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 |
|---|---|---|---|---|---|---|
The "Tuning Day Method" is a procedural algorithm for seeking precise fractional representations of numbers. Suppose the insufficient approximation and the excessive approximation of a real number $x$ are $\dfrac{b}{a}$ and $\dfrac{d}{c}$ ($a,b,c,d \in \mathbb{N}^*$) respectively, then $\dfrac{b+d}{a+c}$ is a more accu... | \dfrac{22}{7} | 0.375 | 6,933.5625 | 6,170.333333 | 7,391.5 | |
A right triangle and a circle are drawn such that the circle is tangent to the legs of the right triangle. The circle cuts the hypotenuse into three segments of lengths 1,24 , and 3 , and the segment of length 24 is a chord of the circle. Compute the area of the triangle. | 192 | Let the triangle be $\triangle A B C$, with $A C$ as the hypotenuse, and let $D, E, F, G$ be on sides $A B, B C, A C$, $A C$, respectively, such that they all lie on the circle. We have $A G=1, G F=24$, and $F C=3$. By power of a point, we have $$\begin{aligned} & A D=\sqrt{A G \cdot A F}=\sqrt{1(1+24)}=5 \\ & C E=\sqr... | 0 | 8,192 | -1 | 8,192 |
Given the function $f(x) = \frac{x}{\ln x}$, and $g(x) = f(x) - mx (m \in \mathbb{R})$,
(I) Find the interval of monotonic decrease for function $f(x)$.
(II) If function $g(x)$ is monotonically decreasing on the interval $(1, +\infty)$, find the range of the real number $m$.
(III) If there exist $x_1, x_2 \in [e, e^... | \frac{1}{2} - \frac{1}{4e^2} | 0 | 7,526.25 | -1 | 7,526.25 | |
Determine the number of different arrangements for assigning 6 repatriated international students to work in Jinan and Qingdao, given that at least 2 students must be assigned to Jinan and at least 3 students must be assigned to Qingdao. | 35 | 0.9375 | 3,572.5625 | 3,264.6 | 8,192 | |
Find the largest 5-digit number \( A \) that satisfies the following conditions:
1. Its 4th digit is greater than its 5th digit.
2. Its 3rd digit is greater than the sum of its 4th and 5th digits.
3. Its 2nd digit is greater than the sum of its 3rd, 4th, and 5th digits.
4. Its 1st digit is greater than the sum of all o... | 95210 | 0 | 8,192 | -1 | 8,192 | |
Given the function $f(x)=ax + a^{-x}$ ($a>0$ and $a\neq1$), and $f(1)=3$, find the value of $f(0)+f(1)+f(2)$. | 12 | 0.125 | 7,747.375 | 4,635 | 8,192 | |
A workshop produces transformers of types $A$ and $B$. One transformer of type $A$ uses 5 kg of transformer iron and 3 kg of wire, while one transformer of type $B$ uses 3 kg of iron and 2 kg of wire. The profit from selling one transformer of type $A$ is 12 thousand rubles, and for type $B$ it is 10 thousand rubles. T... | 1502 | 0 | 7,868.125 | -1 | 7,868.125 | |
A hotel packed breakfast for each of three guests. Each breakfast should have consisted of three types of rolls, one each of nut, cheese, and fruit rolls. The preparer wrapped each of the nine rolls and once wrapped, the rolls were indistinguishable from one another. She then randomly put three rolls in a bag for each ... | 79 | Call the three different types of rolls as A, B, and C. We need to arrange 3As, 3Bs, and 3Cs in a string such that A, B, and C appear in the first three, second three, and the third three like ABCABCABC or BCABACCAB. This can occur in $\left(\frac{3!}{1!1!1!}\right)^3 = 6^3 = 216$ different manners. The total number of... | 0.875 | 5,818.4375 | 5,481.928571 | 8,174 |
Find all values of \( x \) for which the smaller of the numbers \( \frac{1}{x} \) and \( \sin x \) is greater than \( \frac{1}{2} \). In the answer, provide the total length of the resulting intervals on the number line, rounding to the nearest hundredth if necessary. | 2.09 | 0 | 7,327.125 | -1 | 7,327.125 | |
Find the smallest positive integer $N$ such that any "hydra" with 100 necks, where each neck connects two heads, can be defeated by cutting at most $N$ strikes. Here, one strike can sever all the necks connected to a particular head $A$, and immediately after, $A$ grows new necks to connect with all previously unconnec... | 10 | 0 | 8,174.875 | -1 | 8,174.875 | |
