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 |
|---|---|---|---|---|---|---|
Consider two lines $p$ and $q$ in a coordinate plane with equations $y = -3x + 9$ and $y = -6x + 9$, respectively. Determine the probability that a point randomly selected in the first quadrant and below line $p$ will fall between $p$ and $q$. | 0.5 | 0.25 | 5,806.25 | 7,038.25 | 5,395.583333 | |
The function
$$
\begin{aligned}
y= & |x-1|+|2x-1|+|3x-1|+ \\
& |4x-1|+|5x-1|
\end{aligned}
$$
achieves its minimum value when the variable $x$ equals ______. | $\frac{1}{3}$ | 0 | 6,865.1875 | -1 | 6,865.1875 | |
The distances between the points on a line are given as $2, 4, 5, 7, 8, k, 13, 15, 17, 19$. Determine the value of $k$. | 12 | 0 | 8,192 | -1 | 8,192 | |
There are some bullfinches in a pet store. One of the children exclaimed, "There are more than fifty bullfinches!" Another replied, "Don't worry, there are fewer than fifty bullfinches." The mother added, "At least there is one!" The father concluded, "Only one of your statements is true." Can you determine how many bu... | 50 | 0.625 | 6,201.625 | 5,007.4 | 8,192 | |
Given a line $l$ passing through point $A(1,1)$ with a slope of $-m$ ($m>0$) intersects the x-axis and y-axis at points $P$ and $Q$, respectively. Perpendicular lines are drawn from $P$ and $Q$ to the line $2x+y=0$, and the feet of the perpendiculars are $R$ and $S$. Find the minimum value of the area of quadrilateral ... | 3.6 | 0 | 7,884.875 | -1 | 7,884.875 | |
If the graph of the power function $y=mx^{\alpha}$ (where m and $\alpha \in \mathbb{R}$) passes through the point $(8, \frac{1}{4})$, then $\alpha$ equals \_\_\_\_\_\_. | -\frac{2}{3} | 0.9375 | 5,029.75 | 4,818.933333 | 8,192 | |
Let $S$ be the set of positive real numbers. Let $f : S \to \mathbb{R}$ be a function such that
\[f(x) f(y) = f(xy) + 2023 \left( \frac{2}{x} + \frac{2}{y} + 2022 \right)\] for all $x, y > 0.$
Let $n$ be the number of possible values of $f(2)$, and let $s$ be the sum of all possible values of $f(2)$. Find $n \times s.... | 2023 | 0 | 8,192 | -1 | 8,192 | |
Let $a$ and $b$ be positive integers satisfying $\frac{ab+1}{a+b} < \frac{3}{2}$. The maximum possible value of $\frac{a^3b^3+1}{a^3+b^3}$ is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. | 36 | Notice that for $\frac{a^3b^3+1}{a^3+b^3}$ to be maximized, $\frac{ab+1}{a+b}$ has to be maximized. We simplify as above to $2ab + 2 < 3a + 3b$, which is $(a-\frac{3}{2})(b-\frac{3}{2}) < \frac{5}{4}$. To maximize, $a$ has to be as close to $b$ as possible, making $a$ close to $\frac{3+\sqrt{5}}{2}$. Because $a$ and $b... | 0.5625 | 7,149.4375 | 6,338.555556 | 8,192 |
You start with a single piece of chalk of length 1. Every second, you choose a piece of chalk that you have uniformly at random and break it in half. You continue this until you have 8 pieces of chalk. What is the probability that they all have length $\frac{1}{8}$ ? | \frac{1}{63} | There are 7! total ways to break the chalks. How many of these result in all having length $\frac{1}{8}$ ? The first move gives you no choice. Then, among the remaining 6 moves, you must apply 3 breaks on the left side and 3 breaks on the right side, so there are $\binom{6}{3}=20$ ways to order those. On each side, you... | 0 | 7,687.5625 | -1 | 7,687.5625 |
If $a$ and $b$ are positive integers for which $ab - 3a + 4b = 137$, what is the minimal possible value of $|a - b|$? | 13 | 1 | 4,338.5625 | 4,338.5625 | -1 | |
Let $W,X,Y$ and $Z$ be four different digits selected from the set
$\{ 1,2,3,4,5,6,7,8,9\}.$
If the sum $\dfrac{W}{X} + \dfrac{Y}{Z}$ is to be as small as possible, then $\dfrac{W}{X} + \dfrac{Y}{Z}$ must equal | \frac{25}{72} | To minimize the sum $\frac{W}{X} + \frac{Y}{Z}$, where $W, X, Y, Z$ are distinct digits from the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$, we need to choose $W$ and $Y$ to be as small as possible and $X$ and $Z$ to be as large as possible. This is because the smaller the numerator and the larger the denominator, the smaller... | 0 | 8,192 | -1 | 8,192 |
