problem stringlengths 10 5.15k | answer stringlengths 0 1.22k | solution stringlengths 0 11.1k | reward float64 0 1 | length float64 172 8.19k | correct_length float64 -1 8.19k | incorrect_length float64 -1 8.19k |
|---|---|---|---|---|---|---|
In a right triangle, one of the acute angles $\alpha$ satisfies
\[\tan \frac{\alpha}{2} = \frac{1}{\sqrt[3]{2}}.\]Let $\theta$ be the angle between the median and the angle bisector drawn from this acute angle. Find $\tan \theta.$ | \frac{1}{2} | 0 | 8,192 | -1 | 8,192 | |
Given the geometric sequence $\{a_{n}\}$, $3a_{5}-a_{3}a_{7}=0$. If $\{b_{n}\}$ is an arithmetic sequence where $b_{5}=a_{5}$, find the sum of the first 9 terms, $S_{9}$, of $\{b_{n}\}$. | 27 | 1 | 2,828.625 | 2,828.625 | -1 | |
Let $ABCD$ be a trapezoid with $AB \parallel CD$. The bisectors of $\angle CDA$ and $\angle DAB$ meet at $E$, the bisectors of $\angle ABC$ and $\angle BCD$ meet at $F$, the bisectors of $\angle BCD$ and $\angle CDA$ meet at $G$, and the bisectors of $\angle DAB$ and $\angle ABC$ meet at $H$. Quadrilaterals $EABF$ and ... | \frac{256}{7} | Let $M, N$ be the midpoints of $AD, BC$ respectively. Since $AE$ and $DE$ are bisectors of supplementary angles, the triangle $AED$ is right with right angle $E$. Then $EM$ is the median of a right triangle from the right angle, so triangles $EMA$ and $EMD$ are isosceles with vertex $M$. But then $\angle MEA=\angle EAM... | 0 | 8,192 | -1 | 8,192 |
What is the value of $a^3 - b^3$ given that $a+b=12$ and $ab=20$? | 992 | 0.4375 | 6,337.625 | 4,312.142857 | 7,913 | |
Let $a^2 = \frac{9}{25}$ and $b^2 = \frac{(3+\sqrt{7})^2}{14}$, where $a$ is a negative real number and $b$ is a positive real number. If $(a-b)^2$ can be expressed in the simplified form $\frac{x\sqrt{y}}{z}$ where $x$, $y$, and $z$ are positive integers, what is the value of the sum $x+y+z$? | 22 | 0 | 8,192 | -1 | 8,192 | |
Let $P(x) = x^2 - 3x - 9$. A real number $x$ is chosen at random from the interval $5 \le x \le 15$. The probability that $\lfloor\sqrt{P(x)}\rfloor = \sqrt{P(\lfloor x \rfloor)}$ is equal to $\frac{\sqrt{a} + \sqrt{b} + \sqrt{c} - d}{e}$ , where $a$, $b$, $c$, $d$, and $e$ are positive integers. Find $a + b + c + d + ... | 850 | Note that all the "bounds" have to be less than the number+1, otherwise it wouldn't fit the answer format. Therefore, the answer is $\frac{3*3+\sqrt{9+4(4+9)}-10+\sqrt{9+4(16+9)}-12+\sqrt{9+4(144+9)}}{20} \implies \boxed{850}$
~Lcz | 0.375 | 7,307.875 | 6,865.666667 | 7,573.2 |
If $2^a+2^b=3^c+3^d$, the number of integers $a,b,c,d$ which can possibly be negative, is, at most: | 0 | 1. **Assume $c,d \geq 0$ and $a < 0$ (WLOG $a \leq b$):**
- The equation becomes:
\[
\frac{1}{2^{-a}} + 2^b = 3^c + 3^d
\]
- Multiplying both sides by $2^{-a}$, we get:
\[
1 + 2^{b-a} = 2^{-a}(3^c + 3^d)
\]
- Since $2^{-a}(3^c + 3^d)$ must be an integer, and $2^{b-a}$ is an intege... | 0 | 8,192 | -1 | 8,192 |
I have 5 books, two of which are identical copies of the same math book (and all of the rest of the books are different). In how many ways can I arrange them on a shelf? | 60 | 1 | 2,284.4375 | 2,284.4375 | -1 | |
Calculate the area of a rectangle which remains unchanged when it is made $3 \frac{1}{2}$ inches longer and $1 \frac{1}{3}$ inch narrower, or when it is made $3 \frac{1}{2}$ inches shorter and $1 \frac{2}{3}$ inch wider. | 35 | 0 | 4,840.8125 | -1 | 4,840.8125 | |
