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 |
|---|---|---|---|---|---|---|
A sequence of two distinct numbers is extended in two ways: one to form a geometric progression and the other to form an arithmetic progression. The third term of the geometric progression coincides with the tenth term of the arithmetic progression. With which term of the arithmetic progression does the fourth term of ... | 74 | 0.3125 | 6,815.625 | 4,258.8 | 7,977.818182 | |
The following is the process of simplifying fractions by Xiaobai. Please read carefully and complete the corresponding tasks.
Solution: $(\frac{3x+4}{x^2-1}-\frac{2}{x-1})÷\frac{x+2}{x^2-2x+1}$
$=[\frac{3x+4}{(x+1)(x-1)}-\frac{2}{x-1}]÷\frac{x+2}{(x-1)^2}\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ... | \frac{1}{3} | 0.875 | 4,924.6875 | 4,457.928571 | 8,192 | |
A group of women working together at the same rate can build a wall in $45$ hours. When the work started, all the women did not start working together. They joined the worked over a period of time, one by one, at equal intervals. Once at work, each one stayed till the work was complete. If the first woman worked 5 ti... | 75 | 0.1875 | 7,227.6875 | 3,049 | 8,192 | |
Let $T$ be a subset of $\{1,2,3,...,100\}$ such that no pair of distinct elements in $T$ has a sum divisible by $5$. What is the maximum number of elements in $T$? | 60 | 0 | 7,599.6875 | -1 | 7,599.6875 | |
Let $c$ and $d$ be real numbers such that
\[\frac{c}{d} + \frac{c}{d^3} + \frac{c}{d^6} + \dots = 9.\]
Find
\[\frac{c}{c + 2d} + \frac{c}{(c + 2d)^2} + \frac{c}{(c + 2d)^3} + \dotsb.\] | \frac{9}{11} | 0 | 8,192 | -1 | 8,192 | |
Kaashish has written down one integer two times and another integer three times. The sum of the five numbers is $100$, and one of the numbers is $28$. What is the other number? | 8 | 1. Let the first number be $x$ and the second number be $y$. According to the problem, Kaashish has written down one integer ($x$) two times and another integer ($y$) three times. The sum of these five numbers is given by:
\[
2x + 3y = 100
\]
2. We are also given that one of the numbers is $28$. We need to de... | 0.9375 | 2,512.5625 | 2,133.933333 | 8,192 |
A school wishes to understand the psychological state of learning among its senior students and adopts a systematic sampling method to select 40 students out of 800 for a test. The students are randomly assigned numbers from 1 to 800 and then grouped. In the first group, number 18 is selected through simple random samp... | 12 | 0.4375 | 6,611.875 | 4,621.142857 | 8,160.222222 | |
Compute the maximum real value of $a$ for which there is an integer $b$ such that $\frac{ab^2}{a+2b} = 2019$ . Compute the maximum possible value of $a$ . | 30285 | 0.5 | 7,709.9375 | 7,227.875 | 8,192 | |
Suppose $x$ and $y$ are positive real numbers such that $x^2 - 2xy + 3y^2 = 9$. Find the maximum possible value of $x^2 + 2xy + 3y^2$. | 18 + 9\sqrt{3} | 0.0625 | 8,059.1875 | 8,070 | 8,058.466667 | |
At McDonald's restaurants, we can order Chicken McNuggets in packages of 6, 9, or 20 pieces. (For example, we can order 21 pieces because $21=6+6+9$, but there is no way to get 19 pieces.) What is the largest number of pieces that we cannot order? | 43 | 0.125 | 7,884.3125 | 7,327.5 | 7,963.857143 | |
Given that $\alpha \in \left( 0, \pi \right)$ and $3\cos 2\alpha = \sin \left( \frac{\pi}{4} - \alpha \right)$, find the value of $\sin 2\alpha$. | -\frac{17}{18} | 0 | 7,219.25 | -1 | 7,219.25 | |
If $\frac{x^2-bx}{ax-c}=\frac{m-1}{m+1}$ has roots which are numerically equal but of opposite signs, the value of $m$ must be: | \frac{a-b}{a+b} | 1. **Cross-multiplying the given equation:**
\[
\frac{x^2 - bx}{ax - c} = \frac{m-1}{m+1}
\]
Cross-multiplying gives:
\[
(m+1)(x^2 - bx) = (m-1)(ax - c)
\]
Expanding both sides:
\[
(m+1)x^2 - (m+1)bx = (m-1)ax - (m-1)c
\]
Rearranging terms:
\[
(m+1)x^2 - (m+1)bx - (m-1)ax + (m-... | 1 | 3,771.875 | 3,771.875 | -1 |
Joshua chooses five distinct numbers. In how many different ways can he assign these numbers to the variables $p, q, r, s$, and $t$ so that $p<s, q<s, r<t$, and $s<t$? | 8 | Suppose that the five distinct numbers that Joshua chooses are $V, W, X, Y, Z$, and that $V<W<X<Y<Z$. We want to assign these to $p, q, r, s, t$ so that $p<s$ and $q<s$ and $r<t$ and $s<t$. First, we note that $t$ must be the largest of $p, q, r, s, t$. This is because $r<t$ and $s<t$, and because $p<s$ and $q<s$, we g... | 0.0625 | 8,096.3125 | 6,661 | 8,192 |
If an item is sold for $x$ dollars, there is a loss of $15\%$ based on the cost. If, however, the same item is sold for $y$ dollars, there is a profit of $15\%$ based on the cost. The ratio of $y:x$ is: | 23:17 | 1. **Define the cost price**: Let the cost price of the item be denoted as $c$.