Find all the solutions to
\[\frac{1}{x^2 + 11x - 8} + \frac{1}{x^2 + 2x - 8} + \frac{1}{x^2 - 13x - 8} = 0.\]Enter all the solutions, separated by commas. | 8,1,-1,-8 | 0 | 5,465.6875 | -1 | 5,465.6875 | |
The denominators of two irreducible fractions are 600 and 700. What is the smallest possible value of the denominator of their sum (when written as an irreducible fraction)? | 168 | 0.125 | 8,116.1875 | 7,682 | 8,178.214286 | |
In $\triangle PQR$, points $M$ and $N$ lie on $\overline{PQ}$ and $\overline{PR}$, respectively. If $\overline{PM}$ and $\overline{QN}$ intersect at point $S$ so that $PS/SM = 4$ and $QS/SN = 3$, what is $RN/NQ$? | \frac{4}{3} | 0 | 8,192 | -1 | 8,192 | |
In bridge, a standard 52-card deck is dealt in the usual way to 4 players. By convention, each hand is assigned a number of "points" based on the formula $$4 \times(\# \mathrm{~A} \text { 's })+3 \times(\# \mathrm{~K} \text { 's })+2 \times(\# \mathrm{Q} \text { 's })+1 \times(\# \mathrm{~J} \text { 's })$$ Given that ... | \frac{197}{1820} | Obviously, we can ignore the cards lower than J. Simply enumerate the ways to get at least 13 points: AAAA (1), AAAK (16), AAAQ (16), AAAJ (16), AAKK (36), AAKQ (96), AKKK (16). The numbers in parentheses represent the number of ways to choose the suits, given the choices for the values. We see that there are a total o... | 0 | 8,192 | -1 | 8,192 |
Sixteen is 64$\%$ of what number? | 25 | 1 | 1,719.5625 | 1,719.5625 | -1 | |
Square $ABCD$ is inscribed in a circle. Square $EFGH$ has vertices $E$ and $F$ on $\overline{CD}$ and vertices $G$ and $H$ on the circle. If the area of square $ABCD$ is $1$, then the area of square $EFGH$ can be expressed as $\frac {m}{n}$ where $m$ and $n$ are relatively prime positive integers and $m < n$. Find $10n... | 251 | 0.3125 | 7,930.4375 | 7,355 | 8,192 | |
On a circle, there are 2018 points. Each of these points is labeled with an integer. Each number is greater than the sum of the two numbers that immediately precede it in a clockwise direction.
Determine the maximum possible number of positive numbers that can be among the 2018 numbers. | 1008 | 0 | 8,192 | -1 | 8,192 | |
Given $S = \{1, 2, 3, 4\}$. Let $a_{1}, a_{2}, \cdots, a_{k}$ be a sequence composed of numbers from $S$, which includes all permutations of $(1, 2, 3, 4)$ that do not end with 1. That is, if $\left(b_{1}, b_{2}, b_{3}, b_{4}\right)$ is a permutation of $(1, 2, 3, 4)$ and $b_{4} \neq 1$, then there exist indices $1 \le... | 11 | 0 | 7,936.8125 | -1 | 7,936.8125 | |
If \( \sqrt{\frac{3}{x} + 3} = \frac{5}{3} \), solve for \( x \). | -\frac{27}{2} | 1 | 2,383.25 | 2,383.25 | -1 | |
Add $2175_{9} + 1714_{9} + 406_9$. Express your answer in base $9$. | 4406_{9} | 0 | 5,431.5625 | -1 | 5,431.5625 | |
Two dice are made so that the chances of getting an even sum are twice that of getting an odd sum. What is the probability of getting an odd sum in a single roll of these two dice?
(a) \(\frac{1}{9}\)
(b) \(\frac{2}{9}\)
(c) \(\frac{4}{9}\)
(d) \(\frac{5}{9}\) | $\frac{4}{9} | 0 | 8,002.8125 | -1 | 8,002.8125 | |
Three students $A, B$ and $C$ are traveling from a location on the National Highway No. $5$ on direction to Hanoi for participating the HOMC $2018$ . At beginning, $A$ takes $B$ on the motocycle, and at the same time $C$ rides the bicycle. After one hour and a half, $B$ switches to a bicycle and immediate... | 100 | 0.0625 | 7,902.3125 | 4,213 | 8,148.266667 | |
The line \( l: (2m+1)x + (m+1)y - 7m - 4 = 0 \) intersects the circle \( C: (x-1)^{2} + (y-2)^{2} = 25 \) to form the shortest chord length of \(\qquad \). | 4 \sqrt{5} | 0.75 | 6,768.5625 | 6,294.083333 | 8,192 | |
Solve the equation \( 2 \sqrt{2} \sin ^{3}\left(\frac{\pi x}{4}\right) = \sin \left(\frac{\pi}{4}(1+x)\right) \).