In tetrahedron $ABCD,$
\[\angle ADB = \angle ADC = \angle BDC = 90^\circ.\]Also, $x = \sin \angle CAD$ and $y = \sin \angle CBD.$ Express $\cos \angle ACB$ in terms of $x$ and $y.$ | xy | 0.75 | 5,407.0625 | 4,614.083333 | 7,786 | |
Given a function $f(x)$ defined on $\mathbb{R}$ satisfies $f(x+6)=f(x)$. When $-3<x\leq-1$, $f(x)=-(x+2)^2$, and when $-1\leq x\leq 3$, $f(x)=x$. Calculate the value of $f(1) + f(2) + \dots + f(2015)$. | 1680 | 0.625 | 5,958.5625 | 5,456.6 | 6,795.166667 | |
What is $35_8-74_8?$ Express your answer in base 8. | -37_8 | 0.625 | 6,577 | 5,881.4 | 7,736.333333 | |
Using one each of the coins and bills of 1 jiao, 2 jiao, 5 jiao, 1 yuan, 2 yuan, and 5 yuan, how many different monetary values can be formed? | 63 | 0.0625 | 7,875.875 | 3,817 | 8,146.466667 | |
A large chest contains 10 smaller chests. In each of the smaller chests, either 10 even smaller chests are placed or nothing is placed. In each of those smaller chests, either 10 smaller chests are placed or none, and so on. After this, there are exactly 2006 chests with contents. How many are empty? | 18054 | 0 | 8,032.3125 | -1 | 8,032.3125 | |
Find a monic cubic polynomial $P(x)$ with integer coefficients such that
\[P(\sqrt[3]{2} + 1) = 0.\](A polynomial is monic if its leading coefficient is 1.) | x^3 - 3x^2 + 3x - 3 | 1 | 2,133.625 | 2,133.625 | -1 | |
Let $x,$ $y,$ and $z$ be nonnegative real numbers such that $x + y + z = 2.$ Find the maximum value of
\[(x^2 - xy + y^2)(x^2 - xz + z^2)(y^2 - yz + z^2).\] | \frac{256}{243} | 0 | 8,192 | -1 | 8,192 | |
Let $p, q, r$, and $s$ be positive real numbers such that
\[
\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}
p^2+q^2&=&r^2+s^2&=&2500,\\
pr&=&qs&=&1200.
\end{array}
\]
Compute the value of $\lfloor T \rfloor$, where $T=p+q+r+s$. | 140 | 0.9375 | 4,155.1875 | 3,886.066667 | 8,192 | |
Elbert and Yaiza each draw 10 cards from a 20-card deck with cards numbered $1,2,3, \ldots, 20$. Then, starting with the player with the card numbered 1, the players take turns placing down the lowest-numbered card from their hand that is greater than every card previously placed. When a player cannot place a card, the... | 324 | Put each card in order and label them based on if Elbert or Yaiza got them. We will get a string of E's and Y's like EEYYYE ..., and consider the "blocks" of consecutive letters. It is not hard to see that only the first card of each block is played, and the number of cards played is exactly the number of blocks. Thus,... | 0 | 8,192 | -1 | 8,192 |
Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[f(x(x + f(y))) = (x + y)f(x),\]
for all $x, y \in\mathbb{R}$. | f(x) = 0 \text{ and } f(x) = x. |
Let's consider the functional equation \( f(x(x + f(y))) = (x + y)f(x) \) for all \( x, y \in \mathbb{R} \).
### Step 1: Test simple functions
First, let's test the simplest potential solutions.
1. **\( f(x) = 0 \):**
- Substituting \( f(x) = 0 \) into the equation gives:
\[
f(x(x + f(y))) = f(0) = 0, \quad... | 0 | 8,021.8125 | -1 | 8,021.8125 |
Let $f$ be the function defined by $f(x)=ax^2-\sqrt{2}$ for some positive $a$. If $f(f(\sqrt{2}))=-\sqrt{2}$ then $a=$ | \frac{\sqrt{2}}{2} | 1. **Define the function and apply it to $\sqrt{2}$:**
Given the function $f(x) = ax^2 - \sqrt{2}$, we first find $f(\sqrt{2})$:
\[
f(\sqrt{2}) = a(\sqrt{2})^2 - \sqrt{2} = 2a - \sqrt{2}.
