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 |
|---|---|---|---|---|---|---|
In a tetrahedron \(ABCD\), \(\angle ADB = \angle BDC = \angle CDA = 60^\circ\). The areas of \(\triangle ADB\), \(\triangle BDC\), and \(\triangle CDA\) are \(\frac{\sqrt{3}}{2}\), \(2\), and \(1\) respectively. What is the volume of the tetrahedron? | \frac{2\sqrt{6}}{9} | 0 | 7,924.125 | -1 | 7,924.125 | |
Given the ratio of women to men is $7$ to $5$, and the average age of women is $30$ years and the average age of men is $35$ years, determine the average age of the community. | 32\frac{1}{12} | 0.1875 | 2,583.125 | 2,009 | 2,715.615385 | |
There are 85 beads in total on a string, arranged in the pattern "three green, four red, one yellow, three green, four red, one yellow, ...". How many red beads are there? | 42 | 0.375 | 2,143.5 | 2,130.833333 | 2,151.1 | |
If $\tan \theta = 4,$ then find $\tan 3 \theta.$ | \frac{52}{47} | 1 | 2,574.3125 | 2,574.3125 | -1 | |
Two positive integers \( x \) and \( y \) are such that:
\[ \frac{2010}{2011} < \frac{x}{y} < \frac{2011}{2012} \]
Find the smallest possible value for the sum \( x + y \). | 8044 | 0 | 8,192 | -1 | 8,192 | |
In how many ways can you rearrange the letters of "HMMTHMMT" such that the consecutive substring "HMMT" does not appear? | 361 | There are $8!/(4!2!2!)=420$ ways to order the letters. If the permuted letters contain "HMMT", there are $5 \cdot 4!/ 2!=60$ ways to order the other letters, so we subtract these. However, we have subtracted "HMMTHMMT" twice, so we add it back once to obtain 361 possibilities. | 0 | 7,711.5625 | -1 | 7,711.5625 |
Given real numbers $x$ and $y$ satisfy the equation $x^{2}+y^{2}-4x+1=0$.
$(1)$ Find the maximum and minimum values of $\dfrac {y}{x}$;
$(2)$ Find the maximum and minimum values of $y-x$;
$(3)$ Find the maximum and minimum values of $x^{2}+y^{2}$. | 7-4 \sqrt {3} | 0 | 5,279.375 | -1 | 5,279.375 | |
Given a triangle \( ABC \) with sides \( AB=13 \), \( BC=20 \), and \( AC=21 \). Point \( K \) is on side \( AB \), point \( L \) is on side \( AC \), and point \( N \) is on side \( BC \). It is known that \( AK=4 \), \( CN=1 \), and \( CL=\frac{20}{21} \). A line through point \( K \) parallel to \( NL \) intersects ... | 41 | 0 | 8,192 | -1 | 8,192 | |
Define an ordered quadruple of integers $(a, b, c, d)$ as interesting if $1 \le a<b<c<d \le 10$, and $a+d>b+c$. How many interesting ordered quadruples are there? | 80 | We first start out when the value of $a=1$.
Doing casework, we discover that $d=5,6,7,8,9,10$. We quickly find a pattern.
Now, doing this for the rest of the values of $a$ and $d$, we see that the answer is simply:
$(1)+(2)+(1+3)+(2+4)+(1+3+5)+(2+4+6)+(1)+(2)+(1+3)+(2+4)$ $+(1+3+5)+(1)+(2)+(1+3)+(2+4)+(1)+(2)+(1+3)+(... | 0 | 7,684.1875 | -1 | 7,684.1875 |
If two sides of a triangle are 8 and 15 units, and the angle between them is 30 degrees, what is the length of the third side? | \sqrt{289 - 120\sqrt{3}} | 0 | 7,620.125 | -1 | 7,620.125 | |
2 distinct vertices of an octagon are chosen at random. What is the probability that they are adjacent? | \frac{2}{7} | 1 | 1,931.375 | 1,931.375 | -1 | |
In Lhota, there was an election for the mayor. Two candidates ran: Mr. Schopný and his wife, Dr. Schopná. The village had three polling stations. In the first and second stations, Dr. Schopná received more votes. The vote ratios were $7:5$ in the first station and $5:3$ in the second station. In the third polling stati... | 24 : 24 : 25 | 0.5 | 5,948.0625 | 3,949 | 7,947.125 | |
Amerigo Vespucci has a map of America drawn on the complex plane. The map does not distort distances. Los Angeles corresponds to $0$ on this complex plane, and Boston corresponds to $2600i$. Meanwhile, Knoxville corresponds to the point $780+1040i$. With these city-point correspondences, how far is it from Knoxville to... | 1300 | 1 | 3,480.9375 | 3,480.9375 | -1 | |
The Engan alphabet of a fictional region contains 15 letters: A, B, C, D, E, F, G, H, I, J, K, L, M, N, O. Suppose license plates are to be formed with six letters using only the letters in the Engan alphabet. How many license plates of six letters are possible that begin with either A or B, end with O, cannot contain ... | 34320 | 0.0625 | 6,113.25 | 7,290 | 6,034.8 | |
Find the value of $s$ for which the vector
\[\bold{u} = \begin{pmatrix} 1 \\ -2 \\ -4 \end{pmatrix} + s \begin{pmatrix} 5 \\ 3 \\ -2 \end{pmatrix}\] is closest to
\[\bold{b} = \begin{pmatrix} 3 \\ 3 \\ 4 \end{pmatrix}.\] | \frac{9}{38} | 0.875 | 5,352.3125 | 5,068.571429 | 7,338.5 | |