How many solutions of this equation satisfy the condition: \( 2000 \leq x \leq 3000 \)? | 250 | 0.125 | 8,152.125 | 7,873 | 8,192 | |
There are 3 screw-in light bulbs and 5 bayonet light bulbs in the box, and light bulbs are randomly drawn without replacement until the 5th light bulb is drawn to have all the screw-in light bulbs. Calculate the probability of this event. | \frac{3}{28} | 0.1875 | 7,602.875 | 7,355.666667 | 7,659.923077 | |
The equation of a parabola is $y^2 + 6y + 2x + 5 = 0.$ Find the vertex of the parabola. | (2,-3) | 1 | 1,787.1875 | 1,787.1875 | -1 | |
Consider an isosceles right triangle with leg lengths of 1 each. Inscribed in this triangle is a square in such a way that one vertex of the square coincides with the right-angle vertex of the triangle. Another square with side length $y$ is inscribed in an identical isosceles right triangle where one side of the squar... | \sqrt{2} | 0 | 8,192 | -1 | 8,192 | |
Find the coefficient of \(x^8\) in the polynomial expansion of \((1-x+2x^2)^5\). | 80 | 0 | 7,080.25 | -1 | 7,080.25 | |
There are 5 integers written on the board. By summing them in pairs, the following set of 10 numbers is obtained: 2, 6, 10, 10, 12, 14, 16, 18, 20, 24. Determine which numbers are written on the board and write their product as the answer. | -3003 | 0.6875 | 6,289.625 | 5,424.909091 | 8,192 | |
Given that $\alpha \in (-\frac{\pi }{2},\frac{\pi }{2})$, $\beta \in (-\frac{\pi }{2},\frac{\pi }{2})$, and $\tan \alpha$ and $\tan \beta$ are the two real roots of the equation $x^{2}+3\sqrt{3}x+4=0$, find the value of $\alpha + \beta$ = \_\_\_\_\_\_\_\_\_\_\_\_. | - \frac{2\pi}{3} | 0.9375 | 5,887.5 | 5,733.866667 | 8,192 | |
Find the smallest positive integer \( n \) such that a cube with side length \( n \) can be divided into 1996 smaller cubes, each with side length a positive integer. | 13 | 0.1875 | 7,744.125 | 6,234 | 8,092.615385 | |
If $\theta \in (0^\circ, 360^\circ)$ and the terminal side of angle $\theta$ is symmetric to the terminal side of the $660^\circ$ angle with respect to the x-axis, and point $P(x, y)$ is on the terminal side of angle $\theta$ (not the origin), find the value of $$\frac {xy}{x^{2}+y^{2}}.$$ | \frac {\sqrt {3}}{4} | 0 | 6,140.8125 | -1 | 6,140.8125 | |
The base six number $53_{6}$ is equal to the base $b$ number $113_{b}$. What is the positive value of $b$? | 5 | 1 | 2,174 | 2,174 | -1 | |
Li Shuang rides a bike from location $A$ to location $B$ at a speed of 320 meters per minute. On the way, due to a bike malfunction, he pushes the bike and continues walking for 5 minutes to a location 1800 meters from $B$ to repair the bike. Fifteen minutes later, he resumes riding towards $B$ at 1.5 times his origina... | 72 | 0.25 | 6,961.375 | 3,560.5 | 8,095 | |
Given the polar equation of curve $C$ is $\rho=1$, with the pole as the origin of the Cartesian coordinate system, and the polar axis as the positive half-axis of $x$, establish the Cartesian coordinate system. The parametric equation of line $l$ is $\begin{cases} x=-1+4t \\ y=3t \end{cases}$ (where $t$ is the paramete... | \dfrac {8}{5} | 1 | 2,953.125 | 2,953.125 | -1 | |
The price (in euros) of a diamond corresponds to its mass (in grams) squared and then multiplied by 100. The price (in euros) of a crystal corresponds to three times its mass (in grams).