A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might be... | 37 | 1. **Understanding the Sequence Property**: The sequence is such that the tens and units digits of each term become the hundreds and tens digits of the next term. This implies that each digit in a position (hundreds, tens, units) of one term will appear in the next position in the next term and eventually cycle back to... | 0.125 | 8,002.6875 | 6,677.5 | 8,192 |
Find the number of pentominoes (5-square polyominoes) that span a 3-by-3 rectangle, where polyominoes that are flips or rotations of each other are considered the same polyomino. | 6 | By enumeration, the answer is 6. | 0.125 | 7,371 | 6,365 | 7,514.714286 |
Given four points \( O, A, B, C \) on a plane, with \( OA=4 \), \( OB=3 \), \( OC=2 \), and \( \overrightarrow{OB} \cdot \overrightarrow{OC}=3 \), find the maximum area of triangle \( ABC \). | 2 \sqrt{7} + \frac{3\sqrt{3}}{2} | 0 | 8,182.875 | -1 | 8,182.875 | |
Given that $-\frac{\pi}{2}<\alpha<\frac{\pi}{2}, 2 \tan \beta=\tan 2\alpha, \tan (\beta-\alpha)=-2 \sqrt{2}$, find the value of $\cos \alpha$. | \frac{\sqrt{3}}{3} | 0 | 3,960 | -1 | 3,960 | |
Let the focus of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ be $F_{1}$, $F_{2}$, and $P$ be a point on the ellipse with $\angle F_1PF_2=\frac{π}{3}$. If the radii of the circumcircle and incircle of $\triangle F_{1}PF_{2}$ are $R$ and $r$ respectively, and when $R=3r$, the eccentricity of the ellipse is ___... | \frac{3}{5} | 0.9375 | 5,196.6875 | 4,997 | 8,192 | |
Two lines are perpendicular. One line has a direction vector of $\begin{pmatrix} 3 \\ -7 \end{pmatrix}.$ The other line has a direction vector of $\begin{pmatrix} a \\ 2 \end{pmatrix}.$ Find $a.$ | \frac{14}{3} | 1 | 1,565.25 | 1,565.25 | -1 | |
$(-1)^{5^{2}} + 1^{2^{5}} = $ | 0 | 1. **Evaluate the first term** $(-1)^{5^2}$:
- Calculate the exponent: $5^2 = 25$.
- Since $(-1)^{n}$ is $-1$ if $n$ is odd, and $25$ is odd, we have $(-1)^{25} = -1$.
2. **Evaluate the second term** $1^{2^5}$:
- Calculate the exponent: $2^5 = 32$.
- Any number raised to any power is still that number if t... | 0.9375 | 1,352.625 | 1,376.733333 | 991 |
If organisms do not die but only divide, then the population will certainly never die out.
The conditions are satisfied by the function whose graph is highlighted in the image.
$$
x(p)=\left\{\begin{array}{l}
1, \text { if } 0 \leq p \leq \frac{1}{2} \\
\frac{q}{p}, \text { if } \frac{1}{2}<p \leq 1
\end{array}\rig... | \frac{2}{3} | 0.1875 | 6,768 | 4,202 | 7,360.153846 | |
Given the sequence $\{a_{n}\}$, where $a_{n}$ are integers, and for $n \geq 3, n \in \mathbf{N}$, the relation $a_{n} = a_{n-1} - a_{n-2}$ holds. If the sum of the first 1985 terms of the sequence is 1000, and the sum of the first 1995 terms is 4000, then what is the sum of the first 2002 terms of the sequence? | 3000 | 0.8125 | 4,601.5625 | 4,519.230769 | 4,958.333333 | |
Calculate the value of $\sin 68^{\circ} \sin 67^{\circ} - \sin 23^{\circ} \cos 68^{\circ}$. | \frac{\sqrt{2}}{2} | 0 | 4,183.25 | -1 | 4,183.25 | |
There are four cards, each with one of the numbers $2$, $0$, $1$, $5$ written on them. Four people, A, B, C, and D, each take one card.
A says: None of the numbers you three have differ by 1 from the number I have.
B says: At least one of the numbers you three have differs by 1 from the number I have.
C says: The numb... | 5120 | 0.0625 | 7,841.4375 | 5,693 | 7,984.666667 | |
Use $ \log_{10} 2 \equal{} 0.301,\ \log_{10} 3 \equal{} 0.477,\ \log_{10} 7 \equal{} 0.845$ to find the value of $ \log_{10} (10!)$ .
Note that you must answer according to the rules:fractional part of $ 0.5$ and higher is rounded up, and everything strictly less than $ 0.5$ is rounded down,
say $ 1.234\longr... | 22 | 0.25 | 7,839.625 | 7,408.75 | 7,983.25 | |
Let $\{a_k\}_{k=1}^{2011}$ be the sequence of real numbers defined by $a_1=0.201,$ $a_2=(0.2011)^{a_1},$ $a_3=(0.20101)^{a_2},$ $a_4=(0.201011)^{a_3}$, and in general,
\[a_k=\begin{cases}(0.\underbrace{20101\cdots 0101}_{k+2\text{ digits}})^{a_{k-1}}\qquad\text{if }k\text{ is odd,}\\(0.\underbrace{20101\cdots 01011}_{k... | 1341 | To solve this problem, we need to understand the behavior of the sequence $\{a_k\}_{k=1}^{2011}$ and determine when $a_k = b_k$, where $\{b_k\}_{k=1}^{2011}$ is the sequence $\{a_k\}_{k=1}^{2011}$ rearranged in decreasing order.