Find the sum of the first six terms in the geometric sequence $\frac12,\frac14,\frac18,\dots$. Express your answer as a common fraction. | \frac{63}{64} | 1 | 2,535.1875 | 2,535.1875 | -1 | |
In a WeChat group, there are 5 individuals: A, B, C, D, and E, playing a game involving grabbing red envelopes. There are 4 red envelopes, each person may grab at most one, and all red envelopes must be grabbed. Among the 4 red envelopes, there are two 2-yuan envelopes, one 3-yuan envelope, and one 4-yuan envelope (env... | 36 | 0 | 7,938.0625 | -1 | 7,938.0625 | |
Jo and Blair take turns counting from 1, increasing by 1 each turn. However, if the number is a multiple of 3, they skip it. Jo starts by saying "1." Calculate the 53rd number said. | 80 | 0 | 5,600.9375 | -1 | 5,600.9375 | |
Let's call a year interesting if a person turns the number of years equal to the sum of the digits of the year of their birth in that year. A certain year turned out to be interesting for Ivan, who was born in the 20th century, and for Vovochka, who was born in the 21st century. What is the difference in their ages?
N... | 18 | 0.0625 | 7,907.75 | 5,545 | 8,065.266667 | |
There are $100$ students who want to sign up for the class Introduction to Acting. There are three class sections for Introduction to Acting, each of which will fit exactly $20$ students. The $100$ students, including Alex and Zhu, are put in a lottery, and 60 of them are randomly selected to fill up the classes.... | 19/165 | 0.0625 | 7,801.125 | 6,852 | 7,864.4 | |
A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1. What is the area of the rectangle? | 200 | 1. **Identify the relationship between the circle and the rectangle**: The circle is inscribed in the rectangle, which means the diameter of the circle is equal to the height (shorter side) of the rectangle.
2. **Calculate the diameter of the circle**: Given that the radius of the circle is $5$, the diameter is:
\[... | 1 | 3,209.6875 | 3,209.6875 | -1 |
Calculate the probability that all the rational terms are not adjacent to each other when rearranging the terms of the expansion $( \sqrt {x}+ \dfrac {1}{2 \sqrt[4]{x}})^{8}$ in a list. | \frac{5}{12} | 0.375 | 6,473.5 | 4,993.333333 | 7,361.6 | |
For how many integer values of $a$ does the equation $$x^2 + ax + 12a = 0$$ have integer solutions for $x$? | 14 | 0 | 7,730.9375 | -1 | 7,730.9375 | |
Given that the determinant of a $2 \times 2$ matrix $A = \begin{vmatrix} a & b \\ c & d \end{vmatrix}$ is 3, find the values of $\begin{vmatrix} 3a & 3b \\ 3c & 3d \end{vmatrix}$ and $\begin{vmatrix} 4a & 2b \\ 4c & 2d \end{vmatrix}$. | 24 | 0.9375 | 2,304.5625 | 1,912.066667 | 8,192 | |
The repeating decimal for $\frac{3}{11}$ is $0.ababab\dots$ What is the value of the sum $a+b$? | 9 | 1 | 1,874.5625 | 1,874.5625 | -1 | |
Consider a sequence $x_{n}$ such that $x_{1}=x_{2}=1, x_{3}=\frac{2}{3}$. Suppose that $x_{n}=\frac{x_{n-1}^{2} x_{n-2}}{2 x_{n-2}^{2}-x_{n-1} x_{n-3}}$ for all $n \geq 4$. Find the least $n$ such that $x_{n} \leq \frac{1}{10^{6}}$. | 13 | The recursion simplifies to $\frac{x_{n-1}}{x_{n}}+\frac{x_{n-3}}{x_{n-2}}=2 \frac{x_{n-2}}{x_{n-1}}$. So if we set $y_{n}=\frac{x_{n-1}}{x_{n}}$ for $n \geq 2$ then we have $y_{n}-y_{n-1}=y_{n-1}-y_{n-2}$ for $n \geq 3$, which means that $\left\{y_{n}\right\}$ is an arithmetic sequence. From the starting values we hav... | 0 | 8,044.9375 | -1 | 8,044.9375 |
If the graph of the linear function $y=(7-m)x-9$ does not pass through the second quadrant, and the fractional equation about $y$ $\frac{{2y+3}}{{y-1}}+\frac{{m+1}}{{1-y}}=m$ has a non-negative solution, calculate the sum of all integer values of $m$ that satisfy the conditions. | 14 | 0.1875 | 7,144.4375 | 6,634.333333 | 7,262.153846 | |
Let \( A \) be the sum of all non-negative integers \( n \) satisfying
\[ \left\lfloor \frac{n}{27} \right\rfloor = \left\lfloor \frac{n}{28} \right\rfloor. \]
Determine \( A \). | 95004 | 0.0625 | 8,156.8125 | 7,792 | 8,181.133333 | |
An entrepreneur took out a discounted loan of 12 million HUF with a fixed annual interest rate of 8%. What will be the debt after 10 years if they can repay 1.2 million HUF annually? | 8523225 | 0 | 7,634.4375 | -1 | 7,634.4375 | |
Find the greatest positive integer $x$ such that $23^{6+x}$ divides $2000!$ | 83 |
To find the greatest positive integer \( x \) such that \( 23^{6+x} \) divides \( 2000! \), we need to determine how many times the prime factor 23 appears in the prime factorization of \( 2000! \).