Martin and Théodore unearth a treasure consisting of precious stones that are either diamonds or crystals and whose total value is €... | 2000000 | 0.25 | 7,568 | 5,696 | 8,192 | |
In a wooden block shaped like a cube, all the vertices and edge midpoints are marked. The cube is cut along all possible planes that pass through at least four marked points. Let \(N\) be the number of pieces the cube is cut into. Estimate \(N\). An estimate of \(E>0\) earns \(\lfloor 20 \min (N / E, E / N)\rfloor\) po... | 15600 | Answer: 15600 | 0 | 8,192 | -1 | 8,192 |
Two distinct natural numbers end with 7 zeros and have exactly 72 divisors. Find their sum. | 70000000 | 0.3125 | 6,773 | 5,477.4 | 7,361.909091 | |
Given P(A) = 0.65, P(B) = 0.2, and P(C) = 0.1, calculate the probability of the event "the drawn product is not a first-class product". | 0.35 | 0.8125 | 4,245.5625 | 4,488.384615 | 3,193.333333 | |
Given that $\frac{a}{25-a}+\frac{b}{65-b}+\frac{c}{60-c}=7$, evaluate $\frac{5}{25-a}+\frac{13}{65-b}+\frac{12}{60-c}$. | 2 | 1 | 2,934.75 | 2,934.75 | -1 | |
A 9 by 9 checkerboard has alternating black and white squares. How many distinct squares, with sides on the grid lines of the checkerboard (horizontal and vertical) and containing at least 6 black squares, can be drawn on the checkerboard? | 91 | 0.375 | 6,932.5625 | 5,462.5 | 7,814.6 | |
A license plate in a certain state consists of 5 digits, not necessarily distinct, and 3 letters, with the condition that at least one of the letters must be a vowel (A, E, I, O, U). These letters do not need to be next to each other but must be in a sequence. How many distinct license plates are possible if the digits... | 4,989,000,000 | 0 | 7,495.1875 | -1 | 7,495.1875 | |
How many three-digit whole numbers have at least one 5 or consecutively have the digit 1 followed by the digit 2? | 270 | 0 | 7,179.75 | -1 | 7,179.75 | |
The graph of the parabola defined by the equation $y=(x-2)^2+3$ is rotated 180 degrees about its vertex, then shifted 3 units to the left, then shifted 2 units down. The resulting parabola has zeros at $x=a$ and $x=b$. What is $a+b$? | -2 | 1 | 2,761.25 | 2,761.25 | -1 | |
In a large library storage room, there are $1584$ boxes, each containing $45$ books. The library dean asks for these books to be repacked so that each new box contains $47$ books. How many books will be left over after repacking the books into as many full boxes of $47$ books each as possible? | 28 | 0.9375 | 4,243.5625 | 3,980.333333 | 8,192 | |
The graph of $y = f(x)$ is shown below.
[asy]
unitsize(0.3 cm);
real func(real x) {
real y;
if (x >= -3 && x <= 0) {y = -2 - x;}
if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}
if (x >= 2 && x <= 3) {y = 2*(x - 2);}
return(y);
}
int i, n;
for (i = -8; i <= 8; ++i) {
draw((i,-8)--(i,8),gray(0.7));
... | \left( 1, \frac{1}{2}, -4 \right) | 0.3125 | 7,510.6875 | 6,011.8 | 8,192 | |
Given that the mean of $x_1, x_2, x_3, \ldots, x_n$ is 4 and the standard deviation is 7, then the mean of $3x_1+2, 3x_2+2, \ldots, 3x_n+2$ is ______; the standard deviation is ______. | 21 | 1 | 1,873.1875 | 1,873.1875 | -1 | |
Let $ABCD$ be a square with side length $1$. How many points $P$ inside the square (not on its sides) have the property that the square can be cut into $10$ triangles of equal area such that all of them have $P$ as a vertex? | 16 |
Let \(ABCD\) be a square with side length \(1\). We are tasked to determine the number of points \(P\) inside the square such that the square can be partitioned into \(10\) triangles of equal area, all having \(P\) as a common vertex.
To solve this problem, consider the following steps:
1. **Understanding the Equal ... | 0.0625 | 7,869.5 | 6,871 | 7,936.066667 |
Find the angle between the vectors $\begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}$ and $\begin{pmatrix} -1 \\ 1 \\ 0 \end{pmatrix},$ in degrees. | 150^\circ | 1 | 2,503.75 | 2,503.75 | -1 | |
Select 4 students from 7 students, including 4 boys and 3 girls, to participate in an environmental knowledge contest, ensuring both boys and girls are among the selected students. Calculate the number of different selection methods. | 34 | 1 | 2,153 | 2,153 | -1 | |
How many diagonals are in a convex polygon with 25 sides, if we only consider diagonals that skip exactly one vertex? | 50 | 0 | 7,496.625 | -1 | 7,496.625 | |
\( AB \) and \( AC \) are two chords forming an angle \( BAC \) equal to \( 70^\circ \). Tangents are drawn through points \( B \) and \( C \) until they intersect at point \( M \). Find \(\angle BMC\). | 40 | 0.5 | 5,664.375 | 3,698.125 | 7,630.625 | |
Let $x$, $y\in \mathbb{R}$, vectors $\overrightarrow{a}=(2,x)$, $\overrightarrow{b}=(y,1)$, $\overrightarrow{c}=(3,-3)$, and $\overrightarrow{a}⊥\overrightarrow{b}$, $\overrightarrow{b}∥\overrightarrow{c}$.
