, 3, 4$. We are to draw three digits without replacement to form a three-digit number. The number of ways to choose the first digit is $4$, the second digit is $3$ (since one digit has already been drawn), and th... | 0.875 | 5,065.625 | 4,619 | 8,192 |
Simplify first, then evaluate: $(1-\frac{2}{{m+1}})\div \frac{{{m^2}-2m+1}}{{{m^2}-m}}$, where $m=\tan 60^{\circ}-1$. | \frac{3-\sqrt{3}}{3} | 0 | 3,259.625 | -1 | 3,259.625 | |
Anna Lisa bought two dozen apples for $\$$15.60. At that same rate, how much would three dozen of these apples cost? | \$23.40 | 1 | 2,099.375 | 2,099.375 | -1 | |
Find the coefficient of the $x^4$ term in the expansion of the binomial $(4x^{2}-2x+1)(2x+1)^{5}$. | 80 | 0.5625 | 7,125.75 | 6,296.444444 | 8,192 | |
Let $x_1$ satisfy $2x+2^x=5$, and $x_2$ satisfy $2x+2\log_2(x-1)=5$. Calculate the value of $x_1+x_2$. | \frac {7}{2} | 0.0625 | 8,154.5625 | 7,593 | 8,192 | |
A circle with center $O$ has radius $8$ units and circle $P$ has radius $2$ units. The circles are externally tangent to each other at point $Q$. Segment $TS$ is the common external tangent to circle $O$ and circle $P$ at points $T$ and $S$, respectively. What is the length of segment $OS$? Express your answer in s... | 8\sqrt{2} | 0.6875 | 6,554.4375 | 5,810.090909 | 8,192 | |
How many different ways are there to split the number 2004 into natural summands that are approximately equal? There can be one or several summands. Numbers are considered approximately equal if their difference is no more than 1. Ways that differ only by the order of summands are considered the same. | 2004 | 0.3125 | 7,844.4375 | 7,586.8 | 7,961.545455 | |
(1) Find the value of $\cos\frac{5\pi}{3}$.
(2) Given that $\frac{\sin\alpha + 2\cos\alpha}{5\cos\alpha - \sin\alpha} = \frac{5}{16}$, find the value of $\tan\alpha$.
(3) Given that $\sin\theta = \frac{1}{3}$ and $\theta \in (0, \frac{\pi}{2})$, find the value of $\tan 2\theta$.
(4) In triangle $ABC$, the sides oppo... | 2\sqrt{2} | 0.5 | 7,311.9375 | 6,431.875 | 8,192 | |
The Fibonacci numbers are defined by $F_1=F_2=1$ and $F_n=F_{n-1}+F_{n-2}$ for $n>2$ . It is well-known that the sum of any $10$ consecutive Fibonacci numbers is divisible by $11$ . Determine the smallest integer $N$ so that the sum of any $N$ consecutive Fibonacci numbers is divisible by $12$ . | 24 | 0.125 | 7,994.5625 | 6,612.5 | 8,192 | |
Find the value of $\dfrac{2\cos 10^\circ - \sin 20^\circ }{\sin 70^\circ }$. | \sqrt{3} | 0.5625 | 6,382.4375 | 4,975 | 8,192 | |
Given that there are two alloys with different percentages of copper, with alloy A weighing 40 kg and alloy B weighing 60 kg, a piece of equal weight is cut from each of these two alloys, and each cut piece is then melted together with the remaining part of the other alloy, determine the weight of the alloy cut. | 24 | 0.0625 | 7,057.125 | 5,386 | 7,168.533333 | |
It is now 12:00:00 midnight, as read on a 12-hour digital clock. In 122 hours, 39 minutes and 44 seconds the time will be $A:B:C$. What is the value of $A + B + C$? | 85 | 0.5 | 6,360.625 | 5,464.625 | 7,256.625 | |