2. **Calculate the selling price for a loss of 15%**:
- When the item is sold for $x$ dollars, there is a loss of 15%. This means that the selling price $x$ is 85% of the cost price $c$.
- Therefore, we can write the equation:
\[... | 0.5625 | 3,336.125 | 4,142.666667 | 2,299.142857 |
Given $a \gt 0$, $b\in R$, if the inequality $\left(ax-2\right)(-x^{2}-bx+4)\leqslant 0$ holds for all $x \gt 0$, then the minimum value of $b+\frac{3}{a}$ is ______. | 2\sqrt{2} | 0.5 | 6,642.4375 | 5,092.875 | 8,192 | |
In a high school's "Campus Microfilm Festival" event, the school will evaluate the films from two aspects: "viewing numbers" and "expert ratings". If a film A is higher than film B in at least one of these aspects, then film A is considered not inferior to film B. It is known that there are 10 microfilms participating.... | 10 | 0.125 | 7,807.0625 | 5,112.5 | 8,192 | |
Given an equilateral triangle ∆ABC with side length 6, where all three vertices lie on the surface of sphere O with O as the center, and the angle between OA and plane ABC is 45°, find the surface area of sphere O. | 96\pi | 1 | 3,813.625 | 3,813.625 | -1 | |
What is the product of the numerator and the denominator when $0.\overline{012}$ is expressed as a fraction in lowest terms? | 1332 | 1 | 1,737.875 | 1,737.875 | -1 | |
Certain integers, when divided by $\frac{3}{5}, \frac{5}{7}, \frac{7}{9}, \frac{9}{11}$, result in a mixed number where the fractional part is $\frac{2}{3}, \frac{2}{5}, \frac{2}{7}, \frac{2}{9}$, respectively. Find the smallest integer greater than 1 that satisfies these conditions. | 316 | 0.5625 | 5,578.625 | 4,582.666667 | 6,859.142857 | |
(In the coordinate system and parametric equations optional question) In the polar coordinate system, it is known that the line $l: p(\sin\theta - \cos\theta) = a$ divides the region enclosed by the curve $C: p = 2\cos\theta$ into two parts with equal area. Find the value of the constant $a$. | -1 | 0.4375 | 7,619.5625 | 7,131 | 7,999.555556 | |
A team of four students goes to LMT, and each student brings a lunch. However, on the bus, the students’ lunches get mixed up, and during lunch time, each student chooses a random lunch to eat (no two students may eat the same lunch). What is the probability that each student chooses his or her own lunch correctly? | 1/24 | 1 | 1,634.1875 | 1,634.1875 | -1 | |
Let $N$ be the number of functions $f:\{1,2,3,4,5,6,7,8,9,10\} \rightarrow \{1,2,3,4,5\}$ that have the property that for $1\leq x\leq 5$ it is true that $f(f(x))=x$ . Given that $N$ can be written in the form $5^a\cdot b$ for positive integers $a$ and $b$ with $b$ not divisible by $5$ , find $a+b$ .... | 31 | 0.8125 | 4,215.375 | 3,297.692308 | 8,192 | |
Given two positive integers \(x\) and \(y\), \(xy - (x + y) = \operatorname{HCF}(x, y) + \operatorname{LCM}(x, y)\), where \(\operatorname{HCF}(x, y)\) and \(\operatorname{LCM}(x, y)\) are respectively the greatest common divisor and the least common multiple of \(x\) and \(y\). If \(c\) is the maximum possible value o... | 10 | 0.25 | 7,856.125 | 6,848.5 | 8,192 | |
Let $R=gS-4$. When $S=8$, $R=16$. When $S=10$, $R$ is equal to: | 21 | 1. **Identify the equation and given values:**
The equation given is $R = gS - 4$. We know that when $S = 8$, $R = 16$.