#### Step 1: Analyze the sequence $\{a_k\}_{k=1}^{2011}$
The sequence is defined recursive... | 0 | 7,991.8125 | -1 | 7,991.8125 |
Compute the least positive value of $t$ such that
\[\arcsin (\sin \alpha), \ \arcsin (\sin 2 \alpha), \ \arcsin (\sin 7 \alpha), \ \arcsin (\sin t \alpha)\]is a geometric progression for some $\alpha$ with $0 < \alpha < \frac{\pi}{2}.$ | 9 - 4 \sqrt{5} | 0 | 8,182.25 | -1 | 8,182.25 | |
A two-digit number has its unit digit greater than the tens digit by 2. If this two-digit number is less than 30, find the number. | 24 | 0.5625 | 600.3125 | 516.333333 | 708.285714 | |
A particle starts at $(0,0,0)$ in three-dimensional space. Each second, it randomly selects one of the eight lattice points a distance of $\sqrt{3}$ from its current location and moves to that point. What is the probability that, after two seconds, the particle is a distance of $2\sqrt{2}$ from its original loca... | 3/8 | 0.1875 | 7,816.875 | 6,191.333333 | 8,192 | |
a) In how many ways can 9 people arrange themselves I) on a bench II) around a circular table?
b) In how many ways can 5 men and 4 women arrange themselves on a bench such that I) no two people of the same gender sit next to each other? II) the men and women sit in separate groups (only 1 man and 1 woman sit next to ... | 5760 | 0.6875 | 5,811.4375 | 5,325.090909 | 6,881.4 | |
All the roots of the polynomial $z^6-10z^5+Az^4+Bz^3+Cz^2+Dz+16$ are positive integers, possibly repeated. What is the value of $B$? | -88 | 1. **Identify the Roots**: Given that all roots of the polynomial $z^6 - 10z^5 + Az^4 + Bz^3 + Cz^2 + Dz + 16$ are positive integers, and their sum (as coefficients of $z^5$ with a sign change) is 10, we consider possible sets of roots that sum to 10. The roots are $2, 2, 2, 2, 1, 1$.
2. **Symmetric Sums**: The coeffi... | 0.5625 | 6,137.875 | 4,973.888889 | 7,634.428571 |
An ordinary $6$-sided die has a number on each face from $1$ to $6$ (each number appears on one face). How many ways can I paint two faces of a die red, so that the numbers on the red faces don't add up to $7$? | 12 | 0.9375 | 3,453.3125 | 3,137.4 | 8,192 | |
Given \(\frac{\sin (\beta+\gamma) \sin (\gamma+\alpha)}{\cos \alpha \cos \gamma}=\frac{4}{9}\), find the value of \(\frac{\sin (\beta+\gamma) \sin (\gamma+\alpha)}{\cos (\alpha+\beta+\gamma) \cos \gamma}\). | \frac{4}{5} | 0 | 8,192 | -1 | 8,192 | |
Given a sequence $\{a_n\}$ where all terms are positive, and $a_1=2$, $a_{n+1}-a_n= \frac{4}{a_{n+1}+a_n}$. If the sum of the first $n$ terms of the sequence $\left\{ \frac{1}{a_{n-1}+a_n} \right\}$ is $5$, then $n=\boxed{120}$. | 120 | 0.4375 | 7,371.9375 | 6,317.571429 | 8,192 | |
Given an ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with its upper vertex at $(0,2)$ and an eccentricity of $\frac{\sqrt{5}}{3}$.
(1) Find the equation of ellipse $C$;
(2) From a point $P$ on the ellipse $C$, draw two tangent lines to the circle $x^{2}+y^{2}=1$, with the tangent points being ... | \frac{5}{6} | 0.625 | 6,952.875 | 6,461.9 | 7,771.166667 | |
Let $S_{0}=0$ and let $S_{k}$ equal $a_{1}+2 a_{2}+\ldots+k a_{k}$ for $k \geq 1$. Define $a_{i}$ to be 1 if $S_{i-1}<i$ and -1 if $S_{i-1} \geq i$. What is the largest $k \leq 2010$ such that $S_{k}=0$? | 1092 | Suppose that $S_{N}=0$ for some $N \geq 0$. Then $a_{N+1}=1$ because $N+1 \geq S_{N}$. The following table lists the values of $a_{k}$ and $S_{k}$ for a few $k \geq N$: $k$ & $a_{k}$ & $S_{k}$ \hline$N$ & & 0 \$N+1$ & 1 & $N+1$ \$N+2$ & 1 & $2 N+3$ \$N+3$ & -1 & $N$ \$N+4$ & 1 & $2 N+4$ \$N+5$ & -1 & $N-1$ \$N+6$ & 1 &... | 0 | 8,125.75 | -1 | 8,125.75 |
Two congruent cones, each with a radius of 15 cm and a height of 10 cm, are enclosed within a cylinder. The bases of the cones are the bases of the cylinder, and the height of the cylinder is 30 cm. Determine the volume in cubic centimeters of the space inside the cylinder that is not occupied by the cones. Express you... | 5250\pi | 0.875 | 3,635.0625 | 2,984.071429 | 8,192 | |
Find the number of functions of the form $f(x) = ax^3 + bx^2 + cx + d$ such that
\[f(x) f(-x) = f(x^3).\] | 12 | 0 | 5,286.9375 | -1 | 5,286.9375 | |
I have a bag with $8$ marbles numbered from $1$ to $8.$ Mathew has a bag with $16$ marbles numbered from $1$ to $16.$ Mathew chooses one marble from his bag and I choose two from mine. In how many ways can we choose the marbles (where the order of my choices does matter) such that the sum of the numbers on my marbles i... | 22 | 0.5625 | 6,528.4375 | 5,688.222222 | 7,608.714286 | |
In the drawing, there is a grid composed of 25 small equilateral triangles.