\]
2. **Apply the function $f$ to the result of $f(\sqrt{2})$:**
Next, we need to find $f(f(\sqrt{2}))$:
\[
f(... | 0 | 2,969.8125 | -1 | 2,969.8125 |
Find a positive integer that is divisible by 18 and has a square root between 26 and 26.2. | 684 | 0.9375 | 2,651.1875 | 2,771 | 854 | |
When $n$ is divided by 6, a remainder of 1 is given. What is the remainder when $n+2010$ is divided by 6? | 1 | 1 | 2,169.125 | 2,169.125 | -1 | |
The $y$-intercepts of three parallel lines are 2, 3, and 4. The sum of the $x$-intercepts of the three lines is 36. What is the slope of these parallel lines? | -\frac{1}{4} | Suppose the slope of the three parallel lines is $m$. The equation of a line with slope $m$ and $y$-intercept 2 is $y=mx+2$. To find the $x$-intercept in terms of $m$, we set $y=0$ and solve for $x$. Doing this, we obtain $mx+2=0$ or $x=-\frac{2}{m}$. Similarly, the line with slope $m$ and $y$-intercept 3 has $x$-inter... | 1 | 1,793.625 | 1,793.625 | -1 |
Given that an ellipse has the equation $\frac {x^{2}}{a^{2}} + \frac {y^{2}}{b^{2}} = 1$ with $a > b > 0$ and eccentricity $e = \frac {\sqrt {6}}{3}$. The distance from the origin to the line that passes through points $A(0,-b)$ and $B(a,0)$ is $\frac {\sqrt {3}}{2}$.
$(1)$ Find the equation of the ellipse.
$(2)$ Giv... | \frac {7}{6} | 1 | 5,709.0625 | 5,709.0625 | -1 | |
Knot is ready to face Gammadorf in a card game. In this game, there is a deck with twenty cards numbered from 1 to 20. Each player starts with a five card hand drawn from this deck. In each round, Gammadorf plays a card in his hand, then Knot plays a card in his hand. Whoever played a card with greater value gets a poi... | 2982 | Knot can only lose if all of his cards are lower than 10; if not he can win by playing the lowest card that beats Gammadorf's card, or if this is not possible, his lowest card, each turn. There are $\binom{7}{5}=21$ losing hands, so he has $\binom{15}{5}-\binom{7}{5}$ possible winning hands. | 0 | 8,035.5625 | -1 | 8,035.5625 |
Antoine, Benoît, Claude, Didier, Étienne, and Françoise go to the cinéma together to see a movie. The six of them want to sit in a single row of six seats. But Antoine, Benoît, and Claude are mortal enemies and refuse to sit next to either of the other two. How many different arrangements are possible? | 144 | 0.0625 | 8,085.4375 | 8,192 | 8,078.333333 | |
In parallelogram $ABCD$, let $O$ be the intersection of diagonals $\overline{AC}$ and $\overline{BD}$. Angles $CAB$ and $DBC$ are each twice as large as angle $DBA$, and angle $ACB$ is $r$ times as large as angle $AOB$. Find $r.$ | \frac{7}{9} | 0.0625 | 7,145.5625 | 8,192 | 7,075.8 | |
Given that the function $f(x)$ is an odd function defined on $\mathbb{R}$ and $f(x+ \frac{5}{2})=-\frac{1}{f(x)}$, and when $x \in [-\frac{5}{2}, 0]$, $f(x)=x(x+ \frac{5}{2})$, find $f(2016)=$ \_\_\_\_\_\_. | \frac{3}{2} | 0.0625 | 6,945.75 | 7,728 | 6,893.6 | |
If we exchange a 10-dollar bill into dimes and quarters, what is the total number \( n \) of different ways to have two types of coins? | 20 | 0 | 3,844.875 | -1 | 3,844.875 | |
Given the real numbers \( x_1, x_2, \ldots, x_{2001} \) satisfy \( \sum_{k=1}^{2000} \left|x_k - x_{k+1}\right| = 2001 \). Let \( y_k = \frac{1}{k} \left( x_1 + x_2 + \cdots + x_k \right) \) for \( k = 1, 2, \ldots, 2001 \). Find the maximum possible value of \( \sum_{k=1}^{2000} \left| y_k - y_{k+1} \right| \). | 2000 | 0 | 8,192 | -1 | 8,192 | |
In the set \(\{1, 2, 3, \cdots, 99, 100\}\), how many numbers \(n\) satisfy the condition that the tens digit of \(n^2\) is odd?