2. **Substitute the known values to find $g$:**
Substitute $S = 8$ and $R = 16$ into the equation:
\[
16 = g \cdot 8 - 4
\]
Simplify and solve for $g$:
\[
16 + 4 = 8g \imp... | 1 | 1,725.5 | 1,725.5 | -1 |
In the plane rectangular coordinate system $xOy$, the parameter equations of the line $l$ are $\left\{\begin{array}{l}x=1+\frac{{\sqrt{2}}}{2}t\\ y=\frac{{\sqrt{2}}}{2}t\end{array}\right.$ (where $t$ is the parameter). Taking the coordinate origin $O$ as the pole and the positive half-axis of the $x$-axis as the polar ... | \frac{{5\sqrt{2}}}{{11}} | 0 | 5,192.4375 | -1 | 5,192.4375 | |
Let \[f(n) = \left\{
\begin{array}{cl}
n^2-2 & \text{ if }n<0, \\
2n-20 & \text{ if }n \geq 0.
\end{array}
\right.\]What is the positive difference between the two values of $a$ that satisfy the equation $f(-2)+f(2)+f(a)=0$? | 21 | 1 | 1,586.25 | 1,586.25 | -1 | |
Let \( M = \{1, 2, \ldots, 10\} \), and let \( A_1, A_2, \ldots, A_n \) be distinct non-empty subsets of \( M \). For \( i \neq j \), the intersection \( A_i \cap A_j \) contains at most two elements. Find the maximum value of \( n \). | 175 | 0 | 8,192 | -1 | 8,192 | |
A shooter, in a shooting training session, has the probabilities of hitting the 10, 9, 8, and 7 rings as follows:
0.21, 0.23, 0.25, 0.28, respectively. Calculate the probability that the shooter in a single shot:
(1) Hits either the 10 or 9 ring; (2) Scores less than 7 rings. | 0.03 | 0.375 | 4,616.375 | 4,535.5 | 4,664.9 | |
What is the greatest integer less than 150 for which the greatest common factor of that integer and 24 is 3? | 147 | 1 | 3,331.3125 | 3,331.3125 | -1 | |
Given that $\sin x + \cos x = \frac{1}{2}$, where $x \in [0, \pi]$, find the value of $\sin x - \cos x$. | \frac{\sqrt{7}}{2} | 0 | 4,129.625 | -1 | 4,129.625 | |
The base nine numbers $125_9$ and $33_9$ need to be multiplied and the result expressed in base nine. What is the base nine sum of the digits of their product? | 16 | 0.5 | 6,358.125 | 4,711.875 | 8,004.375 | |
Define \( n! = 1 \times 2 \times \ldots \times n \), for example \( 5! = 1 \times 2 \times 3 \times 4 \times 5 \). If \(\frac{n! \times (n+1)!}{2}\) (where \( \mathbf{n} \) is a positive integer and \( 1 \leq n \leq 100 \)) is a perfect square, what is the sum of all such \( \mathbf{n} \)? | 273 | 0.9375 | 4,035 | 3,757.866667 | 8,192 | |
Barry wrote 6 different numbers, one on each side of 3 cards, and laid the cards on a table, as shown. The sums of the two numbers on each of the three cards are equal. The three numbers on the hidden sides are prime numbers. What is the average of the hidden prime numbers? [asy]
path box=(0,0)--(1,0)--(1,1.5)--(0,1.5)... | 14 | 0.875 | 5,077.5625 | 4,632.642857 | 8,192 | |
In the right triangle \(ABC\) with \(\angle B = 90^\circ\), \(P\) is a point on the angle bisector of \(\angle A\) inside \(\triangle ABC\). Point \(M\) (distinct from \(A\) and \(B\)) lies on the side \(AB\). The lines \(AP\), \(CP\), and \(MP\) intersect sides \(BC\), \(AB\), and \(AC\) at points \(D\), \(E\), and \(... | 1/2 | 0 | 8,192 | -1 | 8,192 | |
Consecutive powers of 3 are added to form this sequence: $3^0,3^0+ 3^1, 3^0+ 3^1+ 3^2$, and so on. What is the simplified value of the fourth term of the sequence? | 40 | 1 | 2,169.5 | 2,169.5 | -1 | |
For each positive integer $n$, let $f_1(n)$ be twice the number of positive integer divisors of $n$, and for $j \ge 2$, let $f_j(n) = f_1(f_{j-1}(n))$. For how many values of $n \le 50$ is $f_{50}(n) = 12?$ | 10 | We are given a function $f_1(n)$ which is twice the number of positive integer divisors of $n$, and for $j \geq 2$, $f_j(n) = f_1(f_{j-1}(n))$. We need to find how many values of $n \leq 50$ satisfy $f_{50}(n) = 12$.