How many rhombuses can be formed from two adjacent small triangles? | 30 | 0 | 8,192 | -1 | 8,192 | |
The minimum value of the quotient of a (base ten) number of three different non-zero digits divided by the sum of its digits is | 10.5 | 1. **Expression Setup**: We start by expressing the number with three different non-zero digits $a$, $b$, and $c$ as $100a + 10b + c$. The sum of its digits is $a + b + c$. Thus, the quotient of the number by the sum of its digits is:
\[
\frac{100a + 10b + c}{a + b + c}
\]
2. **Simplification**: We can rewrit... | 0.0625 | 8,192 | 8,192 | 8,192 |
The side lengths of a cyclic quadrilateral are 25, 39, 52, and 60. Calculate the diameter of the circle. | 65 | 0.5625 | 6,824.4375 | 6,019.888889 | 7,858.857143 | |
The function $f(x)=(m^{2}-m-1)x^{m^{2}+m-3}$ is a power function, and when $x\in (0,+\infty)$, $f(x)$ is a decreasing function. Find the real number $m=$____. | -1 | 0 | 8,192 | -1 | 8,192 | |
Aaron takes a square sheet of paper, with one corner labeled $A$ . Point $P$ is chosen at random inside of the square and Aaron folds the paper so that points $A$ and $P$ coincide. He cuts the sheet along the crease and discards the piece containing $A$ . Let $p$ be the probability that the remaining piece is... | 25 | 0 | 8,192 | -1 | 8,192 | |
What is the smallest positive four-digit integer equivalent to 6 mod 7? | 1000 | 0.9375 | 1,672.4375 | 1,761.466667 | 337 | |
Assume that $p$, $q$, $r$, and $s$ are positive integers such that $p^3 = q^2$, $r^5 = s^4$, and $r - p = 31$. Determine $s - q$. | -2351 | 0.9375 | 5,450.6875 | 5,267.933333 | 8,192 | |
A bag contains 4 tan, 3 pink, and 5 violet chips. If the 12 chips are randomly drawn from the bag, one at a time and without replacement, what is the probability that the chips are drawn such that the 4 tan chips are drawn consecutively, the 3 pink chips are drawn consecutively, and the 5 violet chips are drawn consecu... | \frac{1}{4620} | 0.625 | 6,298.375 | 5,162.2 | 8,192 | |
In counting $n$ colored balls, some red and some black, it was found that $49$ of the first $50$ counted were red.
Thereafter, $7$ out of every $8$ counted were red. If, in all, $90$ % or more of the balls counted were red, the maximum value of $n$ is: | 210 | 1. **Understanding the Problem:**
We are given that $49$ of the first $50$ balls are red, and thereafter, $7$ out of every $8$ balls counted are red. We need to find the maximum number of balls, $n$, such that at least $90\%$ of them are red.
2. **Setting Up the Equation:**
Let $x$ be the number of batches of $8... | 0.875 | 5,113.625 | 4,673.857143 | 8,192 |
You have four textbooks for mandatory courses numbered 1 to 4. They are randomly placed on the same shelf.
1. Calculate the probability that Textbook 2 is to the left of Textbook 4.
2. Calculate the probability that Textbook 2 is to the left of Textbook 3, and Textbook 3 is to the left of Textbook 4. | \frac{1}{4} | 0 | 5,482.8125 | -1 | 5,482.8125 | |
Ten test papers are to be prepared for the National Olympiad. Each paper has 4 problems, and no two papers have more than 1 problem in common. At least how many problems are needed? | 13 | 0.25 | 7,201.875 | 5,422.5 | 7,795 | |
A frequency distribution of the scores for Mr. Sampson's algebra class is shown. What percent of the class received a score in the $60\%$-$69\%$ range? \begin{tabular}{|c|c|}
Test Scores & Frequencies\\
\hline
$90\% - 100\%$& IIII\\
$80\% - 89\%$& IIII IIII\\
$70\% - 79\%$& IIII II\\
$60\% - 69\%$ & IIII I\\
Below $60\... | 20\% | 0.875 | 2,298.5625 | 2,056.214286 | 3,995 | |
We call a pair $(a,b)$ of positive integers, $a<391$ , *pupusa* if $$ \textup{lcm}(a,b)>\textup{lcm}(a,391) $$ Find the minimum value of $b$ across all *pupusa* pairs.