$(1)$ Find $|\overrightarrow{a}+\overrightarrow{b}|$;
$(2)$ Find the cosine value of the angle between vector... | \frac{3}{5} | 1 | 2,396.4375 | 2,396.4375 | -1 | |
Five friends earned $18, $23, $28, $35, and $45. If they split their earnings equally among themselves, how much will the friend who earned $45 need to give to the others? | 15.2 | 0 | 1,102.9375 | -1 | 1,102.9375 | |
How many integers $m \neq 0$ satisfy the inequality $\frac{1}{|m|}\geq \frac{1}{8}$? | 16 | 1 | 1,805.125 | 1,805.125 | -1 | |
The Fibonacci sequence starts with two 1s, and each term afterwards is the sum of its two predecessors. Determine the last digit to appear in the units position of a number in the Fibonacci sequence when considered modulo 12. | 11 | 0.5 | 6,431.8125 | 5,406.375 | 7,457.25 | |
Add the following two numbers: $800,000,000,000 + 299,999,999,999$.
A) $1,000,000,000,000$
B) $1,099,999,999,999$
C) $1,100,000,000,000$
D) $900,000,000,000$
E) $2,099,999,999,999$ | 1,099,999,999,999 | 0.0625 | 4,674 | 653 | 4,942.066667 | |
What is the minimum value of $5x^2-20x+1357$? | 1337 | 1 | 2,186.75 | 2,186.75 | -1 | |
Let $m$ be the smallest integer whose cube root is of the form $n+s$, where $n$ is a positive integer and $s$ is a positive real number less than $1/500$. Find $n$. | 13 | 0.5 | 6,825.4375 | 5,458.875 | 8,192 | |
Let $\{a, b, c, d, e, f, g, h\}$ be a permutation of $\{1, 2, 3, 4, 5, 6, 7, 8\}$ . What is the probability that $\overline{abc} +\overline{def}$ is even? | 3/7 | 0.75 | 4,824.3125 | 4,303.666667 | 6,386.25 | |
The sequence $\left\{ a_n \right\}$ is a geometric sequence with a common ratio of $q$, its sum of the first $n$ terms is $S_n$, and the product of the first $n$ terms is $T_n$. Given that $0 < a_1 < 1, a_{2012}a_{2013} = 1$, the correct conclusion(s) is(are) ______.
$(1) q > 1$ $(2) T_{2013} > 1$ $(3) S_{2012}a_{201... | (1)(3)(4)(5) | 0 | 7,537.6875 | -1 | 7,537.6875 | |
Given the function $$f(x)=(2-a)\ln x+ \frac {1}{x}+2ax \quad (a\leq0)$$.
(Ⅰ) When $a=0$, find the extreme value of $f(x)$;
(Ⅱ) When $a<0$, discuss the monotonicity of $f(x)$. | 2-2\ln2 | 0.5 | 6,870.25 | 5,735 | 8,005.5 | |
Define: In the sequence $\{a_{n}\}$, if $\frac{{a}_{n+2}}{{a}_{n+1}}-\frac{{a}_{n+1}}{{a}_{n}}=d(n∈{N}^{*})$, where $d$ is a constant, then the sequence $\{a_{n}\}$ is called a "geometric difference" sequence. Given a "geometric difference" sequence $\{a_{n}\}$ where $a_{1}=a_{2}=1$ and $a_{3}=3$, find $a_{5}=$______; ... | 3363 | 0.75 | 4,612.625 | 3,857 | 6,879.5 | |
Alice and Bob are playing in the forest. They have six sticks of length $1,2,3,4,5,6$ inches. Somehow, they have managed to arrange these sticks, such that they form the sides of an equiangular hexagon. Compute the sum of all possible values of the area of this hexagon. | 33 \sqrt{3} | Let the side lengths, in counterclockwise order, be $a, b, c, d, e, f$. Place the hexagon on the coordinate plane with edge $a$ parallel to the $x$-axis and the intersection between edge $a$ and edge $f$ at the origin (oriented so that edge $b$ lies in the first quadrant). If you travel along all six sides of the hexag... | 0 | 8,192 | -1 | 8,192 |
Given that $8^{-1} \equiv 85 \pmod{97}$, find $64^{-1} \pmod{97}$, as a residue modulo 97. (Give an answer between 0 and 96, inclusive.) | 47 | 1 | 3,281 | 3,281 | -1 | |
During her birthday, her parents have decided to give Laura and her 2 younger brothers new cellphones. However, they are confused between the innumerable service providers. Assuming no child wants a provider that another sibling has, and that there are 20 service providers, in how many ways can the parents grant the c... | 6840 | 1 | 1,512.4375 | 1,512.4375 | -1 | |
Find the least positive integer $k$ for which the equation $\left\lfloor\frac{2002}{n}\right\rfloor=k$ has no integer solutions for $n$. (The notation $\lfloor x\rfloor$ means the greatest integer less than or equal to $x$.) | 49 | Rewriting the given information and simplifying it a bit, we have \begin{align*} k \le \frac{2002}{n} < k+1 &\implies \frac{1}{k} \ge \frac{n}{2002} > \frac{1}{k+1}. \\ &\implies \frac{2002}{k} \ge n > \frac{2002}{k+1}. \end{align*}
Now note that in order for there to be no integer solutions to $n,$ we must have $\lef... | 0 | 8,192 | -1 | 8,192 |
Find all primes $p$ such that $p^2-p+1$ is a perfect cube. | 19 |
To solve the problem of finding all primes \( p \) such that \( p^2 - p + 1 \) is a perfect cube, we want \( p^2 - p + 1 = n^3 \) for some integer \( n \).