#### Step 1: Understanding $f_1(n)$
The function $f_1(n) = 2d(n)$, where $d(n)$ is the number of divis... | 0 | 7,944.875 | -1 | 7,944.875 |
A large batch of tires contains $1.5\%$ defects. What should be the sample size for the probability of finding at least one defective tire in the sample to be more than $0.92 ?$ | 168 | 0.5 | 6,398.625 | 5,536.625 | 7,260.625 | |
Calculate $\sqrt[4]{\sqrt{\frac{32}{10000}}}$. | \frac{\sqrt[8]{2}}{\sqrt{5}} | 0 | 5,814.75 | -1 | 5,814.75 | |
Let $k$ be a positive integer. Scrooge McDuck owns $k$ gold coins. He also owns infinitely many boxes $B_1, B_2, B_3, \ldots$ Initially, bow $B_1$ contains one coin, and the $k-1$ other coins are on McDuck's table, outside of every box.
Then, Scrooge McDuck allows himself to do the following kind of operations, as many... | 2^{k-1} |
Let \( k \) be a positive integer. Scrooge McDuck initially has \( k \) gold coins, with one coin in box \( B_1 \) and the remaining \( k-1 \) coins on his table. He possesses an infinite number of boxes labeled \( B_1, B_2, B_3, \ldots \). McDuck can perform the following operations indefinitely:
1. If both boxes \(... | 0 | 8,192 | -1 | 8,192 |
If \( x_{1} \) satisfies \( 2x + 2^{x} = 5 \) and \( x_{2} \) satisfies \( 2x + 2 \log_{2}(x - 1) = 5 \), then \( x_{1} + x_{2} = \) ? | \frac{7}{2} | 0.0625 | 8,010.5 | 5,288 | 8,192 | |
The average of the numbers 47 and $x$ is 53. Besides finding the positive difference between 47 and $x$, also determine their sum. | 106 | 0.9375 | 919.25 | 947.866667 | 490 | |
A certain intelligence station has four different passwords A, B, C, and D. Each week, one of the passwords is used, and the password for each week is equally likely to be randomly selected from the three passwords not used in the previous week. If password A is used in the first week, what is the probability that pass... | 1/3 | 0 | 6,904.375 | -1 | 6,904.375 | |
Inside rectangle \(ABCD\), points \(E\) and \(F\) are located such that segments \(EA, ED, EF, FB, FC\) are all congruent. The side \(AB\) is \(22 \text{ cm}\) long and the circumcircle of triangle \(AFD\) has a radius of \(10 \text{ cm}\).
Determine the length of side \(BC\). | 16 | 0.1875 | 7,846.4375 | 6,349 | 8,192 | |
The graph of $x^{4}=x^{2} y^{2}$ is a union of $n$ different lines. What is the value of $n$? | 3 | The equation $x^{4}-x^{2} y^{2}=0$ factors as $x^{2}(x+y)(x-y)=0$, so its graph is the union of the three lines $x=0, x+y=0$, and $x-y=0$. | 1 | 1,774.625 | 1,774.625 | -1 |
Let X be a set containing 10 elements, and A, B be two disjoint subsets of X, containing 3 and 4 elements respectively. Calculate the number of subsets of X that contain neither A nor B. | 840 | 0.3125 | 6,337.8125 | 5,390 | 6,768.636364 | |
The graph of $y = f(x)$ is shown below.
[asy]
unitsize(0.5 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 = -5; i <= 5; ++i) {
draw((i,-5)--(i,5),gray(0.7));
... | \text{E} | 0 | 7,959.875 | -1 | 7,959.875 | |
Let $x_1=97$, and for $n>1$, let $x_n=\frac{n}{x_{n-1}}$. Calculate the product $x_1x_2x_3x_4x_5x_6x_7x_8$. | 384 | Since $x_n=\frac{n}{x_{n-1}}$, $x_n \cdot x_{n - 1} = n$. Setting $n = 2, 4, 6$ and $8$ in this equation gives us respectively $x_1x_2 = 2$, $x_3x_4 = 4$, $x_5x_6 = 6$ and $x_7x_8 = 8$ so \[x_1x_2x_3x_4x_5x_6x_7x_8 = 2\cdot4\cdot6\cdot8 = \boxed{384}.\] Notice that the value of $x_1$ was completely unneeded! | 0.5 | 6,721.9375 | 5,251.875 | 8,192 |
The center of sphere $\alpha$ lies on the surface of sphere $\beta$. The ratio of the surface area of sphere $\beta$ that is inside sphere $\alpha$ to the entire surface area of sphere $\alpha$ is $1 / 5$. Find the ratio of the radii of spheres $\alpha$ and $\beta$. | \sqrt{5} | 0 | 6,779.375 | -1 | 6,779.375 | |
The polynomial $\frac{1}{5}{x^2}{y^{|m|}}-(m+1)y+\frac{1}{7}$ is a cubic binomial in terms of $x$ and $y$. Find the value of $m$. | -1 | 0.5 | 3,995.5625 | 2,857.25 | 5,133.875 | |
Calculate the value of the following expressions:
(1) $(2 \frac {7}{9})^{0.5}+0.1^{-2}+(2 \frac {10}{27})^{- \frac {2}{3}}-3\pi^{0}+ \frac {37}{48}$;
(2) $(-3 \frac {3}{8})^{- \frac {2}{3}}+(0.002)^{- \frac {1}{2}}-10(\sqrt {5}-2)^{-1}+(\sqrt {2}- \sqrt {3})^{0}$. | - \frac {167}{9} | 0.5625 | 5,486.75 | 4,902.222222 | 6,238.285714 | |
Solve the following system of equations: \begin{align*}
3x-5y&=-11,\\
7x+2y&=-12.