The exponent of a prime \( p \) in \( n! \) is given by:
\[
\sum_{k=1}^{\infty} \left\lfloor \frac{n}{p^k} \right\rfloo... | 1 | 2,345.625 | 2,345.625 | -1 |
Given that 20% of the birds are geese, 40% are swans, 10% are herons, and 20% are ducks, and the remaining are pigeons, calculate the percentage of the birds that are not herons and are ducks. | 22.22\% | 0.125 | 411.1875 | 445 | 406.357143 | |
In the vertices of a regular 100-gon, 100 chips numbered $1, 2, \ldots, 100$ are placed in exactly that order in a clockwise direction. During each move, it is allowed to swap two chips placed at adjacent vertices if the numbers on these chips differ by no more than $k$. What is the smallest $k$ such that, in a series ... | 50 | 0.125 | 8,122.3125 | 7,782.5 | 8,170.857143 | |
Let triangle $PQR$ be a right triangle with $\angle PRQ = 90^\circ$. A circle is tangent to the sides $PQ$ and $PR$ at points $S$ and $T$ respectively. The points on the circle diametrically opposite $S$ and $T$ both lie on side $QR$. Given that $PQ = 12$, find the area of the portion of the circle that lies outside tr... | 4\pi - 8 | 0 | 8,192 | -1 | 8,192 | |
Regular hexagon $ABCDEF$ is the base of right pyramid $\allowbreak PABCDEF$. If $PAD$ is an equilateral triangle with side length 8, then what is the volume of the pyramid? | 96 | 1 | 3,315.4375 | 3,315.4375 | -1 | |
Calculate the difference between 12.358 and 7.2943, keeping the answer in decimal form. | 5.0637 | 0.9375 | 549.25 | 542.066667 | 657 | |
How many solutions of the equation $\tan x = \tan 2x$ are in the interval $0 \le x \le \tan^{-1} 500$? Assume $\tan \theta > \theta$ for $0 < \theta < \frac{\pi}{2}$. | 159 | 0 | 7,314.625 | -1 | 7,314.625 | |
For every $m \geq 2$, let $Q(m)$ be the least positive integer with the following property: For every $n \geq Q(m)$, there is always a perfect cube $k^3$ in the range $n < k^3 \leq m \cdot n$. Find the remainder when \[\sum_{m = 2}^{2017} Q(m)\]is divided by 1000. | 59 | We claim that $Q(m) = 1$ when $m \ge 8$.
When $m \ge 8$, for every $n \ge Q(m) = 1$, we need to prove there exists an integer $k$, such that $n < k^3 \le m*n$.
That because $\sqrt[3]{m*n} - \sqrt[3]{n} \ge 2\sqrt[3]{n} - \sqrt[3]{n} = \sqrt[3]{n} \ge 1$, so k exists between $\sqrt[3]{m*n}$ and $\sqrt[3]{n}$
$\sqrt[3]... | 0 | 8,192 | -1 | 8,192 |
Let $a$, $b$, and $c$ be positive real numbers. Find the minimum value of
\[
\frac{(a^2 + 4a + 4)(b^2 + 4b + 4)(c^2 + 4c + 4)}{abc}.
\] | 64 | 0 | 5,416.375 | -1 | 5,416.375 | |
A $3 \times 3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated $90^{\circ}$ clockwise about its center, and every white square in a position formerly occupied by a black sq... | \frac{49}{512} | 1. **Identify the invariant**: The center square remains unchanged after a $90^\circ$ rotation. Since the entire grid must end up black, the center square must initially be black. The probability of this happening is $\frac{1}{2}$.