(45th American High School Mathematics Examination, 1994) | 20 | 0.1875 | 8,021.8125 | 7,284.333333 | 8,192 | |
What is the smallest positive multiple of $17$ that is $3$ more than a multiple of $71$? | 1139 | 0.8125 | 5,512.4375 | 4,894.076923 | 8,192 | |
The older brother and the younger brother each bought several apples. The older brother said to the younger brother, "If I give you one apple, we will have the same number of apples." The younger brother thought for a moment and said to the older brother, "If I give you one apple, the number of apples you have will be ... | 12 | 0.3125 | 4,454.75 | 2,013.6 | 5,564.363636 | |
A regular hexagon \( A B C D E K \) is inscribed in a circle of radius \( 3 + 2\sqrt{3} \). Find the radius of the circle inscribed in the triangle \( B C D \). | \frac{3}{2} | 0.5 | 7,371.4375 | 6,810.25 | 7,932.625 | |
Given the equation $a + b = 30$, where $a$ and $b$ are positive integers, how many distinct ordered-pair solutions $(a, b)$ exist? | 29 | 1 | 1,668.625 | 1,668.625 | -1 | |
Simplify first, then evaluate: $(1-\frac{m}{{m+3}})÷\frac{{{m^2}-9}}{{{m^2}+6m+9}}$, where $m=\sqrt{3}+3$. | \sqrt{3} | 0.6875 | 3,874.0625 | 3,051.454545 | 5,683.8 | |
A vessel with a capacity of 6 liters contains 4 liters of a 70% sulfuric acid solution. Another vessel of the same capacity contains 3 liters of a 90% sulfuric acid solution. Some amount of the solution is transferred from the second vessel to the first one so that the first vessel ends up with an \( r\% \) sulfuric ac... | 76 | 0.75 | 4,574.0625 | 4,371.5 | 5,181.75 | |
Let $A$ and $B$ be the endpoints of a semicircular arc of radius $4$. This arc is divided into nine congruent arcs by eight equally spaced points $C_1$, $C_2$, $\dots$, $C_8$. Draw all chords of the form $\overline{AC_i}$ or $\overline{BC_i}$. Find the product of the lengths of these sixteen chords. | 38654705664 | 0 | 8,082.625 | -1 | 8,082.625 | |
\(\triangle ABC\) is isosceles with base \(AC\). Points \(P\) and \(Q\) are respectively in \(CB\) and \(AB\) and such that \(AC=AP=PQ=QB\).
The number of degrees in \(\angle B\) is: | 25\frac{5}{7} | 1. **Identify the properties of the triangles**: Given that $\triangle ABC$ is isosceles with base $AC$, and $AC = AP = PQ = QB$, we can infer that $\triangle APQ$ and $\triangle BQP$ are also isosceles. Additionally, $\triangle APC$ and $\triangle BQC$ are isosceles because $AP = AC$ and $BQ = BC$.
2. **Analyze $\tri... | 0 | 8,060.4375 | -1 | 8,060.4375 |
Give an example of a quadratic function that has zeroes at $x=2$ and $x=4$, and that takes the value $6$ when $x=3$.
Enter your answer in the expanded form "ax^2 + bx + c", where a,b,c are replaced by appropriate numbers. | -6x^2+36x-48 | 0.9375 | 1,133.5625 | 1,163.6 | 683 | |
Compute
\[\frac{1}{\cos 80^\circ} - \frac{\sqrt{3}}{\sin 80^\circ}.\] | 4 | 1 | 3,957.6875 | 3,957.6875 | -1 | |
Two circles of radius 1 are to be constructed as follows. The center of circle $A$ is chosen uniformly and at random from the line segment joining $(0,0)$ and $(2,0)$. The center of circle $B$ is chosen uniformly and at random, and independently of the first choice, from the line segment joining $(0,1)$ to $(2,1)$. Wha... | \frac {4 \sqrt {3} - 3}{4} | To solve this problem, we need to determine the probability that the distance between the centers of circles $A$ and $B$ is less than or equal to 2 (since each circle has a radius of 1, and two circles intersect if the distance between their centers is less than or equal to the sum of their radii).
#### Step 1: Define... | 0 | 7,739.6875 | -1 | 7,739.6875 |
The sides of a non-degenerate isosceles triangle are \(x\), \(x\), and \(24\) units. How many integer values of \(x\) are possible? | 11 | 0.0625 | 7,481.125 | 6,019 | 7,578.6 | |
Billy starts his hike in a park and walks eastward for 7 miles. Then, he turns $45^{\circ}$ northward and hikes another 8 miles. Determine how far he is from his starting point. Express your answer in simplest radical form. | \sqrt{113 + 56\sqrt{2}} | 0 | 7,511.5 | -1 | 7,511.5 | |
Determine how many integer palindromes are between 200 and 700. | 50 | 0.9375 | 4,488.625 | 4,241.733333 | 8,192 | |
A cuckoo clock produces a number of "cuckoo" sounds equal to the hour it indicates (for example, at 19:00, it sounds "cuckoo" 7 times). One morning, Maxim approaches the clock at 9:05 and starts turning the minute hand forward until the clock shows 7 hours later. How many "cuckoo" sounds are made during this time? | 43 | 0 | 5,826.6875 | -1 | 5,826.6875 | |
Let $1$; $4$; $\ldots$ and $9$; $16$; $\ldots$ be two arithmetic progressions. The set $S$ is the union of the first $2004$ terms of each sequence. How many distinct numbers are in $S$? | 3722 | 1. **Identify the sequences and their general terms**:
- The first sequence is $1, 4, 7, \ldots$ which is an arithmetic progression with the first term $a_1 = 1$ and common difference $d_1 = 3$. The $k$-th term of this sequence can be expressed as $a_k = 3k + 1$ for $k \geq 0$.