Fun Fact: OMCC 2017 was held in El Salvador. *Pupusa* is their national dish. It is a corn tortilla filled with cheese, meat, etc. | 18 | 0 | 8,192 | -1 | 8,192 | |
In the parallelogram \(KLMN\), side \(KL\) is equal to 8. A circle tangent to sides \(NK\) and \(NM\) passes through point \(L\) and intersects sides \(KL\) and \(ML\) at points \(C\) and \(D\) respectively. It is known that \(KC : LC = 4 : 5\) and \(LD : MD = 8 : 1\). Find the side \(KN\). | 10 | 0 | 8,192 | -1 | 8,192 | |
Find $f(2)$ given that $f$ is a real-valued function that satisfies the equation $$ 4f(x)+\left(\frac23\right)(x^2+2)f\left(x-\frac2x\right)=x^3+1. $$ | \frac{19}{12} | 0.625 | 5,305.5625 | 3,807.7 | 7,802 | |
Given $\left(a + \frac{1}{a}\right)^2 = 5$, find the value of $a^3 + \frac{1}{a^3}$. | 2\sqrt{5} | 0 | 7,505.75 | -1 | 7,505.75 | |
Express the given value of $22$ nanometers in scientific notation. | 2.2\times 10^{-8} | 0.6875 | 510.625 | 547.181818 | 430.2 | |
Forty cards are placed into a box, each bearing a number 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10, with each number entered on four cards. Four cards are drawn from the box at random and without replacement. Let $p$ be the probability that all four cards bear the same number. Let $q$ be the probability that three of the cards ... | 144 | 0.6875 | 4,460.1875 | 3,713.636364 | 6,102.6 | |
The dilation, centered at $-1 + 4i,$ with scale factor $-2,$ takes $2i$ to which complex number? | -3 + 8i | 0.9375 | 3,639.9375 | 3,336.466667 | 8,192 | |
.4 + .02 + .006 = | .426 | To solve the problem, we add the decimal numbers together. It's helpful to align the numbers by their decimal places to ensure accurate addition:
1. Write each number with the same number of decimal places:
\[
0.4 = 0.400 \\
0.02 = 0.020 \\
0.006 = 0.006
\]
2. Add the numbers vertically, aligning the d... | 0.9375 | 327.375 | 324.266667 | 374 |
Let \( y = (17 - x)(19 - x)(19 + x)(17 + x) \), where \( x \) is a real number. Find the smallest possible value of \( y \). | -1296 | 0.875 | 5,660.625 | 5,299 | 8,192 | |
Let \( x < 0.1 \) be a positive real number. Consider the series \( 4 + 4x + 4x^2 + 4x^3 + \ldots \), and the series \( 4 + 44x + 444x^2 + 4444x^3 + \ldots \). Suppose that the sum of the second series is four times the sum of the first series. Compute \( x \). | 3/40 | 0.1875 | 5,207.0625 | 4,107 | 5,460.923077 | |
Let $N \ge 5$ be given. Consider all sequences $(e_1,e_2,...,e_N)$ with each $e_i$ equal to $1$ or $-1$ . Per move one can choose any five consecutive terms and change their signs. Two sequences are said to be similar if one of them can be transformed into the other in finitely many moves. Find the maximum num... | 16 | 0.0625 | 7,973.75 | 8,192 | 7,959.2 | |
Given \( 5 \sin 2 \alpha = \sin 2^\circ \), find the value of \( \frac{\tan (\alpha + 1^\circ)}{\tan (\alpha - 1^\circ)} \). | -\frac{3}{2} | 0.375 | 7,481.5 | 6,297.333333 | 8,192 | |
Square \( ABCD \) has center \( O \). Points \( P \) and \( Q \) are on \( AB \), \( R \) and \( S \) are on \( BC \), \( T \) and \( U \) are on \( CD \), and \( V \) and \( W \) are on \( AD \), so that \( \triangle APW \), \( \triangle BRQ \), \( \triangle CTS \), and \( \triangle DVU \) are isosceles and \( \triang... | 1:1 | 0 | 8,192 | -1 | 8,192 | |
Given the distribution of the random variable $\xi$ is $P(\xi=x)= \dfrac{xk}{15}$, where $x$ takes values $(1,2,3,4,5)$, find the value of $P\left( \left. \dfrac{1}{2} < \xi < \dfrac{5}{2} \right. \right)$. | \dfrac{1}{5} | 1 | 2,063.9375 | 2,063.9375 | -1 | |
Points $E$ and $F$ lie inside rectangle $ABCD$ with $AE=DE=BF=CF=EF$ . If $AB=11$ and $BC=8$ , find the area of the quadrilateral $AEFB$ . | 32 | 0.5625 | 6,848.4375 | 6,281 | 7,578 | |
The four-member math team at Pecanridge Middle School is chosen from the math club, which has three girls and five boys. How many different teams made up of two girls and two boys could be chosen? | 30 | 1 | 2,076.4375 | 2,076.4375 | -1 | |
Let $f(x)$ have a domain of $R$, $f(x+1)$ be an odd function, and $f(x+2)$ be an even function. When $x\in [1,2]$, $f(x)=ax^{2}+b$. If $f(0)+f(3)=6$, then calculate the value of $f\left(\frac{9}{2}\right)$. | \frac{5}{2} | 0.1875 | 7,360.9375 | 6,242.666667 | 7,619 | |
A quadrilateral is inscribed in a circle. If angles are inscribed in the four arcs cut off by the sides of the quadrilateral, their sum will be: | 180^{\circ} | 1. **Understanding the Problem**: We are given a quadrilateral inscribed in a circle. We need to find the sum of the angles inscribed in the four arcs cut off by the sides of the quadrilateral.
