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 |
|---|---|---|---|---|---|---|
Given a bag contains 28 red balls, 20 green balls, 12 yellow balls, 20 blue balls, 10 white balls, and 10 black balls, determine the minimum number of balls that must be drawn to ensure that at least 15 balls of the same color are selected. | 75 | 0.8125 | 5,534.375 | 5,160.307692 | 7,155.333333 | |
Given a circle $C$ passes through points $A(-1,0)$ and $B(3,0)$, and the center of the circle is on the line $x-y=0$.
$(1)$ Find the equation of circle $C$;
$(2)$ If point $P(x,y)$ is any point on circle $C$, find the maximum and minimum distance from point $P$ to the line $x+2y+4=0$. | \frac{2}{5}\sqrt{5} | 0 | 4,624 | -1 | 4,624 | |
Find the least positive integer $n$ such that when $3^n$ is written in base $143$, its two right-most digits in base $143$ are $01$. | 195 | We have that \[3^n \equiv 1 \pmod{143^2}.\]Now, $3^{110} \equiv 1 \pmod{11^2}$ so by the Fundamental Theorem of Orders, $\text{ord}_{11^2}(3)|110$ and with some bashing, we get that it is $5$. Similarly, we get that $\text{ord}_{13^2}(3)=39$. Now, $\text{lcm}(39,5)=\boxed{195}$ which is our desired solution. | 0.75 | 4,924.75 | 5,503.916667 | 3,187.25 |
Find the largest integer $k$ such that for all integers $x$ and $y$, if $xy + 1$ is divisible by $k$, then $x + y$ is also divisible by $k$. | 24 | 0 | 8,192 | -1 | 8,192 | |
For what value of $k$ does the equation $\frac{x-1}{x-2} = \frac{x-k}{x-6}$ have no solution for $x$? | 5 | 1. **Identify the domain**: The equation given is $\frac{x-1}{x-2} = \frac{x-k}{x-6}$. We must exclude values of $x$ that make the denominators zero, hence the domain is $\mathbb{R} \setminus \{2,6\}$.
2. **Cross-multiply to eliminate fractions**:
\[
(x-1)(x-6) = (x-k)(x-2)
\]
Expanding both sides, we get... | 0.375 | 6,098.25 | 3,686.833333 | 7,545.1 |
Insert two numbers between 1 and 2 to form an arithmetic sequence. What is the common difference? | \frac{1}{3} | 1 | 986.75 | 986.75 | -1 | |
Given that the area of a rectangle is 192 and its length is 24, what is the perimeter of the rectangle? | 64 | Since the area of the rectangle is 192 and its length is 24, then its width is $192 \div 24=8$. Therefore, its perimeter is $2 \times 24 + 2 \times 8 = 64$. | 1 | 855.125 | 855.125 | -1 |
Let $\pi$ be a uniformly random permutation of the set $\{1,2, \ldots, 100\}$. The probability that $\pi^{20}(20)=$ 20 and $\pi^{21}(21)=21$ can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$. (Here, $\pi^{k}$ means $\pi$ iterated $k$ times.) | 1025 | We look at the cycles formed by $\pi$ Let $\operatorname{ord}_{\pi}(n)$ denote the smallest $m$ such that $\pi^{m}(n)=n$. In particular, the condition implies that $\operatorname{ord}_{\pi}(20) \mid 20$ and $\operatorname{ord}_{\pi}(21) \mid 21$. Claim 1. 20 and 21 cannot be in the same cycle. Proof. If 20 and 21 were ... | 0 | 8,192 | -1 | 8,192 |
The number of games won by five basketball teams is shown in a bar chart. The teams' names are not displayed. The following clues provide information about the teams:
1. The Hawks won more games than the Falcons.
2. The Warriors won more games than the Knights, but fewer games than the Royals.
3. The Knights won more ... | 33 | 0.25 | 3,476.5 | 627.75 | 4,426.083333 | |
Two different integers from 1 through 20 inclusive are chosen at random. What is the probability that both numbers are prime? Express your answer as a common fraction. | \dfrac{14}{95} | 1 | 1,795.625 | 1,795.625 | -1 | |
Let $ABCD$ be a rectangle with sides $AB,BC,CD$ and $DA$ . Let $K,L$ be the midpoints of the sides $BC,DA$ respectivily. The perpendicular from $B$ to $AK$ hits $CL$ at $M$ . Find $$ \frac{[ABKM]}{[ABCL]} $$ | 2/3 | 0.4375 | 7,678.3125 | 7,017.857143 | 8,192 | |
A majority of the $30$ students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than $1$. The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils ... | 11 | 1. **Define Variables:**
Let $s$ be the number of students that bought pencils, $c$ be the cost of each pencil in cents, and $n$ be the number of pencils each student bought.
2. **Analyze Given Information:**
- A majority of the 30 students bought pencils, so $s > \frac{30}{2} = 15$.