2. **Consider the effect of rotation on the other squares**: The rotation affects the p... | 0 | 8,129.1875 | -1 | 8,129.1875 |
The general formula of the sequence \\(\{a_n\}\) is \\(a_n=n\cos \frac{n\pi}{2}\\), and the sum of its first \\(n\\) terms is \\(S_n\\). Find \\(S_{2019}\\). | -1010 | 0.3125 | 6,924.5 | 4,793.8 | 7,893 | |
Given that $θ$ is an angle in the second quadrant, if $\tan \left(θ+ \frac {π}{4}\right)= \frac {1}{2}$, calculate the value of $\sin θ-\cos θ$. | \frac {2 \sqrt {10}}{5} | 0 | 5,401.9375 | -1 | 5,401.9375 | |
There are $24$ different complex numbers $z$ such that $z^{24}=1$. For how many of these is $z^6$ a real number? | 12 | 0.5 | 7,008.5 | 5,825 | 8,192 | |
**p4.** What is gcd $(2^6 - 1, 2^9 - 1)$ ?**p5.** Sarah is walking along a sidewalk at a leisurely speed of $\frac12$ m/s. Annie is some distance behind her, walking in the same direction at a faster speed of $s$ m/s. What is the minimum value of $s$ such that Sarah and Annie spend no more than one second within... | \frac{21}{16} | 0 | 7,212.4375 | -1 | 7,212.4375 | |
Determine the sum of all distinct real values of $x$ such that $|||\cdots||x|+x|\cdots|+x|+x|=1$ where there are 2017 $x$ 's in the equation. | -\frac{2016}{2017} | Note that $|x+| x||=2 x$ when $x$ is nonnegative, and is equal to 0 otherwise. Thus, when there are 2017 $x$ 's, the expression equals $2017 x$ when $x \geq 0$ and $-x$ otherwise, so the two solutions to the equation are $x=-1$ and $\frac{1}{2017}$, and their sum is $-\frac{2016}{2017}$. | 0.375 | 6,819.625 | 5,592.5 | 7,555.9 |
The height of a triangle, equal to 2, divides the angle of the triangle in the ratio 2:1, and the base of the triangle into parts, the smaller of which is equal to 1. Find the area of the triangle. | 11/3 | 0 | 7,087.0625 | -1 | 7,087.0625 | |
In a certain book, there were 100 statements written as follows:
1) "In this book, there is exactly one false statement."
2) "In this book, there are exactly two false statements."
...
3) "In this book, there are exactly one hundred false statements."
Which of these statements is true? | 99 | 0.375 | 6,556.9375 | 3,831.833333 | 8,192 | |
In trapezoid \(A B C D\), the base \(A D\) is four times larger than \(B C\). A line passing through the midpoint of diagonal \(B D\) and parallel to \(A B\) intersects side \(C D\) at point \(K\). Find the ratio \(D K : K C\). | 2:1 | 0.3125 | 7,383.5625 | 6,052.4 | 7,988.636364 | |
A math professor stands up in front of a room containing 100 very smart math students and says, 'Each of you has to write down an integer between 0 and 100, inclusive, to guess 'two-thirds of the average of all the responses.' Each student who guesses the highest integer that is not higher than two-thirds of the averag... | 0 | Since the average cannot be greater than 100, no student will write down a number greater than $\frac{2}{3} \cdot 100$. But then the average cannot be greater than $\frac{2}{3} \cdot 100$, and, realizing this, each student will write down a number no greater than $\left(\frac{2}{3}\right)^{2} \cdot 100$. Continuing in ... | 0.9375 | 4,819.5625 | 4,594.733333 | 8,192 |
Among the scalene triangles with natural number side lengths, a perimeter not exceeding 30, and the sum of the longest and shortest sides exactly equal to twice the third side, there are ____ distinct triangles. | 20 | 0.5625 | 7,216.75 | 6,458.222222 | 8,192 | |
Five students play chess matches against each other. Each student plays three matches against each of the other students. How many matches are played in total? | 30 | We label the players as A, B, C, D, and E. The total number of matches played will be equal to the number of pairs of players that can be formed times the number of matches that each pair plays. The possible pairs of players are AB, AC, AD, AE, BC, BD, BE, CD, CE, and DE. There are 10 such pairs. Thus, the total number... | 1 | 383.375 | 383.375 | -1 |
In rectangle $ABCD$, $AB = 12$ and $BC = 10$. Points $E$ and $F$ lie inside rectangle $ABCD$ so that $BE = 9$, $DF = 8$, $\overline{BE} \parallel \overline{DF}$, $\overline{EF} \parallel \overline{AB}$, and line $BE$ intersects segment $\overline{AD}$. The length $EF$ can be expressed in the form $m \sqrt{n} - p$, wher... | 36 | Extend lines $BE$ and $CD$ to meet at point $G$. Draw the altitude $GH$ from point $G$ to line $BA$ extended.
$GE=DF=8,$ $GB=17$
In right $\bigtriangleup GHB$, $GH=10$, $GB=17$, thus by Pythagoras Theorem we have: $HB=\sqrt{17^2-10^2}=3\sqrt{21}$
$HA=EF=3\sqrt{21}-12$
Thus our answer is: $3+21+12=\boxed{36}$ | 0.4375 | 7,480.75 | 6,704.428571 | 8,084.555556 |
Given real numbers $a$ and $b$ satisfying the equation $\sqrt{(a-1)^2} + \sqrt{(a-6)^2} = 10 - |b+3| - |b-2|$, find the maximum value of $a^2 + b^2$. | 45 | 0.8125 | 4,915.5 | 4,268.692308 | 7,718.333333 | |
A circle of radius $r$ has chords $\overline{AB}$ of length $10$ and $\overline{CD}$ of length 7. When $\overline{AB}$ and $\overline{CD}$ are extended through $B$ and $C$, respectively, they intersect at $P$, which is outside of the circle. If $\angle{APD}=60^\circ$ and $BP=8$, then $r^2=$
$\text{(A) } 70\quad \text{(... | 73 | 0 | 7,686.625 | -1 | 7,686.625 | |
(1) Simplify: $\dfrac{\sin(\pi -\alpha)\cos(\pi +\alpha)\sin(\dfrac{\pi}{2}+\alpha)}{\sin(-\alpha)\sin(\dfrac{3\pi}{2}+\alpha)}$.