2. **Total Arc in a Circle**: The total measure of the arcs in a circle is $360^\circ$. This is because a circle is defined a... | 0.9375 | 3,606.6875 | 3,301 | 8,192 |
Inside triangle \(ABC\), a random point \(M\) is chosen. What is the probability that the area of one of the triangles \(ABM\), \(BCM\), and \(CAM\) will be greater than the sum of the areas of the other two? | 0.75 | 0 | 7,896.5 | -1 | 7,896.5 | |
Find all ordered pairs $(m, n)$ of integers such that $231 m^{2}=130 n^{2}$. | (0,0) | The unique solution is $(0,0)$. | 0.9375 | 5,989.5625 | 5,842.733333 | 8,192 |
Find the area in the plane contained by the graph of
\[|x + y| + |x - y| \le 4.\] | 16 | 0.6875 | 5,678.5 | 4,884.272727 | 7,425.8 | |
Through a point $P$ inside the $\triangle ABC$, a line is drawn parallel to the base $AB$, dividing the triangle into two regions where the area of the region containing the vertex $C$ is three times the area of the region adjacent to $AB$. If the altitude to $AB$ has a length of $2$, calculate the distance from $P$ to... | \frac{1}{2} | 0 | 5,534.6875 | -1 | 5,534.6875 | |
The integer 843301 is prime. The primorial of a prime number $p$, denoted $p \#$, is defined to be the product of all prime numbers less than or equal to $p$. Determine the number of digits in $843301 \#$. Your score will be $$\max \left\{\left\lfloor 60\left(\frac{1}{3}-\left|\ln \left(\frac{A}{d}\right)\right|\right)... | 365851 | Remark: 843301\#-1 is the largest known prime number of the form $p \#-1$, where $p$ is prime. | 0 | 8,039.6875 | -1 | 8,039.6875 |
Write the following expression as a polynomial: $$(2x^2+3x+7)(x+1)-(x+1)(x^2+4x-63)+(3x-14)(x+1)(x+5).$$ | 4x^3+4x^2 | 0.8125 | 4,716.25 | 3,914.153846 | 8,192 | |
The table below gives the percent of students in each grade at Annville and Cleona elementary schools:
\[\begin{tabular}{rccccccc}&\textbf{\underline{K}}&\textbf{\underline{1}}&\textbf{\underline{2}}&\textbf{\underline{3}}&\textbf{\underline{4}}&\textbf{\underline{5}}&\textbf{\underline{6}}\\ \textbf{Annville:}& 16\% &... | 15\% | 1. **Calculate the number of 6th graders in Annville:**
- Annville has 100 students in total.
- The percentage of 6th graders in Annville is 11%.