1. **Case Analysis: Small Values of \( p \):**
Start with small values of \( p \):
- For \( p = 2 \):
\[
p^2 - p + 1 = 2^2 - 2 + 1 = 3 \neq ... | 0.0625 | 8,129.875 | 7,198 | 8,192 |
$\triangle ABC$ has area $240$ . Points $X, Y, Z$ lie on sides $AB$ , $BC$ , and $CA$ , respectively. Given that $\frac{AX}{BX} = 3$ , $\frac{BY}{CY} = 4$ , and $\frac{CZ}{AZ} = 5$ , find the area of $\triangle XYZ$ .
[asy]
size(175);
defaultpen(linewidth(0.8));
pair A=(0,15),B=(0,-5),C=(25,0.5),X=origin,Y... | 122 | 0.1875 | 7,956.6875 | 7,138 | 8,145.615385 | |
The price of an article is cut $10 \%$. To restore it to its former value, the new price must be increased by: | $11\frac{1}{9} \%$ | 1. **Assume the original price**: Let the original price of the article be $P$. Without loss of generality, we can set $P = 100$ for simplicity in calculation.
2. **Calculate the reduced price**: The price of the article is reduced by $10\%$. Therefore, the new price after the reduction is:
\[
0.9 \times P = 0.9... | 0 | 3,611.1875 | -1 | 3,611.1875 |
A right prism with height $h$ has bases that are regular hexagons with sides of length $12$. A vertex $A$ of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and th... | 108 | Let $B$ and $C$ be the vertices adjacent to $A$ on the same base as $A$, and let $D$ be the last vertex of the triangular pyramid. Notice that we can already find some lengths. We have $AB=AC=12$ (given) and $BC=BD=\sqrt{144+h^2}$ by the Pythagorean Theorem. Let $M$ be the midpoint of $BC$. Then, we have $AM=6$ (30-60-... | 0.1875 | 7,114.6875 | 5,712 | 7,438.384615 |
Given the function $f(x)= \frac {2}{x+1}$, point $O$ is the coordinate origin, point $A_{n}(n,f(n))(n∈N^{})$, vector $ \overrightarrow{j}=(0,1)$, and $θ_{n}$ is the angle between vector $ \overrightarrow{OA_{n}}$ and $ \overrightarrow{j}$, determine the value of $\frac {cos θ_{1}}{sin θ_{1}}+ \frac {cos θ_{2}}{sin θ_{2... | \frac{4032}{2017} | 0.6875 | 4,513.125 | 3,559 | 6,612.2 | |
At Beaumont High School, there are 12 players on the baseball team. All 12 players are taking at least one of biology or chemistry. If 7 players are taking biology and 2 players are taking both sciences, how many players are taking chemistry? | 7 | 1 | 1,557.6875 | 1,557.6875 | -1 | |
Find all real numbers \( p \) such that the cubic equation \( 5x^3 - 5(p+1)x^2 + (71p-1)x + 1 = 66p \) has two roots that are natural numbers. | 76 | 0 | 8,192 | -1 | 8,192 | |
In how many ways can 10 fillér and 50 fillér coins be placed side by side (with all centers on a straight line) to cover a $1 \mathrm{~m}$ long segment (not more), using at least 50 coins, and considering the order of the two types of coins? (Coins of the same value are not distinguished. The diameter of the 10 fillér ... | 270725 | 0 | 7,880 | -1 | 7,880 | |
Simplify the expression given by \(\frac{a^{-1} - b^{-1}}{a^{-3} + b^{-3}} : \frac{a^{2} b^{2}}{(a+b)^{2} - 3ab} \cdot \left(\frac{a^{2} - b^{2}}{ab}\right)^{-1}\) for \( a = 1 - \sqrt{2} \) and \( b = 1 + \sqrt{2} \). | \frac{1}{4} | 0.25 | 5,504.75 | 4,970.25 | 5,682.916667 | |
1. Given that ${(3x-2)^{6}}={a_{0}}+{a_{1}}(2x-1)+{a_{2}}{(2x-1)^{2}}+ \cdots +{a_{6}}{(2x-1)^{6}}$, find the value of $\dfrac{{a_{1}}+{a_{3}}+{a_{5}}}{{a_{0}}+{a_{2}}+{a_{4}}+{a_{6}}}$.