- Each student bought mor... | 1 | 2,857.5625 | 2,857.5625 | -1 |
Suppose $a$, $b$ and $c$ are integers such that the greatest common divisor of $x^2+ax+b$ and $x^2+bx+c$ is $x+1$ (in the set of polynomials in $x$ with integer coefficients), and the least common multiple of $x^2+ax+b$ and $x^2+bx+c$ is $x^3-4x^2+x+6$. Find $a+b+c$. | -6 | 0.9375 | 4,779.9375 | 4,552.466667 | 8,192 | |
A hyperbola with its center shifted to $(1,1)$ passes through point $(4, 2)$. The hyperbola opens horizontally, with one of its vertices at $(3, 1)$. Determine $t^2$ if the hyperbola also passes through point $(t, 4)$. | 36 | 0.125 | 6,562.5 | 5,954.5 | 6,649.357143 | |
Given set \( A = \{0, 1, 2, 3, 4, 5, 9\} \), and \( a, b \in A \) where \( a \neq b \). The number of functions of the form \( y = -a x^2 + (4 - b)x \) whose vertex lies in the first quadrant is ___.
| 21 | 0.25 | 6,904.6875 | 6,784.25 | 6,944.833333 | |
A shopping mall's main staircase from the 1st floor to the 2nd floor consists of 15 steps. Each step has a height of 16 centimeters and a depth of 26 centimeters. The width of the staircase is 3 meters. If the cost of carpeting is 80 yuan per square meter, how much will it cost to buy the carpet needed for the staircas... | 1512 | 0 | 2,135.9375 | -1 | 2,135.9375 | |
Find \(n\) such that \(2^6 \cdot 3^3 \cdot n = 10!\). | 2100 | 1 | 3,526.875 | 3,526.875 | -1 | |
A manufacturer built a machine which will address $500$ envelopes in $8$ minutes. He wishes to build another machine so that when both are operating together they will address $500$ envelopes in $2$ minutes. The equation used to find how many minutes $x$ it would require the second machine to address $500$ envelopes al... | $\frac{1}{8}+\frac{1}{x}=\frac{1}{2}$ | 1. **Understanding the problem**: The first machine addresses 500 envelopes in 8 minutes. We need to find the time $x$ it would take for a second machine to address 500 envelopes alone such that both machines working together can address 500 envelopes in 2 minutes.
2. **Representing the rate of the first machine**: Th... | 0 | 3,130.8125 | -1 | 3,130.8125 |
Given $f(x)=\sin (\omega x+\dfrac{\pi }{3})$ ($\omega > 0$), $f(\dfrac{\pi }{6})=f(\dfrac{\pi }{3})$, and $f(x)$ has a minimum value but no maximum value in the interval $(\dfrac{\pi }{6},\dfrac{\pi }{3})$, find the value of $\omega$. | \dfrac{14}{3} | 0.0625 | 8,192 | 8,192 | 8,192 | |
Consider the graph in 3-space of $0=xyz(x+y)(y+z)(z+x)(x-y)(y-z)(z-x)$. This graph divides 3-space into $N$ connected regions. What is $N$? | 48 | Note that reflecting for each choice of sign for $x, y, z$, we get new regions. Therefore, we can restrict to the case where $x, y, z>0$. In this case, the sign of the expression only depends on $(x-y)(y-z)(z-x)$. It is easy to see that for this expression, every one of the $3!=6$ orderings for $\{x, y, z\}$ contribute... | 0.125 | 7,998.75 | 7,919 | 8,010.142857 |
What is the area of the polygon whose vertices are the points of intersection of the curves $x^2 + y^2 = 25$ and $(x-4)^2 + 9y^2 = 81?$ | 27 | 1. **Identify the curves and their equations:**
- The first curve is a circle given by \(x^2 + y^2 = 25\).
- The second curve is an ellipse given by \((x-4)^2 + 9y^2 = 81\).
2. **Substitute \(y^2\) from the circle's equation into the ellipse's equation:**
\[
y^2 = 25 - x^2
\]
Substituting into the el... | 0.9375 | 5,511.3125 | 5,332.6 | 8,192 |
Find the range of
\[f(A)=\frac{\sin A(3\cos^{2}A+\cos^{4}A+3\sin^{2}A+\sin^{2}A\cos^{2}A)}{\tan A (\sec A-\sin A\tan A)}\]if $A\neq \dfrac{n\pi}{2}$ for any integer $n.$ Enter your answer using interval notation. | (3,4) | 0.6875 | 3,875.0625 | 3,754 | 4,141.4 | |
There are 5 integers written on a board. By summing them pairwise, the following set of 10 numbers is obtained: \( 5, 9, 10, 11, 12, 16, 16, 17, 21, 23 \). Determine which numbers are written on the board. Provide the product of these numbers in your answer. | 5292 | 0.25 | 7,324.25 | 4,721 | 8,192 | |
The bar graph shows the results of a survey on color preferences. What percent preferred blue? | 24\% | 1. **Identify the total number of responses and the number of responses for blue:**
- The total number of responses is given by the sum of all frequencies for each color.