A pentagon is formed by placing an equilateral triangle on top of a rectangle. The side length of the equilateral triangle is equal to the width of the rectangle, and the height of the rectangle is twice the side length of the triangle. What percent of the area of the pentagon is the area of the equilateral triangle? | \frac{\sqrt{3}}{\sqrt{3} + 8} \times 100\% | 0 | 6,635.875 | -1 | 6,635.875 | |
If $\left( r + \frac{1}{r} \right)^2 = 3,$ then find $r^3 + \frac{1}{r^3}.$ | 0 | 0.875 | 4,950.6875 | 4,487.642857 | 8,192 | |
We draw diagonals in some of the squares on a chessboard in such a way that no two diagonals intersect at a common point. What is the maximum number of diagonals that can be drawn this way? | 36 | 0 | 7,950.5625 | -1 | 7,950.5625 | |
A unit has a total of 620 staff members. To investigate the time workers spend commuting, it was decided to survey 62 workers using a systematic sampling method. The entire staff was divided into 62 segments with equal intervals, and a simple random sampling method was used to determine that the starting number of the ... | 394 | 1 | 3,456.875 | 3,456.875 | -1 | |
Given positive integers \(a\) and \(b\) such that \(15a + 16b\) and \(16a - 15b\) are both perfect squares, find the smallest possible value of these two perfect squares. | 231361 | 0 | 8,192 | -1 | 8,192 | |
Given that the area of $\triangle ABC$ is $\frac{1}{2}$, $AB=1$, $BC=\sqrt{2}$, determine the value of $AC$. | \sqrt{5} | 0.8125 | 6,640.5625 | 6,347.846154 | 7,909 | |
The value of $x$ at the intersection of $y=\frac{8}{x^2+4}$ and $x+y=2$ is: | 0 | 1. **Substitute for $y$ in the second equation:**
Given the equations $y = \frac{8}{x^2 + 4}$ and $x + y = 2$, we can substitute $y$ from the second equation into the first:
\[
x + y = 2 \implies y = 2 - x.
\]
Substituting $y = 2 - x$ into $y = \frac{8}{x^2 + 4}$, we get:
\[
2 - x = \frac{8}{x^2 ... | 1 | 3,889.0625 | 3,889.0625 | -1 |
Some nodes (vertices of unit squares) in a $6 \times 6$ grid are painted red such that on the boundary of any $k \times k$ subgrid ($1 \leqslant k \leqslant 6$) there is at least one red point. Find the minimum number of red points required to satisfy this condition. | 16 | 0 | 8,164.125 | -1 | 8,164.125 | |
Given vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ with magnitudes $|\overrightarrow {a}| = 6\sqrt {3}$ and $|\overrightarrow {b}| = \frac {1}{3}$, and their dot product $\overrightarrow {a} \cdot \overrightarrow {b} = -3$, determine the angle $\theta$ between $\overrightarrow {a}$ and $\overrightarrow {b}$. | \frac{5\pi}{6} | 0.4375 | 2,364.8125 | 2,535.428571 | 2,232.111111 | |
Calculate: $|1-\sqrt{2}|+(\frac{1}{2})^{-2}-\left(\pi -2023\right)^{0}$. | \sqrt{2} + 2 | 0.625 | 822.5625 | 694.7 | 1,035.666667 | |
A fenced, rectangular field measures $24$ meters by $52$ meters. An agricultural researcher has 1994 meters of fence that can be used for internal fencing to partition the field into congruent, square test plots. The entire field must be partitioned, and the sides of the squares must be parallel to the edges of the fie... | 702 | 0 | 7,656.375 | -1 | 7,656.375 | |
What is the value of $[\sqrt{1}] + [\sqrt{2}] + [\sqrt{3}] + \cdots + [\sqrt{1989 \cdot 1990}] + [-\sqrt{1}] + [-\sqrt{2}] + [-\sqrt{3}] + \cdots + [-\sqrt{1989 \cdot 1990}]$?