(2) Given $\alpha \in (\dfrac{\pi}{2}, \pi)$, and $\sin(\pi -\alpha) + \cos \alpha = \dfrac{7}{13}$, find $\tan \alpha$. | -\dfrac{12}{5} | 0.8125 | 5,238.1875 | 4,864 | 6,859.666667 | |
A bug starts at a vertex of a square. On each move, it randomly selects one of the three vertices where it is not currently located, and crawls along a side of the square to that vertex. Determine the probability that the bug returns to its starting vertex on its eighth move and express this probability in lowest terms... | 2734 | 0.3125 | 7,436.25 | 6,108.4 | 8,039.818182 | |
Given an ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, where the upper vertex of $C$ is $A$, and the two foci are $F_{1}$ and $F_{2}$, with an eccentricity of $\frac{1}{2}$. A line passing through $F_{1}$ and perpendicular to $AF_{2}$ intersects $C$ at points $D$ and $E$, where $|DE| = 6$. Find the ... | 13 | 0 | 8,050.0625 | -1 | 8,050.0625 | |
A tetrahedron has all its faces triangles with sides $13,14,15$. What is its volume? | 42 \sqrt{55} | Let $A B C$ be a triangle with $A B=13, B C=14, C A=15$. Let $A D, B E$ be altitudes. Then $B D=5, C D=9$. (If you don't already know this, it can be deduced from the Pythagorean Theorem: $C D^{2}-B D^{2}=\left(C D^{2}+A D^{2}\right)-\left(B D^{2}+A D^{2}\right)=A C^{2}-A B^{2}=56$, while $C D+B D=B C=14$, giving $C D-... | 0 | 8,192 | -1 | 8,192 |
The sum of the squares of three consecutive positive integers is 7805. What is the sum of the cubes of the three original integers? | 398259 | 1 | 3,898 | 3,898 | -1 | |
A train moves at a speed of 60 kilometers per hour, making stops every 48 kilometers. The duration of each stop, except for the fifth stop, is 10 minutes. The fifth stop lasts half an hour. What distance did the train travel if it departed at noon on September 29 and arrived at its destination on October 1 at 22:00? | 2870 | 0 | 7,660.3125 | -1 | 7,660.3125 | |
We consider a white \( 5 \times 5 \) square consisting of 25 unit squares. How many different ways are there to paint one or more of the unit squares black so that the resulting black area forms a rectangle? | 225 | 0.5625 | 5,213.5625 | 4,197.444444 | 6,520 | |
The perimeter of $\triangle ABC$ is equal to the perimeter of rectangle $DEFG$. What is the area of $\triangle ABC$? | 168 | The perimeter of $\triangle ABC$ is equal to $(3x+4)+(3x+4)+2x=8x+8$. The perimeter of rectangle $DEFG$ is equal to $2 \times (2x-2)+2 \times (3x-1)=4x-4+6x-2=10x-6$. Since these perimeters are equal, we have $10x-6=8x+8$ which gives $2x=14$ and so $x=7$. Thus, $\triangle ABC$ has $AC=2 \times 7=14$ and $AB=BC=3 \times... | 0 | 7,679.1875 | -1 | 7,679.1875 |
Given that $f(\alpha) = \left(\sqrt{\frac{1 - \sin{\alpha}}{1 + \sin{\alpha}}} + \sqrt{\frac{1 + \sin{\alpha}}{1 - \sin{\alpha}}}\right)\cos^3{\alpha} + 2\sin{\left(\frac{\pi}{2} + \alpha\right)}\cos{\left(\frac{3\pi}{2} + \alpha\right)}$ (where $\alpha$ is an angle in the third quadrant),
(I) find the value of $f(\alp... | \frac{3}{4} | 0.625 | 6,751.75 | 6,054 | 7,914.666667 | |
Given the state income tax rate is $q\%$ for the first $\$30000$ of yearly income plus $(q + 1)\%$ for any amount above $\$30000$, and Samantha's state income tax amounts to $(q + 0.5)\%$ of her total annual income, determine Samantha's annual income. | 60000 | 0.9375 | 3,126.125 | 2,788.4 | 8,192 | |
Find the area of triangle $JKL$ below.