- From the problem, the frequencies are: Red = 50, Blue = 60, Green = 40, Yellow = 60, Purple = 40.
- Calculate the total frequency:
\... | 0 | 6,306.125 | -1 | 6,306.125 |
John has a 1 liter bottle of pure orange juice. He pours half of the contents of the bottle into a vat, fills the bottle with water, and mixes thoroughly. He then repeats this process 9 more times. Afterwards, he pours the remaining contents of the bottle into the vat. What fraction of the liquid in the vat is now wate... | \frac{5}{6} | All the liquid was poured out eventually. 5 liters of water was poured in, and he started with 1 liter of orange juice, so the fraction is \(\frac{5}{1+5}=\frac{5}{6}\). | 0 | 7,602.9375 | -1 | 7,602.9375 |
Let $b$ and $c$ be real numbers. If the polynomial $x^3 + bx^2 + cx + d$ has exactly one real root and $d = c + b + 1$, find the value of the product of all possible values of $c$. | -1 | 0 | 8,192 | -1 | 8,192 | |
Let \( A = (2,0) \) and \( B = (8,6) \). Let \( P \) be a point on the parabola \( y^2 = 8x \). Find the smallest possible value of \( AP + BP \). | 10 | 0 | 8,058.6875 | -1 | 8,058.6875 | |
A coin that comes up heads with probability $p > 0$ and tails with probability $1 - p > 0$ independently on each flip is flipped $8$ times. Suppose that the probability of three heads and five tails is equal to $\frac {1}{25}$ of the probability of five heads and three tails. Let $p = \frac {m}{n}$, where $m$ and $n$ a... | 11 | Rewrite it as : $(P)^3$$(1-P)^5=\frac {1}{25}$ $(P)^5$$(1-P)^3$
This can be simplified as $24P^2 -50P + 25 = 0$
This can be factored into $(4P-5)(6P-5)$
This yields two solutions: $\frac54$ (ignored because it would result in $1-p<0$ ) or $\frac56$
Therefore, the answer is $5+6$ = $\boxed{011}$
~IceMatrix
~Shreya... | 1 | 3,404.0625 | 3,404.0625 | -1 |
Select two different natural numbers from the set $\{1, 2, 3, ..., 8\}$. What is the probability that the greatest common factor of these two numbers is one? Express your answer as a common fraction. | \frac{3}{4} | 0.25 | 7,990.375 | 7,385.5 | 8,192 | |
A table can seat 6 people. Two tables joined together can seat 10 people. Three tables joined together can seat 14 people. Following this pattern, if 10 tables are arranged in two rows with 5 tables in each row, how many people can sit? | 44 | 0 | 6,446.4375 | -1 | 6,446.4375 | |
A circular dartboard is divided into regions with various central angles, as shown. The probability of a dart randomly landing in a particular region is $\frac16$. What is the corresponding measure, in degrees, of the central angle of this section of the dartboard? [asy]
unitsize(1.5cm);
defaultpen(linewidth(.7pt));
p... | 60 | 1 | 1,394.25 | 1,394.25 | -1 | |
We repeatedly toss a coin until we get either three consecutive heads ($HHH$) or the sequence $HTH$ (where $H$ represents heads and $T$ represents tails). What is the probability that $HHH$ occurs before $HTH$? | 2/5 | 0.1875 | 7,957.9375 | 7,415.666667 | 8,083.076923 | |
In the expression $x \cdot y^z - w$, the values of $x$, $y$, $z$, and $w$ are 1, 2, 3, and 4, although not necessarily in that order. What is the maximum possible value of the expression? | 161 | 0.3125 | 7,837.875 | 7,177.4 | 8,138.090909 | |
Let $f(n)$ denote the largest odd factor of $n$ , including possibly $n$ . Determine the value of
\[\frac{f(1)}{1} + \frac{f(2)}{2} + \frac{f(3)}{3} + \cdots + \frac{f(2048)}{2048},\]
rounded to the nearest integer. | 1365 | 0.0625 | 8,176.25 | 7,940 | 8,192 | |
Given the ellipse C: $$\frac {x^{2}}{25}+ \frac {y^{2}}{9}=1$$, F is the right focus, and l is a line passing through point F (not parallel to the y-axis), intersecting the ellipse at points A and B. l′ is the perpendicular bisector of AB, intersecting the major axis of the ellipse at point D. Then the value of $$\frac... | \frac {2}{5} | 0.625 | 6,014.875 | 5,026.9 | 7,661.5 | |
A cuckoo clock strikes the number of times corresponding to the current hour (for example, at 19:00, it strikes 7 times). One morning, Max approached the clock when it showed 9:05. He started turning the minute hand until it moved forward by 7 hours. How many times did the cuckoo strike during this period? | 43 | 0 | 4,282.875 | -1 | 4,282.875 | |
The points $A$, $B$ and $C$ lie on the surface of a sphere with center $O$ and radius $20$. It is given that $AB=13$, $BC=14$, $CA=15$, and that the distance from $O$ to $\triangle ABC$ is $\frac{m\sqrt{n}}k$, where $m$, $n$, and $k$ are positive integers, $m$ and $k$ are relatively prime, and $n$ is not divisible by t... | 118 | 0.9375 | 4,051.875 | 3,775.866667 | 8,192 | |
Given two sets
$$
\begin{array}{l}
A=\{(x, y) \mid |x|+|y|=a, a>0\}, \\
B=\{(x, y) \mid |xy|+1=|x|+|y|\}.