2. A group of 6 volunteers is to be divided into 4 teams, with 2 teams of 2 people and the other 2 teams of 1 person each, to be sent... | \dfrac{ \sqrt{2}}{2} | 0 | 8,192 | -1 | 8,192 | |
Each of the symbols $\star$ and $*$ represents an operation in the set $\{+,-,\times,\div\}$, and $\frac{12\star 2}{9*3}=2$. What is the value of $\frac{7\star 3}{12*6}$? Express your answer as a common fraction. | \frac{7}{6} | 0.8125 | 4,373.3125 | 3,644.461538 | 7,531.666667 | |
Square $BCFE$ is inscribed in right triangle $AGD$, as shown in the diagram which is the same as the previous one. If $AB = 36$ units and $CD = 72$ units, what is the area of square $BCFE$? | 2592 | 0 | 8,192 | -1 | 8,192 | |
A circle with radius \(5\) has its center on the \(x\)-axis, and the \(x\)-coordinate of the center is an integer. The circle is tangent to the line \(4x+3y-29=0\).
(Ⅰ) Find the equation of the circle;
(Ⅱ) Let the line \(ax-y+5=0\) (\(a > 0\)) intersect the circle at points \(A\) and \(B\), find the range of values for... | a = \dfrac {3}{4} | 0.8125 | 5,712.625 | 5,140.461538 | 8,192 | |
How many continuous paths from $A$ to $B$, along segments of the figure, do not revisit any of the six labeled points?
[asy]
draw((0,0)--(3,0)--(3,2)--(0,2)--(0,0)--cycle,linewidth(2));
draw((0,2)--(1,0)--(3,2)--(0,2)--cycle,linewidth(2));
draw((0,2)--(1.5,3.5)--(3,2),linewidth(2));
label("$A$",(1.5,3.5),N);
label("$... | 10 | 0 | 7,746.75 | -1 | 7,746.75 | |
Two cards are dealt from a standard deck of 52 cards. What is the probability that the first card dealt is a $\heartsuit$ and the second card dealt is a $\clubsuit$? | \frac{13}{204} | 0.75 | 5,918.1875 | 5,160.25 | 8,192 | |
In triangle \( A B C \) with the side ratio \( A B: A C = 5:4 \), the angle bisector of \( \angle B A C \) intersects side \( B C \) at point \( L \). Find the length of segment \( A L \), given that the length of the vector \( 4 \cdot \overrightarrow{A B} + 5 \cdot \overrightarrow{A C} \) is 2016. | 224 | 0.75 | 5,258.5625 | 4,280.75 | 8,192 | |
If $x+\frac{1}{y}=1$ and $y+\frac{1}{z}=1$, what is the value of the product $xyz$? | -1 | 1 | 3,960.1875 | 3,960.1875 | -1 | |
Let $a,$ $b,$ $c,$ $d$ be distinct real numbers such that the roots of $x^2 - 10ax - 11b = 0$ are $c$ and $d,$ and the roots of $x^2 - 10cx - 11d = 0$ are $a$ and $b.$ Find the value of $a + b + c + d.$ | 1210 | 0.3125 | 7,691.875 | 6,591.6 | 8,192 | |
Two watermelons and one banana together weigh 8100 grams, and two watermelons and three bananas together weigh 8300 grams, calculate the weight of one watermelon and one banana. | 100 | 0.375 | 672.875 | 681 | 668 | |
In Chemistry class, Samantha finds that she can make a certain solution by mixing $.04$ liters of chemical A with $.02$ liters of water (this gives her $.06$ liters of the solution). She wants to make a total of $.48$ liters of this new solution. To do so, how many liters of water will she use? | 0.16 | 1 | 1,624.5625 | 1,624.5625 | -1 | |
Given the function $f\left(x\right)=4^{x}+m\cdot 2^{x}$, where $m\in R$.
$(1)$ If $m=-3$, solve the inequality $f\left(x\right) \gt 4$ with respect to $x$.