- The second sequence is $9, 16, 23... | 0.6875 | 6,249.9375 | 5,445.181818 | 8,020.4 |
Given that $\binom{18}{8}=31824$, $\binom{18}{9}=48620$, and $\binom{18}{10}=43758$, calculate $\binom{20}{10}$. | 172822 | 0 | 8,190.4375 | -1 | 8,190.4375 | |
What is the total number of digits used when the first 2002 positive even integers are written? | 7456 | 0.6875 | 5,846.125 | 4,825.363636 | 8,091.8 | |
Find the smallest positive real number $\lambda$ such that for every numbers $a_1,a_2,a_3 \in \left[0, \frac{1}{2} \right]$ and $b_1,b_2,b_3 \in (0, \infty)$ with $\sum\limits_{i=1}^3a_i=\sum\limits_{i=1}^3b_i=1,$ we have $$ b_1b_2b_3 \le \lambda (a_1b_1+a_2b_2+a_3b_3). $$ | 1/8 | 0.0625 | 8,171.4375 | 7,863 | 8,192 | |
Point \( M \) lies on the edge \( AB \) of cube \( ABCD A_1 B_1 C_1 D_1 \). Rectangle \( MNLK \) is inscribed in square \( ABCD \) in such a way that one of its vertices is at point \( M \), and the other three vertices are located on different sides of the base square. Rectangle \( M_1N_1L_1K_1 \) is the orthogonal pr... | 1:4 | 0 | 8,192 | -1 | 8,192 | |
Find the maximum value of the expression \( x + y \) if \( (2 \sin x - 1)(2 \cos y - \sqrt{3}) = 0 \), \( x \in [0, \frac{3\pi}{2}] \), \( y \in [\pi, 2\pi] \). | \frac{8\pi}{3} | 0.5625 | 6,114.5 | 4,909.666667 | 7,663.571429 | |
Given that the terminal side of angle $a$ passes through point P(4, -3), find:
1. The value of $2\sin{a} - \cos{a}$
2. The coordinates of point P where the terminal side of angle $a$ intersects the unit circle. | -2 | 0 | 1,898.875 | -1 | 1,898.875 | |
Determine the following number:
\[
\frac{12346 \cdot 24689 \cdot 37033 + 12347 \cdot 37034}{12345^{2}}
\] | 74072 | 0.5625 | 5,751.5625 | 3,853.444444 | 8,192 | |
Given vectors $a$ and $b$ that satisfy $|a|=2$, $|b|=1$, and $a\cdot (a-b)=3$, find the angle between $a$ and $b$. | \frac{\pi }{3} | 0 | 1,635.3125 | -1 | 1,635.3125 | |
How many even integers between 4000 and 7000 have four different digits?