\end{array}
$$
If \( A \cap B \) is the set of vertices of a regular octagon in the plane, determine the value of \( a \). | 2 + \sqrt{2} | 0.125 | 7,788.0625 | 5,871.5 | 8,061.857143 | |
What are the first three digits to the right of the decimal point in the decimal representation of $\left(10^{2005}+1\right)^{11/8}$? | 375 | 0 | 8,192 | -1 | 8,192 | |
Let \( S-ABC \) be a triangular prism with the base being an isosceles right triangle \( ABC \) with \( AB \) as the hypotenuse, and \( SA = SB = SC = 2 \) and \( AB = 2 \). If \( S \), \( A \), \( B \), and \( C \) are points on a sphere centered at \( O \), find the distance from point \( O \) to the plane \( ABC \). | \frac{\sqrt{3}}{3} | 0 | 6,399.0625 | -1 | 6,399.0625 | |
In a regular quadrilateral pyramid \(P-ABCD\) with a volume of 1, points \(E\), \(F\), \(G\), and \(H\) are the midpoints of segments \(AB\), \(CD\), \(PB\), and \(PC\), respectively. Find the volume of the polyhedron \(BEG-CFH\). | 5/16 | 0 | 8,192 | -1 | 8,192 | |
Connecting the centers of adjacent faces of a cube forms a regular octahedron. What is the volume ratio of this octahedron to the cube? | $\frac{1}{6}$ | 0 | 6,242.125 | -1 | 6,242.125 | |
Given the set of $n$ numbers; $n > 1$, of which one is $1 - \frac {1}{n}$ and all the others are $1$. The arithmetic mean of the $n$ numbers is: | 1 - \frac{1}{n^2} | 1. **Identify the numbers in the set**: We are given a set of $n$ numbers where $n > 1$. One of these numbers is $1 - \frac{1}{n}$, and the remaining $n-1$ numbers are all $1$.
2. **Calculate the sum of all numbers in the set**:
- The sum of the $n-1$ numbers that are each $1$ is $1 \times (n-1) = n-1$.
- The s... | 1 | 2,287.25 | 2,287.25 | -1 |
How many four-digit numbers are composed of four distinct digits such that one digit is the average of any two other digits? | 216 | 0 | 8,192 | -1 | 8,192 | |
Find the distance from point $M_{0}$ to the plane passing through three points $M_{1}, M_{2}, M_{3}$.
$M_{1}(2, 3, 1)$
$M_{2}(4, 1, -2)$
$M_{3}(6, 3, 7)$
$M_{0}(-5, -4, 8)$ | 11 | 0.875 | 4,591.0625 | 4,076.642857 | 8,192 | |
Given that $α, β ∈ (0, \frac{π}{2})$, and $\frac{\sin β}{\sin α} = \cos(α + β)$,
(1) If $α = \frac{π}{6}$, then $\tan β =$ _______;
(2) The maximum value of $\tan β$ is _______. | \frac{\sqrt{2}}{4} | 0 | 5,307.25 | -1 | 5,307.25 | |
What is \( \frac{3}{10} \) more than \( 57.7 \)? | 58 | 0.625 | 1,930.25 | 2,109.8 | 1,631 | |
Compute the number of functions $f:\{1,2, \ldots, 9\} \rightarrow\{1,2, \ldots, 9\}$ which satisfy $f(f(f(f(f(x)))))=$ $x$ for each $x \in\{1,2, \ldots, 9\}$. | 3025 | All cycles lengths in the permutation must divide 5 , which is a prime number. Either $f(x)=x$ for all $x$, or there exists exactly one permutation cycle of length 5 . In the latter case, there are $\binom{9}{5}$ ways to choose which numbers are in the cycle and 4 ! ways to create the cycle. The answer is thus $1+\bino... | 0.0625 | 4,920.125 | 3,922 | 4,986.666667 |
To assess the shooting level of a university shooting club, an analysis group used stratified sampling to select the shooting scores of $6$ senior members and $2$ new members for analysis. After calculation, the sample mean of the shooting scores of the $6$ senior members is $8$ (unit: rings), with a variance of $\frac... | \frac{9}{2} | 0.125 | 7,289.3125 | 5,507 | 7,543.928571 | |