- Therefore, the number of 6th graders in Annville is:
\[
\frac{11\% \times 100}{100} = 11 \text{ students}
\]
2. **Calculate the number of 6th grade... | 0.9375 | 2,048.9375 | 1,999.133333 | 2,796 |
Let $\mathcal{T}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{abcd}$ where $a, b, c, d$ are distinct digits. Find the sum of the elements of $\mathcal{T}.$ | 2520 | 0.1875 | 7,554.25 | 4,790.666667 | 8,192 | |
Given \( \frac{\pi}{4} < \theta < \frac{\pi}{2} \), find the maximum value of \( S = \sin 2\theta - \cos^2 \theta \). | \frac{\sqrt{5} - 1}{2} | 0 | 6,615.875 | -1 | 6,615.875 | |
What is the largest possible distance between two points, one on the sphere of radius 15 with center $(3, -5, 7),$ and the other on the sphere of radius 95 with center $(-10, 20, -25)$? | 110 + \sqrt{1818} | 0.0625 | 4,719.4375 | 6,729 | 4,585.466667 | |
How many diagonals can be drawn for a hexagon? | 9 | 0.9375 | 2,375.625 | 1,987.866667 | 8,192 | |
Given \( x \) and \( y \) are in \( (0, +\infty) \), and \(\frac{19}{x} + \frac{98}{y} = 1\). What is the minimum value of \( x + y \)? | 117 + 14\sqrt{38} | 0.4375 | 7,923.1875 | 7,577.571429 | 8,192 | |
In the six-digit integer $3A6,\!792$, what is the largest digit $A$ so that the six-digit integer will be divisible by 3? | 9 | 1 | 2,078.25 | 2,078.25 | -1 | |
Medians $\overline{DP}$ and $\overline{EQ}$ of $\triangle DEF$ are perpendicular. If $DP= 15$ and $EQ = 20$, then what is ${DF}$? | \frac{20\sqrt{13}}{3} | 0 | 7,297.9375 | -1 | 7,297.9375 | |
A triangle has side lengths 7, 8, and 9. There are exactly two lines that simultaneously bisect the perimeter and area of the triangle. Let $\theta$ be the acute angle between these two lines. Find $\tan \theta.$
[asy]
unitsize(0.5 cm);
pair A, B, C, P, Q, R, S, X;
B = (0,0);
C = (8,0);
A = intersectionpoint(arc(... | 3 \sqrt{5} + 2 \sqrt{10} | 0.0625 | 8,192 | 8,192 | 8,192 | |
Let \( M = \{1, 2, 3, \cdots, 1995\} \) and \( A \subseteq M \), with the constraint that if \( x \in A \), then \( 19x \notin A \). Find the maximum value of \( |A| \). | 1890 | 0 | 8,192 | -1 | 8,192 | |
A traffic light cycles as follows: green for 45 seconds, yellow for 5 seconds, then red for 50 seconds. Felix chooses a random five-second interval to observe the light. What is the probability that the color changes while he is observing? | \frac{3}{20} | 0.4375 | 6,709.5625 | 5,544.714286 | 7,615.555556 | |
In rectangle $ABCD$, diagonal $DB$ is divided into three segments of lengths $1$, $2$, and $3$ by parallel lines $L$ and $L'$, which pass through vertices $A$ and $C$ and are perpendicular to diagonal $DB$. Calculate the area of rectangle $ABCD$.
A) $6\sqrt{5}$
B) $12$
C) $12\sqrt{5}$
D) $30$
E) $18\sqrt{3}$ | 6\sqrt{5} | 0 | 8,192 | -1 | 8,192 | |
Sabrina has a fair tetrahedral die whose faces are numbered 1, 2, 3, and 4, respectively. She creates a sequence by rolling the die and recording the number on its bottom face. However, she discards (without recording) any roll such that appending its number to the sequence would result in two consecutive terms that ... | 10 | 0 | 8,177.625 | -1 | 8,177.625 | |
What percentage of a seven-by-seven grid is shaded?
In a seven-by-seven grid, alternate squares are shaded starting with the top left square similar to a checkered pattern. However, an entire row (the fourth row from the top) and an entire column (the fourth column from the left) are left completely unshaded. | 73.47\% | 0.0625 | 8,077.5 | 7,322 | 8,127.866667 | |
Consider a $6\times 6\times h$ rectangular box containing a larger sphere of radius $3$ and four smaller spheres, each with radius $2$. The smaller spheres are placed at each corner of the bottom square face of the box and are tangent to two adjacent sides of the box. The larger sphere is tangent to all four smaller sp... | 5 + \sqrt{23} | 0 | 7,484.4375 | -1 | 7,484.4375 | |
Find the average of the quantity
\[(a_1 - a_2)^2 + (a_2 - a_3)^2 +\cdots + (a_{n-1} -a_n)^2\]
taken over all permutations $(a_1, a_2, \dots , a_n)$ of $(1, 2, \dots , n).$ | \frac{(n-1)n(n+1)}6 |
To find the average of the expression
\[
(a_1 - a_2)^2 + (a_2 - a_3)^2 + \cdots + (a_{n-1} - a_n)^2
\]
over all permutations \((a_1, a_2, \dots, a_n)\) of \((1, 2, \dots, n)\), we need to consider the contribution of each term \((a_i - a_{i+1})^2\) in the sum.
### Step 1: Understanding the Contribution of Each Pai... | 0 | 7,543.75 | -1 | 7,543.75 |
The expression $\frac{2021}{2020} - \frac{2020}{2021}$ is equal to the fraction $\frac{p}{q}$ in which $p$ and $q$ are positive integers whose greatest common divisor is 1. What is $p?$ | 2021 | 1. Let $a = 2020$. We start by rewriting the given expression using $a$:
\[
\frac{2021}{2020} - \frac{2020}{2021} = \frac{a+1}{a} - \frac{a}{a+1}
\]
2. To combine these fractions, find a common denominator, which is $a(a+1)$:
\[
\frac{(a+1)^2 - a^2}{a(a+1)}
\]
3. Simplify the numerator:
\[
(a+... | 0 | 5,572.625 | -1 | 5,572.625 |
Let \( P \) be a regular 2006-sided polygon. A diagonal of \( P \) is called a "good edge" if both parts of \( P \)'s boundary divided by this diagonal contain an odd number of edges. Each side of \( P \) is also defined as a "good edge."