\end{align*}Express your answer as an ordered pair $(x,y).$ | (-2,1) | 1 | 2,137.0625 | 2,137.0625 | -1 | |
In rectangle $ABCD$, $P$ is a point on $BC$ so that $\angle APD=90^{\circ}$. $TS$ is perpendicular to $BC$ with $BP=PT$, as shown. $PD$ intersects $TS$ at $Q$. Point $R$ is on $CD$ such that $RA$ passes through $Q$. In $\triangle PQA$, $PA=20$, $AQ=25$ and $QP=15$. [asy]
size(7cm);defaultpen(fontsize(9));
real sd = ... | 12,9 | 0 | 7,697.25 | -1 | 7,697.25 | |
Simplify $(9 \times 10^{12}) \div (3 \times 10^4) + (2 \times 10^8) \div (4 \times 10^2)$. | 300,500,000 | 0 | 5,353.9375 | -1 | 5,353.9375 | |
The area of a triangle is 600 square feet. Find the altitude, in feet, of the triangle if the length of the corresponding base is 30 feet. | 40 | 1 | 1,132.5 | 1,132.5 | -1 | |
In the 100th year of his reign, the Immortal Treasurer decided to start issuing new coins. This year, he issued an unlimited supply of coins with a denomination of \(2^{100} - 1\), next year with a denomination of \(2^{101} - 1\), and so on. As soon as the denomination of a new coin can be obtained without change usin... | 200 | 0 | 8,192 | -1 | 8,192 | |
There are real numbers $a$ and $b$ for which the function $f$ has the properties that $f(x) = ax + b$ for all real numbers $x$, and $f(bx + a) = x$ for all real numbers $x$. What is the value of $a+b$? | -2 | Since $f(x) = ax + b$ for all real numbers $x$, then $f(t) = at + b$ for some real number $t$. When $t = bx + a$, we obtain $f(bx + a) = a(bx + a) + b = abx + (a^{2} + b)$. We also know that $f(bx + a) = x$ for all real numbers $x$. This means that $abx + (a^{2} + b) = x$ for all real numbers $x$ and so $(ab - 1)x + (a... | 1 | 1,713.4375 | 1,713.4375 | -1 |
Given that the polar coordinate equation of curve $C\_1$ is $ρ=2\sin θ$, and the polar coordinate equation of curve $C\_2$ is $θ =\dfrac{π }{3}(ρ \in R)$, curves $C\_1$ and $C\_2$ intersect at points $M$ and $N$. The length of chord $MN$ is _______. | \sqrt {3} | 0 | 5,335.5625 | -1 | 5,335.5625 | |
The first three stages of a pattern are shown below, where each line segment represents a matchstick. If the pattern continues such that at each successive stage, four matchsticks are added to the previous arrangement, how many matchsticks are necessary to create the arrangement for the 100th stage? | 400 | 0.625 | 3,920.75 | 4,013.7 | 3,765.833333 | |
The line $y = \frac{5}{3} x - \frac{17}{3}$ is to be parameterized using vectors. Which of the following options are valid parameterizations?
(A) $\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 4 \\ 1 \end{pmatrix} + t \begin{pmatrix} -3 \\ -5 \end{pmatrix}$
(B) $\begin{pmatrix} x \\ y \end{pmatrix} = \begin... | \text{A,C} | 0 | 3,764.375 | -1 | 3,764.375 | |
How many triangles are in the figure to the right? [asy]
defaultpen(linewidth(0.7));
pair hexcoords (real over, real upover)
{
return dir(0)*over+dir(60)*upover;
}
real r = 0.3;
int i,j;
for(i=0;i<=2;++i)
{
for(j=0;j<=2-i;++j)
{
draw(hexcoords(i,j)--hexcoords(i+1,j));
draw(hexcoords(i,j)--hexcoords(i,j+1));
draw... | 16 | 0.125 | 6,774.125 | 5,851.5 | 6,905.928571 | |
Students from three middle schools worked on a summer project.