[asy]
unitsize(1inch);
pair P,Q,R;
P = (0,0);
Q= (sqrt(3),0);
R = (0,1);
draw (P--Q--R--P,linewidth(0.9));
draw(rightanglemark(Q,P,R,3));
label("$J$",P,S);
label("$K$",Q,S);
label("$L$",R,N);
label("$20$",(Q+R)/2,NE);
label("$60^\circ$",(0,0.75),E);
[/asy] | 50\sqrt{3} | 0.8125 | 1,566.5 | 1,794.769231 | 577.333333 | |
Given \\(|a|=1\\), \\(|b|= \sqrt{2}\\), and \\(a \perp (a-b)\\), the angle between vector \\(a\\) and vector \\(b\\) is ______. | \frac{\pi}{4} | 0.375 | 2,253.6875 | 3,037.833333 | 1,783.2 | |
Let $S_n$ be the sum of the first $n$ terms of the sequence $\{a_n\}$. If $2a_n + (-1)^n \cdot a_n = 2^n + (-1)^n \cdot 2^n$ ($n \in \mathbb{N}^*$), then $S_{10}=$ \_\_\_\_\_\_. | \dfrac{2728}{3} | 0.8125 | 6,234.375 | 5,782.615385 | 8,192 | |
In a certain competition, two players of equal level meet in the final. The final is played in a best-of-five format, with the winner receiving all the prize money. When the score was $2:1$ after the first three games, the competition was terminated due to unforeseen circumstances. If the total prize money is $12000$ y... | 9000 | 0 | 5,196.8125 | -1 | 5,196.8125 | |
What is the smallest positive integer $n$ such that $\sqrt{n}-\sqrt{n-1}<0.02$?
A) 624
B) 625
C) 626
D) 627
E) 628 | 626 | 0 | 7,091.75 | -1 | 7,091.75 | |
Square $ABCD$ has area $200$. Point $E$ lies on side $\overline{BC}$. Points $F$ and $G$ are the midpoints of $\overline{AE}$ and $\overline{DE}$, respectively. Given that quadrilateral $BEGF$ has area $34$, what is the area of triangle $GCD$? | 41 | 0.6875 | 6,500.4375 | 6,073.090909 | 7,440.6 | |
Determine which positive integers $n$ have the following property: For all integers $m$ that are relatively prime to $n$, there exists a permutation $\pi\colon \{1,2,\dots,n\} \to \{1,2,\dots,n\}$ such that $\pi(\pi(k)) \equiv mk \pmod{n}$ for all $k \in \{1,2,\dots,n\}$. | n = 1 \text{ or } n \equiv 2 \pmod{4} | The desired property holds if and only if $n = 1$ or $n \equiv 2 \pmod{4}$. Let $\sigma_{n,m}$ be the permutation of $\ZZ/n\ZZ$ induced by multiplication by $m$; the original problem asks for which $n$ does $\sigma_{n,m}$ always have a square root. For $n=1$, $\sigma_{n,m}$ is the identity permutation and hence has a s... | 0 | 8,192 | -1 | 8,192 |
If $a$ and $b$ are integers with $a > b$, what is the smallest possible positive value of $\frac{a+b}{a-b} + \frac{a-b}{a+b}$? | 2 | 0.5 | 7,244.6875 | 6,297.375 | 8,192 | |
In a circle, parallel chords of lengths 2, 3, and 4 determine central angles of $\alpha$, $\beta$, and $\alpha + \beta$ radians, respectively, where $\alpha + \beta < \pi$. If $\cos \alpha$, which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominator... | 49 | Using the first diagram above, \[\sin \frac{\alpha}{2} = \frac{1}{r}\] \[\sin \frac{\beta}{2} = \frac{1.5}{r}\] \[\sin(\frac{\alpha}{2}+\frac{\beta}{2})=\frac{2}{r}\] by the Pythagorean trig identities, \[\cos\frac{\alpha}{2}=\sqrt{1-\frac{1}{r^2}}\] \[\cos\frac{\beta}{2}=\sqrt{1-\frac{2.25}{r^2}}\] so by the composite... | 0.875 | 4,898.75 | 4,428.285714 | 8,192 |
Evaluate \(\sqrt{114 + 44\sqrt{6}}\) and express it in the form \(x + y\sqrt{z}\), where \(x\), \(y\), and \(z\) are integers and \(z\) has no square factors other than 1. Find \(x + y + z\). | 13 | 0 | 8,192 | -1 | 8,192 | |
Start by writing the integers $1,2,4,6$ on the blackboard. At each step, write the smallest positive integer $n$ that satisfies both of the following properties on the board. - $n$ is larger than any integer on the board currently. - $n$ cannot be written as the sum of 2 distinct integers on the board. Find the 100-th ... | 388 | The sequence goes $1,2,4,6,9,12,17,20,25, \ldots$. Common differences are $5,3,5,3,5,3, \ldots$, starting from 12. Therefore, the answer is $12+47 \times 8=388$. | 0 | 8,192 | -1 | 8,192 |
In trapezoid $PQRS$, the lengths of the bases $PQ$ and $RS$ are 10 and 20, respectively. The height of the trapezoid from $PQ$ to $RS$ is 6 units. The legs of the trapezoid are extended beyond $P$ and $Q$ to meet at point $T$. What is the ratio of the area of triangle $TPQ$ to the area of trapezoid $PQRS$? | \frac{1}{3} | 0.5 | 6,267.25 | 4,658.875 | 7,875.625 | |
Let the two foci of the conic section $C$ be $F_1$ and $F_2$, respectively. If there exists a point $P$ on curve $C$ such that the ratio $|PF_1| : |F_1F_2| : |PF_2| = 4 : 3 : 2$, determine the eccentricity of curve $C$. | \frac{3}{2} | 0.125 | 6,251.4375 | 5,005.5 | 6,429.428571 | |