Consider a $2 \times n$ grid of points and a path consisting of $2 n-1$ straight line segments connecting all these $2 n$ points, starting from the bottom left corner and ending at the upper right corner. Such a path is called efficient if each point is only passed through once and no two line segments intersect. How m... | \binom{4030}{2015} | The general answer is $\binom{2(n-1)}{n-1}$ : Simply note that the points in each column must be taken in order, and anything satisfying this avoids intersections, so just choose the steps during which to be in the first column. | 0 | 6,130.3125 | -1 | 6,130.3125 |
How many positive integers less than $500$ can be written as the sum of two positive perfect cubes? | 26 | 0.25 | 7,985.5 | 7,394.75 | 8,182.416667 | |
Petya and Vasya competed in a 60-meter race. When Petya finished, Vasya was 9 meters behind him. In the second race, Petya stood exactly 9 meters behind Vasya at the starting point. Who finished first in the second race and by how many meters did he lead his opponent? (Assume that each boy ran both times at his own con... | 1.35 | 0.125 | 6,613.375 | 4,380.5 | 6,932.357143 | |
Given $a \neq 0,$ solve for $x$ in
\[\begin{vmatrix} x + a & x & x \\ x & x + a & x \\ x & x & x + a \end{vmatrix} = 0.\]Give your answer in terms of $a.$ | -\frac{a}{3} | 0.75 | 4,815.4375 | 4,128.75 | 6,875.5 | |
If the proposition "$\exists x\in [-2,1], ax^{2}+2ax+3a > 1$" is false, determine the maximum value of $a$. | \frac{1}{6} | 0.8125 | 5,550.3125 | 4,940.692308 | 8,192 | |
For positive reals $p$ and $q$, define the remainder when $p$ is divided by $q$ as the smallest nonnegative real $r$ such that $\frac{p-r}{q}$ is an integer. For an ordered pair $(a, b)$ of positive integers, let $r_{1}$ and $r_{2}$ be the remainder when $a \sqrt{2}+b \sqrt{3}$ is divided by $\sqrt{2}$ and $\sqrt{3}$ r... | 16 | The remainder when we divide $a \sqrt{2}+b \sqrt{3}$ by $\sqrt{2}$ is defined to be the smallest non-negative real $r_{1}$ such that $\frac{a \sqrt{2}+b \sqrt{3}-r_{1}}{\sqrt{2}}$ is integral. As $\frac{x}{\sqrt{2}}$ is integral iff $x$ is an integral multiple of $\sqrt{2}$, it follows that $r_{1}=b \sqrt{3}-c \sqrt{2}... | 0 | 8,192 | -1 | 8,192 |
Compute $156 + 44 + 26 + 74$. | 300 | 0.9375 | 2,382.1875 | 1,994.866667 | 8,192 | |
Let $V$ be a rectangular prism with integer side lengths. The largest face has area 240 and the smallest face has area 48. A third face has area $x$, where $x$ is not equal to 48 or 240. What is the sum of all possible values of $x$? | 260 | Let the length, width, and height of the prism be $s_{1}, s_{2}, s_{3}$. Without loss of generality, assume that $s_{1} \leq s_{2} \leq s_{3}$. Then, we have that $s_{1} s_{2}=48$ and $s_{2} s_{3}=240$. Noting that $s_{1} \leq s_{2}$, we must have $\left(s_{1}, s_{2}\right)=(1,48),(2,24),(3,16),(4,12),(6,8)$. We must a... | 0.4375 | 6,266.625 | 4,666.857143 | 7,510.888889 |
An equilateral triangle has sides 8 units long. An equilateral triangle with sides 4 units long is cut off at the top, leaving an isosceles trapezoid. What is the ratio of the area of the smaller triangle to the area of the trapezoid? Express your answer as a common fraction. | \frac{1}{3} | 0.9375 | 3,341.375 | 3,018 | 8,192 | |
Xiaopang, Xiaodingding, Xiaoya, and Xiaoqiao have a total of 8 parents and 4 children in their four families. They are going to an amusement park together. The ticket pricing is as follows: Adult tickets are 100 yuan per person, children's tickets are 50 yuan per person. If there are 10 or more people, they can buy gro... | 800 | 0 | 7,056.0625 | -1 | 7,056.0625 | |
What is the sum of the digits of the base-2 expression for $222_{10}$? | 6 | 0.875 | 4,178.375 | 3,605 | 8,192 | |
In triangle $ABC$, it is known that $\sqrt{3}\sin{2B} = 1 - \cos{2B}$.