| 728 | 0.375 | 7,339.4375 | 5,923.833333 | 8,188.8 | |
I have a bag with $5$ marbles numbered from $1$ to $5.$ Mathew has a bag with $12$ marbles numbered from $1$ to $12.$ Mathew picks one marble from his bag and I pick two from mine, with the choice order being significant. In how many ways can we choose the marbles such that the sum of the numbers on my marbles exceeds ... | 24 | 0 | 7,605.4375 | -1 | 7,605.4375 | |
Divide 6 volunteers into 4 groups for service at four different venues of the 2012 London Olympics. Among these groups, 2 groups will have 2 people each, and the other 2 groups will have 1 person each. How many different allocation schemes are there? (Answer with a number) | 540 | 0 | 7,949.625 | -1 | 7,949.625 | |
Given that $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=\sqrt{2}$, and $\overrightarrow{a} \bot (\overrightarrow{a}-\overrightarrow{b})$, find the projection of vector $\overrightarrow{a}$ in the direction of vector $\overrightarrow{b}$. | \frac{\sqrt{2}}{2} | 0 | 3,017.625 | -1 | 3,017.625 | |
If $a$ is the smallest positive two-digit multiple of $3$, and $b$ is the smallest positive three-digit multiple of $4$, then find $a+b$. | 112 | 1 | 1,378.875 | 1,378.875 | -1 | |
Compute the length of the segment tangent from the origin to the circle that passes through the points $(4,5)$, $(8,10)$, and $(10,25)$. | \sqrt{82} | 0.375 | 8,083.3125 | 7,902.166667 | 8,192 | |
Lottery. (For 7th grade, 3 points) It so happened that Absent-Minded Scientist has only 20 rubles left, but he needs to buy a bus ticket to get home. The bus ticket costs 45 rubles. Nearby the bus stop, instant lottery tickets are sold for exactly 10 rubles each. With a probability of $p = 0.1$, a ticket contains a win... | 0.19 | 0.125 | 7,412.5 | 4,350.5 | 7,849.928571 | |
In his spare time, Thomas likes making rectangular windows. He builds windows by taking four $30\text{ cm}\times20\text{ cm}$ rectangles of glass and arranging them in a larger rectangle in wood. The window has an $x\text{ cm}$ wide strip of wood between adjacent glass pieces and an $x\text{ cm}$ wide strip of wo... | \frac{20}{3} | 0.0625 | 6,028.375 | 7,383 | 5,938.066667 | |
Given that $a_1$, $a_2$, $a_3$, $a_4$, $a_5$ are five different integers satisfying the condition $a_1 + a_2 + a_3 + a_4 + a_5 = 9$, if $b$ is an integer root of the equation $(x - a_1)(x - a_2)(x - a_3)(x - a_4)(x - a_5) = 2009$, then the value of $b$ is. | 10 | 0.5625 | 6,636.1875 | 5,426.111111 | 8,192 | |
In a set of 15 different-colored markers, how many ways can Jane select five markers if the order of selection does not matter? | 3003 | 0.875 | 1,305.25 | 1,359.071429 | 928.5 | |
Find the integer \( n \), \( -180 \le n \le 180 \), such that \( \cos n^\circ = \cos 745^\circ \). | -25 | 0.0625 | 5,524.1875 | 5,806 | 5,505.4 | |
Let $a$ and $b$ be real numbers greater than 1 such that $a b=100$. The maximum possible value of $a^{(\log_{10} b)^{2}}$ can be written in the form $10^{x}$ for some real number $x$. Find $x$. | \frac{32}{27} | Let $p=\log_{10} a, q=\log_{10} b$. Since $a, b>1, p$ and $q$ are positive. The condition $a b=100$ translates to $p+q=2$. We wish to maximize $$x=\log_{10} a^{(\log_{10} b)^{2}}=(\log_{10} a)(\log_{10} b)^{2}=p q^{2}$$ By AM-GM, $$\frac{27}{4} p q^{2} \leq\left(p+\frac{q}{2}+\frac{q}{2}\right)^{3}=8$$ Hence $p q^{2} \... | 1 | 3,901.0625 | 3,901.0625 | -1 |
Determine the number of 8-tuples of nonnegative integers $\left(a_{1}, a_{2}, a_{3}, a_{4}, b_{1}, b_{2}, b_{3}, b_{4}\right)$ satisfying $0 \leq a_{k} \leq k$, for each $k=1,2,3,4$, and $a_{1}+a_{2}+a_{3}+a_{4}+2 b_{1}+3 b_{2}+4 b_{3}+5 b_{4}=19$. | 1540 | For each $k=1,2,3,4$, note that set of pairs $\left(a_{k}, b_{k}\right)$ with $0 \leq a_{k} \leq k$ maps bijectively to the set of nonnegative integers through the map $\left(a_{k}, b_{k}\right) \mapsto a_{k}+(k+1) b_{k}$, as $a_{k}$ is simply the remainder of $a_{k}+(k+1) b_{k}$ upon division by $k+1$. By letting $x_{... | 0 | 8,192 | -1 | 8,192 |
On her first day of work, Janabel sold one widget. On day two, she sold three widgets. On day three, she sold five widgets, and on each succeeding day, she sold two more widgets than she had sold on the previous day. How many widgets in total had Janabel sold after working $20$ days? | 400 |
1. **Identify the pattern of widgets sold each day**: Janabel's sales pattern starts with 1 widget on the first day and increases by 2 widgets each subsequent day. This forms an arithmetic sequence where the first term \(a_1 = 1\) and the common difference \(d = 2\).