$(2)$ If the minimum value of the function $y=f\left(x\right)+f\left(-x\right)$ is $-4$, find the value of $m$. | -3 | 1 | 3,690.8125 | 3,690.8125 | -1 | |
Let $W, S$ be as in problem 32. Let $A$ be the least positive integer such that an acute triangle with side lengths $S, A$, and $W$ exists. Find $A$. | 7 | There are two solutions to the alphametic in problem 32: $36 \times 686=24696$ and $86 \times 636=54696$. So $(W, S)$ may be $(3,2)$ or $(8,5)$. If $(W, S)=(3,2)$, then by problem (3) $A=3$, but then by problem $31 W=4$, a contradiction. So, $(W, S)$ must be $(8,5)$. By problem $33, A=7$, and this indeed checks in prob... | 0 | 8,037.1875 | -1 | 8,037.1875 |
Let $A B C D$ be a tetrahedron such that its circumscribed sphere of radius $R$ and its inscribed sphere of radius $r$ are concentric. Given that $A B=A C=1 \leq B C$ and $R=4 r$, find $B C^{2}$. | 1+\sqrt{\frac{7}{15}} | Let $O$ be the common center of the two spheres. Projecting $O$ onto each face of the tetrahedron will divide it into three isosceles triangles. Unfolding the tetrahedron into its net, the reflection of any of these triangles about a side of the tetrahedron will coincide with another one of these triangles. Using this ... | 0 | 8,192 | -1 | 8,192 |
Points $A$, $B$, and $C$ lie in that order along a straight path where the distance from $A$ to $C$ is $1800$ meters. Ina runs twice as fast as Eve, and Paul runs twice as fast as Ina. The three runners start running at the same time with Ina starting at $A$ and running toward $C$, Paul starting at $B$ and running towa... | 800 | Let $x$ be the distance from $A$ to $B$. Then the distance from $B$ to $C$ is $1800-x$. Since Eve is the slowest, we can call her speed $v$, so that Ina's speed is $2v$ and Paul's speed is $4v$.
For Paul and Eve to meet, they must cover a total distance of $1800-x$ which takes them a time of $\frac{1800-x}{4v+v}$. Pau... | 0.1875 | 7,141.6875 | 4,447.666667 | 7,763.384615 |
Without using a calculator, find the largest prime factor of $17^4 + 2 \times 17^2 + 1 - 16^4$. | 17 | 0.9375 | 4,274.375 | 4,042.733333 | 7,749 | |
Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $105, Dorothy paid $125, and Sammy paid $175. In order to share costs equally, find the difference between the amount of money Tom gave Sammy and the amount of money Dorothy gave Sammy. | 20 | 0.875 | 1,514.4375 | 1,452.428571 | 1,948.5 | |
A set containing three real numbers can be represented as $\{a, \frac{b}{a}, 1\}$, and also as $\{a^2, a+b, 0\}$. Find the value of $a^{2003} + b^{2004}$. | -1 | 0.75 | 4,835.5 | 3,721.25 | 8,178.25 | |
Given the equation of a circle $(x-1)^{2}+(y-1)^{2}=9$, point $P(2,2)$ lies inside the circle. The longest and shortest chords passing through point $P$ are $AC$ and $BD$ respectively. Determine the product $AC \cdot BD$. | 12\sqrt{7} | 0.4375 | 6,871.25 | 5,173.142857 | 8,192 | |
\(\cos \frac{\pi}{11} - \cos \frac{2 \pi}{11} + \cos \frac{3 \pi}{11} - \cos \frac{4 \pi}{11} + \cos \frac{5 \pi}{11} = \) (Answer with a number). | \frac{1}{2} | 0 | 8,192 | -1 | 8,192 | |
The sides of a regular hexagon are trisected, resulting in 18 points, including vertices. These points, starting with a vertex, are numbered clockwise as $A_{1}, A_{2}, \ldots, A_{18}$. The line segment $A_{k} A_{k+4}$ is drawn for $k=1,4,7,10,13,16$, where indices are taken modulo 18. These segments define a region co... | 9/13 | Let us assume all sides are of side length 3. Consider the triangle $A_{1} A_{4} A_{5}$. Let $P$ be the point of intersection of $A_{1} A_{5}$ with $A_{4} A_{8}$. This is a vertex of the inner hexagon. Then $\angle A_{4} A_{1} A_{5}=\angle A_{5} A_{4} P$, by symmetry. It follows that $A_{1} A_{4} A_{5} \sim A_{4} P A_{... | 0 | 8,192 | -1 | 8,192 |
The numbers from 1 to 9 are arranged in the cells of a $3 \times 3$ table such that the sum of the numbers on one diagonal is 7, and on the other diagonal, it is 21. What is the sum of the numbers in the five shaded cells?
![Table with shaded cells](https://cdn.mathpix.com/cropped/2024_05_06_ff369b3e8ca7495bdf12g-28.j... | 25 | 0 | 6,922.6875 | -1 | 6,922.6875 | |
Given a configuration of four unit squares arranged in a 2x2 grid, find the area of triangle $\triangle ABC$, where $A$ is the midpoint of the top side of the top-left square, $B$ is the bottom-right corner of the bottom-right square, and $C$ is the midpoint of the right side of the bottom-right square. | 0.375 | 0 | 4,111.625 | -1 | 4,111.625 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.