(The 1st "Hope Cup" Mathematics Contest, 1990) | -3956121 | 0.1875 | 7,294.875 | 6,221.666667 | 7,542.538462 | |
The function $f(n)$ is defined on the positive integers such that $f(f(n)) = 2n$ and $f(4n + 1) = 4n + 3$ for all positive integers $n.$ Find $f(1000).$ | 1016 | 0 | 8,192 | -1 | 8,192 | |
A circular sheet of paper with radius of $6$ cm is cut into three congruent sectors. What is the height of the cone in centimeters that can be created by rolling one of the sections until the edges meet? Express your answer in simplest radical form. | 4 \sqrt{2} | 1 | 1,379 | 1,379 | -1 | |
In the third season of "China Poetry Conference", there were many highlights under the theme of "Life has its own poetry". In each of the ten competitions, there was a specially designed opening poem recited in unison by a hundred people under the coordination of lights and dances. The poems included "Changsha Spring i... | 144 | 0 | 7,912.3125 | -1 | 7,912.3125 | |
The set of vectors $\mathbf{u}$ such that
\[\mathbf{u} \cdot \mathbf{u} = \mathbf{u} \cdot \begin{pmatrix} 8 \\ -28 \\ 12 \end{pmatrix}\]forms a solid in space. Find the volume of this solid. | \frac{4}{3} \pi \cdot 248^{3/2} | 0 | 6,769.25 | -1 | 6,769.25 | |
The sum of three numbers \( a \), \( b \), and \( c \) is 150. If we increase \( a \) by 10, decrease \( b \) by 5, and multiply \( c \) by 7, the three resulting numbers are equal. What is the value of \( b \)? | 77.\overline{3} | 0 | 3,277.875 | -1 | 3,277.875 | |
What is the largest integer that must divide the product of any $5$ consecutive integers? | 30 | 0 | 7,158.875 | -1 | 7,158.875 | |
In Ms. Smith's class, each student averages one day absent out of thirty. What is the probability that out of any two students chosen at random, one student will be absent while the other is present? Express your answer as a percent rounded to the nearest tenth. | 6.4 | 1 | 2,555.3125 | 2,555.3125 | -1 | |
June and Julia live 1 mile apart. It takes June 4 minutes to ride her bike directly to Julia's house. At the same rate, how many minutes would it take June to ride the 3.5 miles from her own house to Bernard's house? | 14 | 1 | 1,173.125 | 1,173.125 | -1 | |
Let the random variable $\xi$ follow the normal distribution $N(1, \sigma^2)$ ($\sigma > 0$). If $P(0 < \xi < 1) = 0.4$, then find the value of $P(\xi > 2)$. | 0.2 | 0 | 5,802.3125 | -1 | 5,802.3125 | |
The sides and vertices of a pentagon are labelled with the numbers $1$ through $10$ so that the sum of the numbers on every side is the same. What is the smallest possible value of this sum? | 14 | 0 | 8,192 | -1 | 8,192 | |
What is the remainder when \(2,468,135,790\) is divided by \(99\)? | 54 | 0.125 | 7,256.25 | 5,268 | 7,540.285714 | |
Let \( ABCD \) be a trapezoid such that \( (AB) \) is parallel to \( (CD) \), \( AB = 3 \), \( CD = 3 \), \( DA = 3 \) and \( \widehat{ADC} = 120^\circ \). Determine the angle \( \widehat{CBA} \) in degrees. | 30 | 0 | 7,701.9375 | -1 | 7,701.9375 | |
Given an acute triangle $\triangle ABC$ where the internal angles $A$, $B$, and $C$ are opposite to sides $a$, $b$, and $c$ respectively, and $a=3$, $b=\sqrt{13}$. If $a\sin 2B=b\sin A$. Find:<br/>
$(1)$ the value of angle $B$;<br/>
$(2)$ if point $D$ satisfies $\overrightarrow{BD}=\frac{1}{3}\overrightarrow{BA}+\frac{... | \frac{2\sqrt{19}}{3} | 0 | 5,098 | -1 | 5,098 | |
Emily and John each solved three-quarters of the homework problems individually and the remaining one-quarter together. Emily correctly answered 70% of the problems she solved alone, achieving an overall accuracy of 76% on her homework. John had an 85% success rate with the problems he solved alone. Calculate John's ov... | 87.25\% | 0.375 | 4,559.5625 | 4,693.166667 | 4,479.4 | |
In triangle $XYZ$, the medians $\overline{XT}$ and $\overline{YS}$ are perpendicular. If $XT = 15$ and $YS = 20$, find the length of side $XZ$. | \frac{50}{3} | 0 | 7,884.125 | -1 | 7,884.125 | |
The sides of triangle $DEF$ are in the ratio $3:4:5$. Segment $EG$ is the angle bisector drawn to the shortest side, dividing it into segments $DG$ and $GE$. If the length of side $DE$ (the base) is 12 inches, what is the length, in inches, of the longer segment of side $EF$ once the bisector is drawn from $E$ to $EF$? | \frac{80}{7} | 0.0625 | 7,237.875 | 6,842 | 7,264.266667 | |
Let $n>3$ be a positive integer. Equilateral triangle ABC is divided into $n^2$ smaller congruent equilateral triangles (with sides parallel to its sides). Let $m$ be the number of rhombuses that contain two small equilateral triangles and $d$ the number of rhombuses that contain eight small equilateral triangles. Find... | \[
6n - 9
\] | First we will show that the side lengths of the small triangles are $\tfrac{1}{n}$ of the original length. Then we can count the two rhombuses.