2. **General formula for the nth term of an arithm... | 1 | 1,880.75 | 1,880.75 | -1 |
Exactly at noon, Anna Kuzminichna looked out the window and saw Klava, the rural store salesperson, going on break. At two minutes past noon, Anna Kuzminichna looked out the window again, and there was still no one in front of the closed store. Klava was gone for exactly 10 minutes, and when she returned, she found Iva... | 0.75 | 0 | 7,001.8125 | -1 | 7,001.8125 | |
The operation $\odot$ is defined as $a \odot b = a + \frac{3a}{2b}$. What is the value of $8 \odot 6$? | 10 | 0.9375 | 2,138.1875 | 1,734.6 | 8,192 | |
If the graph of the function $f(x)=\sin 2x+\cos 2x$ is translated to the left by $\varphi (\varphi > 0)$ units, and the resulting graph is symmetric about the $y$-axis, then find the minimum value of $\varphi$. | \frac{\pi}{8} | 0.9375 | 4,877.6875 | 4,656.733333 | 8,192 | |
Find the focus of the the parabola $y = x^2.$ | \left( 0, \frac{1}{4} \right) | 1 | 2,350.4375 | 2,350.4375 | -1 | |
Find the minimum positive period of the function \( f(x) = |\tan 2x| \). | \frac{\pi}{2} | 0.6875 | 6,076.75 | 5,448.818182 | 7,458.2 | |
In a WeChat group, there are five people playing the red envelope game: A, B, C, D, and E. There are 4 red envelopes, each person can grab at most one, and all red envelopes must be grabbed. Among the 4 red envelopes, there are two worth 2 yuan, one worth 3 yuan, and one worth 4 yuan (red envelopes with the same amount... | 36 | 0.125 | 7,447.0625 | 5,586 | 7,712.928571 | |
If \( 9210 - 9124 = 210 - \square \), the value represented by the \( \square \) is: | 124 | 1 | 1,545.5 | 1,545.5 | -1 | |
Let $f(x)=2x+1$. Find the sum of all $x$ that satisfy the equation $f^{-1}(x)=f(x^{-1})$. | 3 | 1 | 1,934.8125 | 1,934.8125 | -1 | |
In $\triangle ABC$, angle bisectors $BD$ and $CE$ intersect at $I$, with $D$ and $E$ located on $AC$ and $AB$ respectively. A perpendicular from $I$ to $DE$ intersects $DE$ at $P$, and the extension of $PI$ intersects $BC$ at $Q$. If $IQ = 2 IP$, find $\angle A$. | 60 | 0.0625 | 8,192 | 8,192 | 8,192 | |
A computer screen shows a $98 \times 98$ chessboard, colored in the usual way. One can select with a mouse any rectangle with sides on the lines of the chessboard and click the mouse button: as a result, the colors in the selected rectangle switch (black becomes white, white becomes black). Find, with proof, the minim... | \[ 98 \] | Answer: $98$ .
There are $4\cdot97$ adjacent pairs of squares in the border and each pair has one black and one white square. Each move can fix at most $4$ pairs, so we need at least $97$ moves. However, we start with two corners one color and two another, so at least one rectangle must include a corner square. But suc... | 0 | 8,137.4375 | -1 | 8,137.4375 |
Let $n$ be a positive integer. A child builds a wall along a line with $n$ identical cubes. He lays the first cube on the line and at each subsequent step, he lays the next cube either on the ground or on the top of another cube, so that it has a common face with the previous one. How many such distinct walls exist? | 2^{n-1} |
To solve this problem, we need to determine how many distinct ways the child can build a wall with \( n \) identical cubes. Each cube can be placed in such a way that it shares a face with the previous cube. This can be done either by placing the new cube on the ground or on top of the previously placed cube.