Seven students from Allen school worked for 3 days.
Four students from Balboa school worked for 5 days.
Five students from Carver school worked for 9 days.
The total amount paid for the students' work was 744. Assuming each student received the same amoun... | 180.00 | 1. **Calculate the total number of student-days worked**:
Each student-day is a unit representing one student working for one day. We calculate the total student-days for each school and sum them up:
- Allen school: $7$ students $\times 3$ days $= 21$ student-days.
- Balboa school: $4$ students $\times 5$ day... | 0 | 2,372.375 | -1 | 2,372.375 |
On the board, there are natural numbers from 1 to 1000, each written once. Vasya can erase any two numbers and write one of the following in their place: their greatest common divisor or their least common multiple. After 999 such operations, one number remains on the board, which is equal to a natural power of ten. Wh... | 10000 | 0 | 8,047.5625 | -1 | 8,047.5625 | |
Between A and B, there are 6 parallel network cables, with their maximum information capacities being 1, 1, 2, 2, 3, and 4, respectively. Now, if we randomly select 3 of these network cables, in how many ways can we ensure that the sum of the maximum information capacities of these 3 cables is not less than 6? | 15 | 0 | 8,101 | -1 | 8,101 | |
A fair coin is flipped $8$ times. What is the probability that at least $6$ consecutive flips come up heads? | \frac{17}{256} | 0 | 8,158.5 | -1 | 8,158.5 | |
What is the product of the numerator and the denominator when $0.\overline{018}$ is expressed as a fraction in lowest terms? | 222 | 0.9375 | 2,077.125 | 1,669.466667 | 8,192 | |
The prime numbers are added in order starting with $2$: $2$, $2 + 3$, $2 + 3 + 5$, and so on. How many of the first 12 such sums are also prime? | 5 | 0.875 | 4,515.375 | 4,444.5 | 5,011.5 | |
The Fahrenheit temperature ( $F$ ) is related to the Celsius temperature ( $C$ ) by $F = \tfrac{9}{5} \cdot C + 32$ . What is the temperature in Fahrenheit degrees that is one-fifth as large if measured in Celsius degrees? | -4 | 0.6875 | 2,253.4375 | 2,299.636364 | 2,151.8 | |
-15 + 9 \times (6 \div 3) = | 3 | To solve the expression $-15+9\times (6\div 3)$, we follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).
1. **Evaluate the expression inside the parentheses**:
\[
6 \div 3... | 1 | 215.9375 | 215.9375 | -1 |
For any $x \in \mathbb{R}$, the function $f(x)$ represents the minimum value among the three function values $y_{1}=4x+1$, $y_{2}=x+2$, $y_{3}=-2x+4$. The maximum value of $f(x)$ is \_\_\_\_\_\_. | \frac{8}{3} | 0.6875 | 5,628.8125 | 4,986 | 7,043 | |
Let \\(n\\) be a positive integer, and \\(f(n) = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}\\). It is calculated that \\(f(2) = \frac{3}{2}\\), \\(f(4) > 2\\), \\(f(8) > \frac{5}{2}\\), and \\(f(16) > 3\\). Observing the results above, according to the pattern, it can be inferred that \\(f(128) > \_\_\_\_\_\_... | \frac{9}{2} | 0.125 | 7,695.3125 | 7,138 | 7,774.928571 | |
Grisha has 5000 rubles. Chocolate bunnies are sold in a store at a price of 45 rubles each. To carry the bunnies home, Grisha will have to buy several bags at 30 rubles each. One bag can hold no more than 30 chocolate bunnies. Grisha bought the maximum possible number of bunnies and enough bags to carry all the bunnies... | 20 | 0.4375 | 6,801.875 | 6,931.142857 | 6,701.333333 | |
Given that in $\triangle ABC$, $B= \frac{\pi}{4}$ and the height to side $BC$ is equal to $\frac{1}{3}BC$, calculate the value of $\sin A$. | \frac{3\sqrt{10}}{10} | 0 | 6,378.1875 | -1 | 6,378.1875 | |
Find all natural numbers with the property that, when the first digit is moved to the end, the resulting number is $\dfrac{7}{2}$ times the original one. | 153846 | 0.375 | 7,666.6875 | 6,791.166667 | 8,192 | |
Given the point \( P \) on the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \left(a>b>0, c=\sqrt{a^{2}-b^{2}}\right)\), and the equation of the line \( l \) is \(x=-\frac{a^{2}}{c}\), and the coordinate of the point \( F \) is \((-c, 0)\). Draw \( PQ \perp l \) at point \( Q \). If the points \( P \), \( Q \), a... | \frac{\sqrt{2}}{2} | 0 | 8,192 | -1 | 8,192 | |
In triangle \(ABC\), the height \(BD\) is equal to 6, the median \(CE\) is equal to 5, and the distance from point \(K\) (the intersection of segments \(BD\) and \(CE\)) to side \(AC\) is 1.