Let $\{a_n\}$ be an arithmetic sequence. If we select any 4 different numbers from $\{a_1, a_2, a_3, \ldots, a_{10}\}$ such that these 4 numbers still form an arithmetic sequence, then there are at most \_\_\_\_\_\_ such arithmetic sequences. | 24 | 0 | 7,644.3125 | -1 | 7,644.3125 | |
Given $\overrightarrow{m}=(2\sqrt{3},1)$, $\overrightarrow{n}=(\cos^2 \frac{A}{2},\sin A)$, where $A$, $B$, and $C$ are the interior angles of $\triangle ABC$;
$(1)$ When $A= \frac{\pi}{2}$, find the value of $|\overrightarrow{n}|$;
$(2)$ If $C= \frac{2\pi}{3}$ and $|AB|=3$, when $\overrightarrow{m} \cdot \overrigh... | \sqrt{3} | 0.875 | 3,691.8125 | 3,678.714286 | 3,783.5 | |
The data set $[6, 19, 33, 33, 39, 41, 41, 43, 51, 57]$ has median $Q_2 = 40$, first quartile $Q_1 = 33$, and third quartile $Q_3 = 43$. An outlier in a data set is a value that is more than $1.5$ times the interquartile range below the first quartle ($Q_1$) or more than $1.5$ times the interquartile range above the thi... | 1 | 1. **Identify the Interquartile Range (IQR):**
The interquartile range (IQR) is calculated as $Q_3 - Q_1$. Given $Q_1 = 33$ and $Q_3 = 43$, we have:
\[
\text{IQR} = Q_3 - Q_1 = 43 - 33 = 10.
\]
2. **Calculate the Outlier Thresholds:**
Outliers are defined as values that are more than $1.5 \times \text{I... | 1 | 2,205.1875 | 2,205.1875 | -1 |
Peter Ivanovich, along with 49 other men and 50 women, are seated in a random order around a round table. We call a man satisfied if a woman is sitting next to him. Find:
a) The probability that Peter Ivanovich is satisfied.
b) The expected number of satisfied men. | \frac{1250}{33} | 0.125 | 6,631.75 | 5,709 | 6,763.571429 | |
For points P and Q on the curve $y = 1 - x^2$, which are situated on opposite sides of the y-axis, find the minimum area of the triangle formed by the tangents at P and Q and the x-axis. | \frac{8 \sqrt{3}}{9} | 0 | 7,607.9375 | -1 | 7,607.9375 | |
Ten distinct points are identified on the circumference of a circle. How many different convex quadrilaterals can be formed if each vertex must be one of these 10 points? | 210 | 1 | 1,819.625 | 1,819.625 | -1 | |
A strictly increasing sequence of positive integers $a_1$, $a_2$, $a_3$, $\cdots$ has the property that for every positive integer $k$, the subsequence $a_{2k-1}$, $a_{2k}$, $a_{2k+1}$ is geometric and the subsequence $a_{2k}$, $a_{2k+1}$, $a_{2k+2}$ is arithmetic. Suppose that $a_{13} = 2016$. Find $a_1$. | 504 | We first create a similar sequence where $a_1=1$ and $a_2=2$. Continuing the sequence,
\[1, 2,4,6,9,12,16,20,25,30,36,42,49,\cdots\]
Here we can see a pattern; every second term (starting from the first) is a square, and every second term (starting from the third) is the end of a geometric sequence. This can be proven... | 0 | 8,192 | -1 | 8,192 |
Let \(\mathbf{A} = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\) and \(\mathbf{B} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\) be two matrices such that \(\mathbf{A} \mathbf{B} = \mathbf{B} \mathbf{A}\). Assuming \(4b \neq c\), find \(\frac{a - 2d}{c - 4b}\). | \frac{3}{10} | 0 | 7,692.875 | -1 | 7,692.875 | |
When a class of math students lined up for a class picture on Picture Day, they found that when they stood in rows of four there was one person left over. When they formed rows of five students, there were two extra students, and when they lined up in rows of seven, there were three students left over. What is the fewe... | 17 | 1 | 3,033.75 | 3,033.75 | -1 | |
Define $f(x)=3x-8$. If $f^{-1}$ is the inverse of $f$, find the value(s) of $x$ for which $f(x)=f^{-1}(x)$. | 4 | 1 | 1,977.125 | 1,977.125 | -1 | |
Let $ABCD$ be a convex quadrilateral with $\angle ABD = \angle BCD$ , $AD = 1000$ , $BD = 2000$ , $BC = 2001$ , and $DC = 1999$ . Point $E$ is chosen on segment $DB$ such that $\angle ABD = \angle ECD$ . Find $AE$ . | 1000 | 0 | 8,192 | -1 | 8,192 | |
Find the value of the expression \(\cos ^{4} \frac{7 \pi}{24}+\sin ^{4} \frac{11 \pi}{24}+\sin ^{4} \frac{17 \pi}{24}+\cos ^{4} \frac{13 \pi}{24}\). | \frac{3}{2} | 0.0625 | 7,923.875 | 5,999 | 8,052.2 | |
An isosceles right triangle with legs of length $8$ is partitioned into $16$ congruent triangles as shown. The shaded area is | 20 |
#### Step 1: Understand the problem
We are given an isosceles right triangle with legs of length $8$. This triangle is partitioned into $16$ congruent smaller triangles, and we need to find the area of the shaded region which consists of $10$ of these smaller triangles.