Let's ... | 0.4375 | 6,777.4375 | 5,736.428571 | 7,587.111111 |
A right pyramid has a square base with perimeter 24 inches. Its apex is 9 inches from each of the other vertices. What is the height of the pyramid from its peak to the center of its square base, in inches? | 3\sqrt{7} | 1 | 1,665 | 1,665 | -1 | |
If $a$, $b$, $c$, $d$, $e$, and $f$ are integers for which $8x^3 + 125 = (ax^2 + bx + c)(d x^2 + ex + f)$ for all $x$, then what is $a^2 + b^2 + c^2 + d^2 + e^2 + f^2$? | 770 | 0.25 | 7,127.5 | 5,600.75 | 7,636.416667 | |
Through the vertex \(C\) of the base of a regular triangular pyramid \(SABC\), a plane is drawn perpendicular to the lateral edge \(SA\). This plane forms an angle with the base plane, the cosine of which is \( \frac{2}{3} \). Find the cosine of the angle between two lateral faces. | \frac{1}{7} | 0.1875 | 7,611.0625 | 5,989 | 7,985.384615 | |
A digital music library contains 12 days of music and takes up 16,000 megabytes of disk space. On average, how many megabytes of disk space does one hour of music in this library take up? Express your answer to the nearest whole number. | 56 | 0.875 | 4,045 | 3,452.571429 | 8,192 | |
A rhombus $ABCD$ is given with $\angle BAD = 60^o$ . Point $P$ lies inside the rhombus such that $BP = 1$ , $DP = 2$ , $CP = 3$ . Determine the length of the segment $AP$ . | \sqrt{7} | 0 | 7,462.8125 | -1 | 7,462.8125 | |
If \(1 + 1.1 + 1.11 + \square = 4.44\), what number should be put in the box to make the equation true? | 1.23 | 1 | 2,480.875 | 2,480.875 | -1 | |
An equilateral triangle with side length $12$ is completely filled with non-overlapping equilateral triangles of side length $2$. Calculate the number of small triangles required. | 36 | 0.875 | 4,893.8125 | 4,422.642857 | 8,192 | |
How many paths are there from point $C$ to point $D$ on a grid, if every step must be either to the right or upwards, and the grid dimensions are now 7 steps to the right and 9 steps upward? | 11440 | 1 | 3,205.625 | 3,205.625 | -1 | |
A line containing the points $(2, 16)$ and $(-8, -4)$ intersects the $y$-axis at what point? | (0, 12) | 1 | 1,567.625 | 1,567.625 | -1 | |
If $x=\sqrt2+\sqrt3+\sqrt6$ is a root of $x^4+ax^3+bx^2+cx+d=0$ where $a,b,c,d$ are integers, what is the value of $|a+b+c+d|$ ? | 93 | 0.5 | 7,249.5625 | 6,307.125 | 8,192 | |
A point $P$ is chosen at random in the interior of equilateral triangle $ABC$. What is the probability that $\triangle ABP$ has a greater area than each of $\triangle ACP$ and $\triangle BCP$? | \frac{1}{3} | 0.0625 | 8,178.5625 | 7,977 | 8,192 | |
Right triangles \(ABC\) and \(ABD\) share a common hypotenuse \(AB = 5\). Points \(C\) and \(D\) are located on opposite sides of the line passing through points \(A\) and \(B\), with \(BC = BD = 3\). Point \(E\) lies on \(AC\), and \(EC = 1\). Point \(F\) lies on \(AD\), and \(FD = 2\). Find the area of the pentagon \... | 9.12 | 0 | 7,813.8125 | -1 | 7,813.8125 | |
In the fictional country of Novaguard, they use the same twelve-letter Rotokas alphabet for their license plates, which are also five letters long. However, for a special series of plates, the following rules apply:
- The plate must start with either P or T.
- The plate must end with R.
- The letter U cannot appear an... | 1440 | 0.125 | 3,646.25 | 700.5 | 4,067.071429 | |
Let $N$ denote the number of $7$ digit positive integers have the property that their digits are in increasing order. Determine the remainder obtained when $N$ is divided by $1000$. (Repeated digits are allowed.)
| 435 | 0.9375 | 3,717.8125 | 3,419.533333 | 8,192 | |
In the 2013 Zhejiang College Entrance Examination, arrange the six letters A, B, C, D, E, F in a row, with both A and B on the same side of C. How many different arrangements are there? (Answer with a number.) | 480 | 0 | 8,192 | -1 | 8,192 | |
Suppose that $f(x)=\frac{1}{2x+b}$. For what value of $b$ does $f^{-1}(x)=\frac{1-2x}{2x}$? | 2 | 1 | 2,208.8125 | 2,208.8125 | -1 | |
In the triangular prism \(P-ABC\), \(\triangle ABC\) is an equilateral triangle with side length \(2\sqrt{3}\), \(PB = PC = \sqrt{5}\), and the dihedral angle \(P-BC-A\) is \(45^\circ\). Find the surface area of the circumscribed sphere around the triangular prism \(P-ABC\). | 25\pi | 0.25 | 7,772.0625 | 6,512.25 | 8,192 | |
What is the least integer whose square is 36 more than three times its value? | -6 | 0 | 8,170.25 | -1 | 8,170.25 | |
In triangle \(ABC\), it is known that \(AB = 3\), \(AC = 3\sqrt{7}\), and \(\angle ABC = 60^\circ\). The bisector of angle \(ABC\) is extended to intersect at point \(D\) with the circle circumscribed around the triangle. Find \(BD\). | 4\sqrt{3} | 0.5 | 7,130.0625 | 6,603.5 | 7,656.625 | |
A penny, nickel, dime, and quarter are simultaneously flipped. What is the expected value of the amount of the coins which come up heads? Express your answer in cents, rounded to the nearest tenth of a cent. | 20.5 | 0.9375 | 2,250.625 | 1,854.533333 | 8,192 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.