Given that 2003 non-intersecting diagonals inside \( P \) divide it into several... | 1003 | 0.1875 | 8,167.8125 | 8,063 | 8,192 | |
Given a tetrahedron \( A B C D \) with side lengths \( A B = 41 \), \( A C = 7 \), \( A D = 18 \), \( B C = 36 \), \( B D = 27 \), and \( C D = 13 \), let \( d \) be the distance between the midpoints of edges \( A B \) and \( C D \). Find the value of \( d^{2} \). | 137 | 0.125 | 7,905.6875 | 7,145.5 | 8,014.285714 | |
Regular hexagon $PQRSTU$ has vertices $P$ and $R$ at $(0,0)$ and $(8,2)$, respectively. What is its area? | 102\sqrt{3} | 0 | 7,735.0625 | -1 | 7,735.0625 | |
In the number $2 * 0 * 1 * 6 * 0 * 2 *$, each of the 6 asterisks must be replaced with any of the digits $0, 2, 4, 5, 7, 9$ (digits may be repeated) so that the resulting 12-digit number is divisible by 12. How many ways can this be done? | 5184 | 0.0625 | 8,102.6875 | 7,997 | 8,109.733333 | |
Equilateral triangle $T$ is inscribed in circle $A$, which has radius $10$. Circle $B$ with radius $3$ is internally tangent to circle $A$ at one vertex of $T$. Circles $C$ and $D$, both with radius $2$, are internally tangent to circle $A$ at the other two vertices of $T$. Circles $B$, $C$, and $D$ are all externally ... | 32 | Let $X$ be the intersection of the circles with centers $B$ and $E$, and $Y$ be the intersection of the circles with centers $C$ and $E$. Since the radius of $B$ is $3$, $AX =4$. Assume $AE$ = $p$. Then $EX$ and $EY$ are radii of circle $E$ and have length $4+p$. $AC = 8$, and angle $CAE = 60$ degrees because we are gi... | 0.6875 | 6,056.3125 | 5,151.909091 | 8,046 |
If $f(x)$ is a monic quartic polynomial such that $f(-1)=-1, f(2)=-4, f(-3)=-9$, and $f(4)=-16$, find $f(1)$. | 23 | The given data tells us that the roots of $f(x)+x^{2}$ are $-1,2,-3$, and 4. Combining with the fact that $f$ is monic and quartic we get $f(x)+x^{2}=(x+1)(x-2)(x+3)(x-4)$. Hence $f(1)=(2)(-1)(4)(-3)-1=\mathbf{23}$. | 0.75 | 5,111 | 4,084 | 8,192 |
In $\triangle ABC$, the three internal angles are $A$, $B$, and $C$. If $\dfrac{\sqrt{3}\cos A + \sin A}{\sqrt{3}\sin A - \cos A} = \tan(-\dfrac{7}{12}\pi)$, find the maximum value of $2\cos B + \sin 2C$. | \dfrac{3}{2} | 0.9375 | 4,975.3125 | 4,760.866667 | 8,192 | |
How many whole numbers are between $\sqrt{8}$ and $\sqrt{80}$? | 6 | 1. **Identify the range of numbers between $\sqrt{8}$ and $\sqrt{80}$:**
- We start by approximating $\sqrt{8}$ and $\sqrt{80}$. We know that $\sqrt{9} = 3$ and $\sqrt{4} = 2$, and since $8$ is between $4$ and $9$, it follows that $\sqrt{8}$ is between $2$ and $3$. Therefore, $\sqrt{8}$ is closer to $3$.
- Simila... | 1 | 3,369 | 3,369 | -1 |
In the quadrilateral \(ABCD\), it is given that \(\cos \angle BAD = \frac{3}{4}\), \(\angle BAC = \angle DAC\), \(AD < AB\), and \(AB = 5\), \(AC = BD = \sqrt{14}\). If \(\overrightarrow{AC} = \lambda \overrightarrow{AB} + \mu \overrightarrow{AD}\) (\(\lambda, \mu \in \mathbf{R}\)), find \(\lambda + \mu\). | \frac{7}{5} | 0.0625 | 7,797.6875 | 8,192 | 7,771.4 | |
A square has vertices at \((-2a, -2a), (2a, -2a), (-2a, 2a), (2a, 2a)\). The line \( y = -\frac{x}{2} \) cuts this square into two congruent quadrilaterals. Calculate the perimeter of one of these quadrilaterals divided by \( 2a \). Express your answer in simplified radical form. | 4 + \sqrt{5} | 0.5 | 6,663.25 | 5,293.75 | 8,032.75 | |
Compute the sum of \(0.\overline{25} + 0.\overline{87}\) as a common fraction. | \frac{112}{99} | 0.9375 | 3,200.125 | 2,867.333333 | 8,192 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.