#### Step 2: Calculate the area of the large tr... | 0 | 6,002.3125 | -1 | 6,002.3125 |
Let $n$ be given, $n \geq 4$, and suppose that $P_1, P_2, \dots, P_n$ are $n$ randomly, independently and uniformly, chosen points on a circle. Consider the convex $n$-gon whose vertices are the $P_i$. What is the probability that at least one of the vertex angles of this polygon is acute? | n(n-2) 2^{-n+1} | The angle at a vertex $P$ is acute if and only if all of the other points lie on an open semicircle. We first deduce from this that if there are any two acute angles at all, they must occur consecutively. Suppose the contrary; label the vertices $Q_1, \dots, Q_n$ in counterclockwise order (starting anywhere), and suppo... | 0 | 8,168.625 | -1 | 8,168.625 |
Given that the line passing through the focus of the parabola $y^2=x$ intersects the parabola at points A and B, and O is the origin of the coordinates, calculate $\overrightarrow {OA}\cdot \overrightarrow {OB}$. | -\frac{3}{16} | 0.75 | 6,520.75 | 5,963.666667 | 8,192 | |
Due to a snow and ice disaster, a citrus orchard suffered severe damage. To address this, experts proposed a rescue plan for the fruit trees, which needs to be implemented over two years and is independent each year. The plan estimates that in the first year, the probability of the citrus yield recovering to 1.0, 0.9, ... | 0.2 | 0.0625 | 4,168.1875 | 5,283 | 4,093.866667 | |
Given the sets \( A = \{2, 0, 1, 7\} \) and \( B = \{ x \mid x^2 - 2 \in A, \, x - 2 \notin A \} \), the product of all elements in set \( B \) is: | 36 | 0.875 | 4,927.6875 | 4,812.642857 | 5,733 | |
Which terms must be removed from the sum
$\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\frac{1}{10}+\frac{1}{12}$
if the sum of the remaining terms is to equal $1$? | \frac{1}{8} \text{ and } \frac{1}{10} | 1. **Calculate the total sum of all terms**: We start by finding the sum of all the terms given in the problem:
\[
\frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8} + \frac{1}{10} + \frac{1}{12}
\]
To add these fractions, we need a common denominator. The least common multiple of the denominators (2, 4,... | 0.1875 | 6,595 | 5,753 | 6,789.307692 |
A teacher wants to arrange 3 copies of Introduction to Geometry and 4 copies of Introduction to Number Theory on a bookshelf. In how many ways can he do that? | 35 | 1 | 1,980.6875 | 1,980.6875 | -1 | |
China was the first country in the world to use negative numbers. Li Heng, in the book "Fa Jing" written during the Warring States period, already used negative numbers. If the year 500 BC is written as $-500$ years, then the year 2024 AD should be written as ______ years. | +2024 | 0.125 | 2,555.0625 | 410 | 2,861.5 | |
Let $C$ be a unit cube and let $p$ denote the orthogonal projection onto the plane. Find the maximum area of $p(C)$ . | \sqrt{3} | 0.1875 | 8,177.4375 | 8,114.333333 | 8,192 | |
If $4^6=8^n$, what is $n$? | 4 | 1 | 1,859.25 | 1,859.25 | -1 | |
Rachel and Steven play games of chess. If either wins two consecutive games, they are declared the champion. The probability that Rachel will win any given game is 0.6, the probability that Steven will win any given game is 0.3, and the probability that any given game is drawn is 0.1. Find the value of \(1000P\), where... | 343 | 0 | 8,173.3125 | -1 | 8,173.3125 | |
A round table has radius $4$. Six rectangular place mats are placed on the table. Each place mat has width $1$ and length $x$ as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length $x$. Further, the mats are positioned so th... | $\frac{3\sqrt{7}-\sqrt{3}}{2}$ | 1. **Understanding the Geometry**: The problem describes a round table with radius $4$ and six rectangular place mats each with width $1$ and length $x$. Each place mat has two corners on the edge of the table, and the inner corners of each mat touch the inner corner of an adjacent mat.
2. **Analyzing the Placement**:... | 0 | 7,858.75 | -1 | 7,858.75 |
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