(1) Find the value of angle $B$;
(2) If $BC = 2$ and $A = \frac{\pi}{4}$, find the area of triangle $ABC$. | \frac{3 + \sqrt{3}}{2} | 0 | 5,243.5 | -1 | 5,243.5 | |
Moe has a new, larger rectangular lawn measuring 120 feet by 180 feet. He uses a mower with a swath width of 30 inches. However, he overlaps each cut by 6 inches to ensure no grass is missed. Moe walks at a rate of 6000 feet per hour while pushing the mower. What is the closest estimate of the number of hours it will t... | 1.8 | 0.0625 | 5,411.5 | 5,777 | 5,387.133333 | |
Circle inscribed in square $ABCD$ , is tangent to sides $AB$ and $CD$ at points $M$ and $K$ respectively. Line $BK$ intersects this circle at the point $L, X$ is the midpoint of $KL$ . Find the angle $\angle MXK $ . | 135 | 0.625 | 6,311.125 | 5,182.6 | 8,192 | |
In any isosceles triangle $PQR$ with $PQ = PR$, the altitude $PS$ bisects the base $QR$ so that $QS = RS$. Given that the triangle sides are $PQ = PR = 15$ and the entire base length $QR = 20$.
[asy]
draw((0,0)--(20,0)--(10,36)--cycle,black+linewidth(1));
draw((10,36)--(10,0),black+linewidth(1)+dashed);
draw((10,0)--(... | 50\sqrt{5} | 0 | 3,195.125 | -1 | 3,195.125 | |
It is known that the numbers \( x, y, z \) form an arithmetic progression in the given order with a common difference \( \alpha = \arccos \left(-\frac{3}{7}\right) \), and the numbers \( \frac{1}{\cos x}, \frac{7}{\cos y}, \frac{1}{\cos z} \) also form an arithmetic progression in the given order. Find \( \cos^{2} y \... | \frac{10}{13} | 0.25 | 7,270.0625 | 4,504.25 | 8,192 | |
Find the sum of the distinct prime factors of $5^5 - 5^3$. | 10 | 0.875 | 1,923.5625 | 1,557.571429 | 4,485.5 | |
The product of the two $99$-digit numbers
$303,030,303,...,030,303$ and $505,050,505,...,050,505$
has thousands digit $A$ and units digit $B$. What is the sum of $A$ and $B$? | 8 | To solve the problem, we need to determine the units digit and the thousands digit of the product of two $99$-digit numbers:
\[ 303030303\ldots030303 \quad \text{and} \quad 505050505\ldots050505. \]
1. **Identify the pattern in each number:**
- The first number repeats the digits $303$.
- The second number repea... | 0.25 | 7,269.0625 | 6,830.25 | 7,415.333333 |
Positive numbers \(a\), \(b\), and \(c\) satisfy the equations
\[
a^{2} + ab + b^{2} = 1, \quad b^{2} + bc + c^{2} = 3, \quad c^{2} + ca + a^{2} = 4
\]
Find the value of \(a + b + c\). | \sqrt{7} | 0.6875 | 6,640.3125 | 5,935 | 8,192 | |
Three coplanar squares with sides of lengths two, four and six units, respectively, are arranged side-by-side, as shown so that one side of each square lies on line $AB$ and a segment connects the bottom left corner of the smallest square to the upper right corner of the largest square. What is the area of the shaded q... | 8 | 0.9375 | 4,937.1875 | 4,720.2 | 8,192 | |
Compute the value of $0.25 \cdot 0.8 - 0.12$. | 0.08 | 0.9375 | 1,882.75 | 1,462.133333 | 8,192 | |
Anna enjoys dinner at a restaurant in Washington, D.C., where the sales tax on meals is 10%. She leaves a 15% tip on the price of her meal before the sales tax is added, and the tax is calculated on the pre-tip amount. She spends a total of 27.50 dollars for dinner. What is the cost of her dinner without tax or tip in ... | 22 | Let $x$ be the cost of Anna's dinner before tax and tip.
1. **Calculate the tax:** The tax rate is 10%, so the tax amount is $\frac{10}{100}x = 0.1x$.
2. **Calculate the tip:** The tip is 15% of the pre-tax meal cost, so the tip amount is $\frac{15}{100}x = 0.15x$.
3. **Total cost:** The total cost of the meal inclu... | 1 | 1,316.0625 | 1,316.0625 | -1 |
Given the polar equation of curve $C_{1}$ is $\rho=2\sin \theta$, and the polar equation of curve $C_{2}$ is $\theta= \frac {\pi}{3}$ ($\rho\in\mathbb{R}$), curves $C_{1}$ and $C_{2}$ intersect at points $M$ and $N$, then the length of chord $MN$ is ______. | \sqrt {3} | 0 | 6,426.9375 | -1 | 6,426.9375 | |
Two candles of the same length are made of different materials so that one burns out completely at a uniform rate in $3$ hours and the other in $4$ hours. At what time P.M. should the candles be lighted so that, at 4 P.M., one stub is twice the length of the other? | 1:36 | 1. **Identify the rate of burning for each candle**:
- The first candle burns out in $3$ hours, i.e., $180$ minutes.
- The second candle burns out in $4$ hours, i.e., $240$ minutes.
2. **Set up the function for the length of each candle's stub**:
- Let $\ell$ be the initial length of each candle.