Find the side \(AB\). | \frac{2 \sqrt{145}}{3} | 0 | 6,951.125 | -1 | 6,951.125 | |
Five towns are connected by a system of roads. There is exactly one road connecting each pair of towns. Find the number of ways there are to make all the roads one-way in such a way that it is still possible to get from any town to any other town using the roads (possibly passing through other towns on the way). | 544 | As noted before, you can see that there are 2 ways that the condition cannot be met; it either has all roads leading into one city, or all roads leading out of one city. We then use complementary counting to count these cases, with PIE. Obviously there are $2^5*2^5$ ways to make roads (just draw a pentagon with all of ... | 0 | 7,959.75 | -1 | 7,959.75 |
Given the sequence $\left\{a_{n}\right\}$ that satisfies $a_{1}=1$ and $S_{n+1}=2 S_{n}-\frac{n(n+1)}{2}+1$, where $S_{n}=a_{1}+a_{2}+\cdots+a_{n}$ $(n=1,2, \cdots)$. If $\Delta a_{n}=a_{n+1}-a_{n}$, find the number of elements in the set $S=\left\{n \in \mathbf{N}^{*} \mid \Delta\left(\Delta a_{n}\right) \geqslant-201... | 11 | 0.75 | 5,164.0625 | 4,472.833333 | 7,237.75 | |
Given that the function $f(x) = x^3 + ax^2 + bx + a^2$ has an extreme value of 10 at $x = 1$, find the slope of the tangent to the function at $x = 2$. | 17 | 0.9375 | 4,819.3125 | 4,694.2 | 6,696 | |
Below is the graph of $y = a \tan bx$ for some positive constants $a$ and $b.$ Find $ab.$
[asy]import TrigMacros;
size(250);
real g(real x)
{
return 2*tan(3/2*x);
}
draw(graph(g,-pi + 0.01,-pi/3 - 0.01),red);
draw(graph(g,-pi/3 + 0.01,pi/3 - 0.01),red);
draw(graph(g,pi/3 + 0.01,pi - 0.01),red);
limits((-pi,-4),(p... | 3 | 0.9375 | 3,010.8125 | 2,665.4 | 8,192 | |
For a permutation $p = (a_1,a_2,\ldots,a_9)$ of the digits $1,2,\ldots,9$, let $s(p)$ denote the sum of the three $3$-digit numbers $a_1a_2a_3$, $a_4a_5a_6$, and $a_7a_8a_9$. Let $m$ be the minimum value of $s(p)$ subject to the condition that the units digit of $s(p)$ is $0$. Let $n$ denote the number of permutations ... | 162 | To minimize $s(p)$, the numbers $1$, $2$, and $3$ (which sum to $6$) must be in the hundreds places. For the units digit of $s(p)$ to be $0$, the numbers in the ones places must have a sum of either $10$ or $20$. However, since the tens digit contributes more to the final sum $s(p)$ than the ones digit, and we are look... | 0.0625 | 7,836.8125 | 5,888 | 7,966.733333 |
We are given a cone with height 6, whose base is a circle with radius $\sqrt{2}$ . Inside the cone, there is an inscribed cube: Its bottom face on the base of the cone, and all of its top vertices lie on the cone. What is the length of the cube's edge?