Lemma: Small Triangle is Length of Original Triangle
Let the side length of the triangle be $x$ , so the total area is $\tfrac{x^2 \sqrt{3}}{4}$ .
Since the big triangle is ... | 0 | 8,030.75 | -1 | 8,030.75 |
Find the sum of all three-digit natural numbers that do not contain the digit 0 or the digit 5. | 284160 | 0.5625 | 5,866.25 | 4,057.333333 | 8,192 | |
Find the perimeter of a triangle with points $A(1,2)$, $B(1,8)$, and $C(5,5)$ on a Cartesian coordinate plane. | 16 | 1 | 1,993.3125 | 1,993.3125 | -1 | |
Find all possible values of $\frac{d}{a}$ where $a^{2}-6 a d+8 d^{2}=0, a \neq 0$. | \frac{1}{2}, \frac{1}{4} | Dividing $a^{2}-6 a d+8 d^{2}=0$ by $a^{2}$, we get $1-6 \frac{d}{a}+8\left(\frac{d}{a}\right)^{2}=0$. The roots of this quadratic are $\frac{1}{2}, \frac{1}{4}$. | 0 | 2,126.375 | -1 | 2,126.375 |
In isosceles trapezoid $ABCD$, parallel bases $\overline{AB}$ and $\overline{CD}$ have lengths $500$ and $650$, respectively, and $AD=BC=333$. The angle bisectors of $\angle{A}$ and $\angle{D}$ meet at $P$, and the angle bisectors of $\angle{B}$ and $\angle{C}$ meet at $Q$. Find $PQ$.
Diagram
[asy] /* Made by MRENTHUSI... | 242 | This will be my first solution on AoPS. My apologies in advance for any errors.
Angle bisectors can be thought of as the locus of all points equidistant from the lines whose angle they bisect. It can thus be seen that $P$ is equidistant from $AB, AD,$ and $CD$ and $Q$ is equidistant from $AB, BC,$ and $CD.$ If we let ... | 0.0625 | 8,192 | 8,192 | 8,192 |
How many distinct prime factors does the sum of the positive divisors of $400$ have? | 1 | 0.9375 | 2,565.875 | 2,603.4 | 2,003 | |
Compute $\arccos \frac{\sqrt{3}}{2}.$ Express your answer in radians. | \frac{\pi}{6} | 1 | 1,474.3125 | 1,474.3125 | -1 | |
Given the general term of the sequence $\{a_n\}$ is $a_n=n^2\left(\cos^2 \frac{n\pi}{3}-\sin^2 \frac{n\pi}{3}\right)$, calculate the value of $S_{30}$. | 470 | 0.375 | 7,161 | 6,329.333333 | 7,660 | |
Let $\overline{AD},$ $\overline{BE},$ $\overline{CF}$ be the altitudes of acute triangle $ABC.$ If
\[9 \overrightarrow{AD} + 4 \overrightarrow{BE} + 7 \overrightarrow{CF} = \mathbf{0},\]then compute $\angle ACB,$ in degrees.
[asy]
unitsize (0.6 cm);
pair A, B, C, D, E, F, H;
A = (2,5);
B = (0,0);
C = (8,0);
D = (A ... | 60^\circ | 0.0625 | 8,144.3125 | 7,429 | 8,192 | |
A hotel packed breakfast for each of three guests. Each breakfast should have consisted of three types of rolls, one each of nut, cheese, and fruit rolls. The preparer wrapped each of the nine rolls and once wrapped, the rolls were indistinguishable from one another. She then randomly put three rolls in a bag for each ... | 79 | 0.875 | 6,081.25 | 5,920.928571 | 7,203.5 | |
How many distinct four-digit numbers are divisible by 5 and have 45 as their last two digits? | 90 | 0.9375 | 2,722.0625 | 2,357.4 | 8,192 | |
From 6 athletes, 4 are to be chosen to participate in a 4×100 meters relay race, given that athletes A and B both cannot run the first leg, calculate the number of different methods to select the participants. | 240 | 0.5 | 6,604.5625 | 5,139.875 | 8,069.25 | |
The entire contents of the jug can exactly fill 9 small glasses and 4 large glasses of juice, and also fill 6 small glasses and 6 large glasses. If the entire contents of the jug is used to fill only large glasses, determine the maximum number of large glasses that can be filled. | 10 | 1 | 2,921.875 | 2,921.875 | -1 | |