- For th... | 0.6875 | 4,542.3125 | 4,247.454545 | 5,191 |
Find how many integer values of \( x \) are there such that \( \lceil{\sqrt{x}}\rceil=18 \)? | 35 | 0.4375 | 3,746.5625 | 3,619.428571 | 3,845.444444 | |
Find the maximum possible number of three term arithmetic progressions in a monotone sequence of $n$ distinct reals. | \[
f(n) = \left\lfloor \frac{(n-1)^2}{2} \right\rfloor
\] | Consider the first few cases for $n$ with the entire $n$ numbers forming an arithmetic sequence \[(1, 2, 3, \ldots, n)\] If $n = 3$ , there will be one ascending triplet (123). Let's only consider the ascending order for now.
If $n = 4$ , the first 3 numbers give 1 triplet, the addition of the 4 gives one more, for 2 i... | 0 | 8,059.5625 | -1 | 8,059.5625 |
Given that an integer is either only even or only odd and must be divisible by 4, calculate the number of 4-digit positive integers that satisfy these conditions. | 240 | 0 | 6,741.5 | -1 | 6,741.5 | |
A certain stationery store stipulates that if one purchases 250 or more exercise books at a time (including 250 books), they can pay at the wholesale price; if one purchases fewer than 250 books, they must pay at the retail price. Li, a teacher, went to the store to buy exercise books for the 8th-grade students. If he ... | 200 | 0.5625 | 6,674.9375 | 5,495 | 8,192 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $C= \dfrac {\pi}{6}$, $a=1$, $b= \sqrt {3}$, find the measure of $B$. | \dfrac {2\pi}{3} | 0.9375 | 3,654.875 | 3,576.266667 | 4,834 | |
How many natural numbers greater than 10 but less than 100 are relatively prime to 21? | 51 | 0.0625 | 6,427.8125 | 8,192 | 6,310.2 | |
Persons A, B, and C set out from location $A$ to location $B$ at the same time. Their speed ratio is 4: 5: 12, respectively, where A and B travel by foot, and C travels by bicycle. C can carry one person with him on the bicycle (without changing speed). In order for all three to reach $B$ at the same time in the shorte... | 7/10 | 0 | 7,795.5 | -1 | 7,795.5 | |
Write the numbers $0, 1, 2, 3, 4, 5, 6, 7$ on the eight vertices of a cube (each vertex has one number, and all numbers must be used once), such that the sum of the two numbers on each edge is a prime number. Then, find the maximum sum of the four numbers on one face of the cube. | 18 | 0 | 8,192 | -1 | 8,192 | |
Given the function $f(x)=e^{x}$, $g(x)=-x^{2}+2x+b(b\in\mathbb{R})$, denote $h(x)=f(x)- \frac {1}{f(x)}$
(I) Determine the parity of $h(x)$ and write down the monotonic interval of $h(x)$, no proof required;
(II) For any $x\in[1,2]$, there exist $x_{1}$, $x_{2}\in[1,2]$ such that $f(x)\leqslant f(x_{1})$, $g(x)\leq... | e^{2}-1 | 0 | 2,938.6875 | -1 | 2,938.6875 | |
Find the smallest positive integer $n$ such that $1^{2}+2^{2}+3^{2}+4^{2}+\cdots+n^{2}$ is divisible by 100. | 24 | The sum of the first $n$ squares equals $n(n+1)(2 n+1) / 6$, so we require $n(n+1)(2 n+1)$ to be divisible by $600=24 \cdot 25$. The three factors are pairwise relatively prime, so one of them must be divisible by 25 . The smallest $n$ for which this happens is $n=12$ $(2 n+1=25)$, but then we do not have enough factor... | 0.1875 | 8,135.9375 | 7,893 | 8,192 |
Suppose you have an unlimited number of pennies, nickels, dimes, and quarters. Determine the number of ways to make 30 cents using these coins. | 17 | 0 | 7,868.4375 | -1 | 7,868.4375 | |
Compute the domain of the real-valued function $$f(x)=\sqrt{3-\sqrt{5-\sqrt{x}}}.$$ | [0, 25] | 1 | 3,428.0625 | 3,428.0625 | -1 | |
Find the number of ordered pairs $(x,y)$ of real numbers such that
\[16^{x^2 + y} + 16^{x + y^2} = 1.\] | 1 | 0.1875 | 7,994.5625 | 7,139 | 8,192 | |
Determine all real numbers $ a$ such that the inequality $ |x^2 + 2ax + 3a|\le2$ has exactly one solution in $ x$. | 1, 2 | 0 | 6,666.75 | -1 | 6,666.75 | |
A circle is inscribed in a triangle with side lengths $8, 13$, and $17$. Let the segments of the side of length $8$, made by a point of tangency, be $r$ and $s$, with $r<s$. What is the ratio $r:s$? | 1:3 | 1. **Label the Triangle and Points of Tangency**: Consider triangle $ABC$ with $AB = 17$, $AC = 13$, and $BC = 8$. Let $L$, $J$, and $K$ be the points of tangency on sides $BC$, $AC$, and $AB$, respectively.