 | \frac{3}{2} | 0.5625 | 6,542.4375 | 5,796 | 7,502.142857 | |
Find the greatest root of the polynomial $f(x) = 16x^4 - 8x^3 + 9x^2 - 3x + 1$. | 0.5 | 0 | 8,155.875 | -1 | 8,155.875 | |
For positive integers $n$, let $f(n)$ return the smallest positive integer $k$ such that $\frac{1}{k}$ has exactly $n$ digits after the decimal point. How many positive integer divisors does $f(2010)$ have? | 2011 | 0.6875 | 6,645 | 6,146 | 7,742.8 | |
A positive integer $n$ between $1$ and $N=2007^{2007}$ inclusive is selected at random. If $a$ and $b$ are natural numbers such that $a/b$ is the probability that $N$ and $n^3-36n$ are relatively prime, find the value of $a+b$ . | 1109 | 0.8125 | 5,283.625 | 4,802.615385 | 7,368 | |
A hairdresser moved from Vienna to Debrecen to continue his trade. Over the course of 3 years, he became impoverished despite having some money originally. In the first year, he had to spend half of his money. In the second year, he spent a third of what he initially took with him. In the third year, he spent 200 forin... | 1500 | 0.125 | 3,753.4375 | 1,843 | 4,026.357143 | |
Find both the sum and the product of the coordinates of the midpoint of the segment with endpoints $(8, 15)$ and $(-2, -3)$. | 18 | 1 | 1,260.9375 | 1,260.9375 | -1 | |
Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a $50\%$ chance of winning each game it plays, and the outcomes of the games are independent. In each game, the winner is awarded a point and the loser gets 0 points. The total points are accumilate... | 831 | 0.5625 | 7,002.9375 | 6,078.111111 | 8,192 | |
Evaluate \[\frac 3{\log_5{3000^5}} + \frac 4{\log_7{3000^5}},\] giving your answer as a fraction in lowest terms. | \frac{1}{5} | 0 | 8,192 | -1 | 8,192 | |
In right $\Delta ABC$, $\angle CAB$ is a right angle. Point $M$ is the midpoint of $\overline{BC}$. What is the number of centimeters in the length of median $\overline{AM}$? Express your answer as a decimal to the nearest tenth. [asy] pair A,B,C,M;
A = (0,0); B = (4,0); C = (0,3); M = (B+C)/2;
draw(M--A--B--C--A);
lab... | 2.5 | 0.9375 | 2,392.5 | 2,005.866667 | 8,192 | |
The eccentricity of the ellipse given that the slope of line $l$ is $2$, and it intersects the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ $(a > b > 0)$ at two different points, where the projections of these two intersection points on the $x$-axis are exactly the two foci of the ellipse. | \sqrt{2}-1 | 1 | 4,737.1875 | 4,737.1875 | -1 | |
Let $p$, $q$, and $r$ be the roots of the polynomial $x^3 - x - 1 = 0$. Find the value of $\frac{1}{p-2} + \frac{1}{q-2} + \frac{1}{r-2}$. | \frac{11}{7} | 0 | 5,655.8125 | -1 | 5,655.8125 | |
Let $[x]$ denote the greatest integer not exceeding $x$, and let $\{x\} = x - [x]$. Find the value of the sum $\left\{\frac{2012+1}{5}\right\} + \left\{\frac{2012+2}{5}\right\} + \left\{\frac{2012+3}{5}\right\} + \cdots + \left\{\frac{2012+2012}{5}\right\}$. | 805.4 | 0 | 7,271.75 | -1 | 7,271.75 | |
Let \( p \) and \( q \) be positive integers such that
\[
\frac{6}{11} < \frac{p}{q} < \frac{5}{9}
\]
and \( q \) is as small as possible. What is \( p+q \)? | 31 | 0.25 | 8,034 | 7,560 | 8,192 | |
There are 20 points, each pair of adjacent points are equally spaced. By connecting four points with straight lines, you can form a square. Using this method, you can form _ squares. | 20 | 0.1875 | 7,305 | 6,471 | 7,497.461538 | |
Let $ABCD$ be a trapezoid of bases $AB$ and $CD$ . Let $O$ be the intersection point of the diagonals $AC$ and $BD$ . If the area of the triangle $ABC$ is $150$ and the area of the triangle $ACD$ is $120$ , calculate the area of the triangle $BCO$ . | \frac{200}{3} | 0.6875 | 6,375.875 | 5,550.363636 | 8,192 | |
In $\triangle PQR$, we have $PQ = QR = 46$ and $PR = 40$. Point $M$ is the midpoint of $\overline{QR}$. Find the length of segment $PM$. | \sqrt{1587} | 0 | 7,799.25 | -1 | 7,799.25 | |
In Pascal's Triangle, each number is the sum of the number just above it and to the left and the number just above it and to the right. So the middle number in Row 2 is $2$ because $1+1=2.$ What is the sum of the numbers in Row 8 of Pascal's Triangle?
\begin{tabular}{rccccccccccc}
Row 0:& & & & & & 1\\\noalign{\smalls... | 256 | 1 | 719.8125 | 719.8125 | -1 | |
Each vertex of this parallelogram has integer coordinates. The perimeter of the parallelogram is \( p \) units, and the area is \( a \) square units. If the parallelogram is defined by vertices \((2, 3)\), \((7, 3)\), \((x, y)\), and \((x-5, y)\), where \(x\) and \(y\) are integers, find the value of \(p + a\). | 38 | 0 | 8,192 | -1 | 8,192 | |
For how many values of $k$ is $18^{18}$ the least common multiple of the positive integers $6^9$, $9^9$, and $k$? | 19 | 0.4375 | 5,555.6875 | 3,283.285714 | 7,323.111111 | |
In acute triangle \( ABC \), \( M \) and \( N \) are the midpoints of sides \( AB \) and \( BC \), respectively. The tangents to the circumcircle of triangle \( BMN \) at \( M \) and \( N \) meet at \( P \). Suppose that \( AP \) is parallel to \( BC \), \( AP = 9 \), and \( PN = 15 \). Find \( AC \). | 20\sqrt{2} | 0 | 8,192 | -1 | 8,192 |
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