In a particular right triangle, the lengths of the two legs are 30 inches and 24 inches. Calculate both the area of the triangle and the length of its hypotenuse. | \sqrt{1476} | 0 | 2,498.4375 | -1 | 2,498.4375 | |
Pools $A$ and $B$ are both rectangular cuboids with a length of 3 meters, a width of 2 meters, and a depth of 1.2 meters. Valve 1 is used to fill pool $A$ with water and can fill an empty pool $A$ in 18 minutes. Valve 2 is used to transfer water from pool $A$ to pool $B$, taking 24 minutes to transfer a full pool $A$. ... | 7.2 | 0.375 | 6,329.6875 | 5,123 | 7,053.7 | |
Given Abby finished the softball season with a total of 45 hits, among which were 2 home runs, 3 triples, and 7 doubles, calculate the percentage of her hits that were singles. | 73.33\% | 0.75 | 448.375 | 465.5 | 397 | |
A whole block of modeling clay is a right rectangular prism six inches by two inches by one inch. How many whole blocks need to be unwrapped to mold a cylindrical sculpture seven inches high and four inches in diameter? | 8 | 1 | 2,202.0625 | 2,202.0625 | -1 | |
Given a $24$-inch by $30$-inch pan of brownies, cut into pieces that measure $3$ inches by $4$ inches. Calculate the number of pieces of brownie the pan contains. | 60 | 0.625 | 5,737.6875 | 4,265.1 | 8,192 | |
Express $\frac{165_7}{11_2}+\frac{121_6}{21_3}$ in base 10. | 39 | 1 | 2,170.4375 | 2,170.4375 | -1 | |
In the spring round of the 2000 Cities Tournament, high school students in country $N$ were presented with six problems. Each problem was solved by exactly 1000 students, but no two students together solved all six problems. What is the minimum possible number of high school students in country $N$ who participated in ... | 2000 | 0 | 8,192 | -1 | 8,192 | |
Let $b_1, b_2, \dots$ be a sequence defined by $b_1 = b_2 = 2$ and $b_{n+2} = b_{n+1} + b_n$ for $n \geq 1$. Find
\[
\sum_{n=1}^\infty \frac{b_n}{3^{n+1}}.
\] | \frac{2}{5} | 0.5 | 6,735.9375 | 5,279.875 | 8,192 | |
Given two lines $l_1: ax+3y-1=0$ and $l_2: 2x+(a^2-a)y+3=0$, and $l_1$ is perpendicular to $l_2$, find the value of $a$. | a = \frac{1}{3} | 0.5625 | 6,911.1875 | 5,942.888889 | 8,156.142857 | |
Given points $A(-1,1)$, $B(1,2)$, $C(-2,-1)$, $D(2,2)$, the projection of vector $\overrightarrow{AB}$ in the direction of $\overrightarrow{CD}$ is ______. | \dfrac{11}{5} | 0.875 | 2,822.3125 | 2,667.642857 | 3,905 | |
Equilateral triangle $ABP$ (see figure) with side $AB$ of length $2$ inches is placed inside square $AXYZ$ with side of length $4$ inches so that $B$ is on side $AX$. The triangle is rotated clockwise about $B$, then $P$, and so on along the sides of the square until $P$ returns to its original position. The length of ... | \frac{40\pi}{3} | To solve this problem, we need to understand the path traced by vertex $P$ of the equilateral triangle $ABP$ as it rotates around the square $AXYZ$.
1. **Understanding the Rotation**:
- The triangle $ABP$ is equilateral with side length $2$ inches.
- The square $AXYZ$ has a side length of $4$ inches.
- Verte... | 0 | 7,941.625 | -1 | 7,941.625 |
A tangent line is drawn from a point on the line $y=x$ to the circle $(x-4)^2 + (y+2)^2 = 1$. Find the minimum length of the tangent line. | \sqrt{17} | 1 | 4,964.8125 | 4,964.8125 | -1 | |
Let $n$ be a positive integer. The sequence $(x_k)$ is defined by $x_0 = 0,$ $x_1 = 1,$ and
\[x_{k + 2} = \frac{(n - 1) x_{k + 1} - (n - k) x_k}{k + 1}\]for $k \ge 0.$ Find $x_0 + x_1 + x_2 + \dotsb$ as a function of $n.$ | 2^{n - 1} | 0.125 | 8,032.25 | 6,914 | 8,192 |
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