2. **Assign Variables to Segments**: Let $AK = x$. By the property of tangents from a point to a circle, $AJ = ... | 0.5625 | 4,610.375 | 4,681.888889 | 4,518.428571 |
The sequence \(\{a_n\}\) has consecutive terms \(a_n\) and \(a_{n+1}\) as the roots of the equation \(x^2 - c_n x + \left(\frac{1}{3}\right)^n = 0\), with initial term \(a_1 = 2\). Find the sum of the infinite series \(c_1, c_2, \cdots, c_n, \cdots\). | \frac{9}{2} | 0.3125 | 7,686.75 | 6,575.2 | 8,192 | |
When $n$ is an odd number, $\frac{1+(-1)^{n}}{4}=$______; when $n$ is an even number, $\frac{1+(-1)^{n}}{4}=\_\_\_\_\_\_$. | \frac{1}{2} | 1 | 481.625 | 481.625 | -1 | |
If the difference between each number in a row and the number immediately to its left in the given diagram is the same, and the quotient of each number in a column divided by the number immediately above it is the same, then $a + b \times c =\quad$ | 540 | 0 | 8,192 | -1 | 8,192 | |
On an island, there are red, yellow, green, and blue chameleons.
- On a cloudy day, either one red chameleon changes its color to yellow, or one green chameleon changes its color to blue.
- On a sunny day, either one red chameleon changes its color to green, or one yellow chameleon changes its color to blue.
In Septe... | 11 | 0.3125 | 6,616.1875 | 3,160.4 | 8,187 | |
To shift the graph of the function $y=\sin \left(2x- \frac{\pi}{4}\right)$ to the graph of the function $y = \sin(2x)$, determine the horizontal shift required. | \frac{\pi}{8} | 1 | 3,087 | 3,087 | -1 | |
What is the maximum number of balls of clay of radius $2$ that can completely fit inside a cube of side length $6$ assuming the balls can be reshaped but not compressed before they are packed in the cube? | 6 | 1. **Calculate the volume of the cube**:
The side length of the cube is given as $6$. Therefore, the volume of the cube ($V_{\text{cube}}$) is calculated as:
\[
V_{\text{cube}} = 6^3 = 216.
\]
2. **Calculate the volume of a ball of clay**:
The radius of each ball of clay is $2$. The volume of a sphere (... | 0.5 | 7,518.75 | 7,068.375 | 7,969.125 |
On three faces of a cube, diagonals are drawn such that a triangle is formed. Find the angles of this triangle. | 60 | 0.375 | 6,878.25 | 5,964.5 | 7,426.5 | |
What is the greatest three-digit positive integer $n$ for which the sum of the first $n$ positive integers is not a divisor of the product of the first $n$ positive integers? | 996 |
1. **Understanding the Problem:**
We need to find the largest three-digit integer $n$ such that the sum of the first $n$ positive integers is not a divisor of the product of the first $n$ positive integers.
2. **Sum and Product Formulas:**
- The sum of the first $n$ positive integers is given by the formula:
... | 0.4375 | 6,757.4375 | 4,913 | 8,192 |
Given the following four conclusions:
\\((1)\\) The center of symmetry of the function \\(f(x)= \dfrac {x-1}{2x+1}\\) is \\((- \dfrac {1}{2},- \dfrac {1}{2})\\);
\\((2)\\) If the equation in \\(x\\), \\(x- \dfrac {1}{x}+k=0\\), has no real roots for \\(x \in (0,1)\\), then the range of \\(k\\) is \\(k \geqslant 2\\... | (3)(4) | 0 | 8,025.9375 | -1 | 8,025.9375 | |
Given the coordinates of the vertices of triangle $\triangle O A B$ are $O(0,0), A(4,4 \sqrt{3}), B(8,0)$, with its incircle center being $I$. Let the circle $C$ pass through points $A$ and $B$, and intersect the circle $I$ at points $P$ and $Q$. If the tangents drawn to the two circles at points $P$ and $Q$ are perpen... | 2\sqrt{7} | 0.3125 | 7,447.8125 | 5,810.6 | 8,192 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and they satisfy $(3b-c)\cos A - a\cos C = 0$.
(1) Find $\cos A$;
(2) If $a = 2\sqrt{3}$ and the area of $\triangle ABC$ is $S_{\triangle ABC} = 3\sqrt{2}$, determine the shape of $\triangle ABC$ and explain the reason;
(3) I... | 4\sqrt{2} | 0.6875 | 6,523 | 6,003.181818 | 7,666.6 | |
A group of dancers are arranged in a rectangular formation. When they are arranged in 12 rows, there are 5 positions unoccupied in the formation. When they are arranged in 10 rows, there are 5 positions unoccupied. How many dancers are in the group if the total number is between 200 and 300? | 295 | 0.0625 | 7,606.4375 | 8,192 | 7,567.4 |
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