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 how many ways can the natural numbers from 1 to 10 (each used exactly once) be arranged in a $2 \times 5$ table so that the sum of the numbers in each of the five columns is odd? | 460800 | 0.4375 | 6,868 | 6,059.571429 | 7,496.777778 | |
Given $a \in \{0, 1, 2\}$ and $b \in \{-1, 1, 3, 5\}$, find the probability that the function $f(x) = ax^2 - 2bx$ is an increasing function on the interval $(1, +\infty)$. | \frac{5}{12} | 0.75 | 5,463.75 | 4,993 | 6,876 | |
Given point O in the plane of △ABC, such that $|$$\overrightarrow {OA}$$|=|$$\overrightarrow {OB}$$|=|$$\overrightarrow {OC}$$|=1, and 3$$\overrightarrow {OA}$$+4$$\overrightarrow {OB}$$+5$$\overrightarrow {OC}$$= $$\overrightarrow {0}$$, find the value of $$\overrightarrow {AB}\cdot \overrightarrow {AC}$$. | \frac {4}{5} | 0.8125 | 5,631.875 | 5,041.076923 | 8,192 | |
Alice and Bob play on a $20 \times 20$ grid. Initially, all the cells are empty. Alice starts and the two players take turns placing stones on unoccupied cells. On her turn, Alice places a red stone on an empty cell that is not at a distance of $\sqrt{5}$ from any other cell containing a red stone. On his turn, Bob pla... | 100 | 0.375 | 7,709.1875 | 6,904.5 | 8,192 | |
Two parabolas have equations $y= x^2 + ax +b$ and $y= x^2 + cx +d$, where $a, b, c,$ and $d$ are integers, each chosen independently by rolling a fair six-sided die. What is the probability that the parabolas will have at least one point in common? | \frac{31}{36} |
1. **Set the equations equal**: Given the equations of the parabolas $y = x^2 + ax + b$ and $y = x^2 + cx + d$, set them equal to find the condition for intersection:
\[
x^2 + ax + b = x^2 + cx + d.
\]
Simplifying this, we get:
\[
ax + b = cx + d.
\]
Rearranging terms, we have:
\[
ax - cx... | 0.8125 | 6,119.5625 | 5,641.307692 | 8,192 |
Given the function \[f(x) = \left\{ \begin{aligned} x+3 & \quad \text{ if } x < 2 \\ x^2 & \quad \text{ if } x \ge 2 \end{aligned} \right.\] determine the value of \(f^{-1}(-5) + f^{-1}(-4) + \dots + f^{-1}(2) + f^{-1}(3). \) | -35 + \sqrt{2} + \sqrt{3} | 0 | 5,388.125 | -1 | 5,388.125 | |
What is the base $2$ representation of $84_{10}$? | 1010100_2 | 0.6875 | 3,035.0625 | 2,925.363636 | 3,276.4 | |
Given that α is an angle in the fourth quadrant, and $sin \left( \frac{\pi}{2} + \alpha \right) = \frac{4}{5}$, calculate the value of $tan(\alpha)$. | -\frac{3}{4} | 1 | 1,934.5625 | 1,934.5625 | -1 | |
On side \(AD\) of rectangle \(ABCD\), a point \(E\) is marked. On segment \(EC\) there is a point \(M\) such that \(AB = BM\) and \(AE = EM\). Find the length of side \(BC\), given that \(ED = 16\) and \(CD = 12\). | 20 | 0.5625 | 6,928 | 5,944.888889 | 8,192 | |
Given two positive numbers $a$, $b$ such that $a<b$. Let $A.M.$ be their arithmetic mean and let $G.M.$ be their positive geometric mean. Then $A.M.$ minus $G.M.$ is always less than: | \frac{(b-a)^2}{8a} | Given two positive numbers $a$ and $b$ such that $a < b$. We need to find the difference between their arithmetic mean (A.M.) and geometric mean (G.M.) and compare it to the given options.
1. **Calculate A.M. and G.M.:**
- The arithmetic mean (A.M.) of $a$ and $b$ is given by:
\[
A.M. = \frac{a+b}{2}
... | 0.0625 | 8,169.8125 | 7,837 | 8,192 |
Given that the graph of the exponential function $y=f(x)$ passes through the point $(\frac{1}{2}, \frac{\sqrt{2}}{2})$, determine the value of $\log_{2}f(2)$. | -2 | 0.5625 | 6,273.3125 | 5,208.111111 | 7,642.857143 | |
Given that \(\alpha\) is an acute angle and \(\beta\) is an obtuse angle, and \(\sec (\alpha - 2\beta)\), \(\sec \alpha\), and \(\sec (\alpha + 2\beta)\) form an arithmetic sequence, find the value of \(\frac{\cos \alpha}{\cos \beta}\). | \sqrt{2} | 0 | 7,877.875 | -1 | 7,877.875 | |
Anton writes down all positive integers that are divisible by 2. Berta writes down all positive integers that are divisible by 3. Clara writes down all positive integers that are divisible by 4. The orderly Dora notes the numbers written by the others. She arranges these numbers in ascending order and does not write an... | 3026 | 0.5 | 6,366.75 | 5,231.625 | 7,501.875 | |
Let $\mathbf{u},$ $\mathbf{v},$ and $\mathbf{w}$ be nonzero vectors, no two of which are parallel, such that
\[(\mathbf{u} \times \mathbf{v}) \times \mathbf{w} = \frac{1}{4} \|\mathbf{v}\| \|\mathbf{w}\| \mathbf{u}.\]
Let $\phi$ be the angle between $\mathbf{v}$ and $\mathbf{w}.$ Find $\sin \phi.$ | \frac{\sqrt{15}}{4} | 0 | 4,706.4375 | -1 | 4,706.4375 | |
In a certain class of Fengzhong Junior High School, some students participated in a study tour and were assigned to several dormitories. If each dormitory accommodates 6 people, there are 10 students left without a room. If each dormitory accommodates 8 people, one dormitory has more than 4 people but less than 8 peopl... | 46 | 0.5625 | 4,976.1875 | 2,905.111111 | 7,639 | |
Given $A=\{a^{2},a+1,-3\}$ and $B=\{a-3,3a-1,a^{2}+1\}$, if $A∩B=\{-3\}$, find the value of the real number $a$. | - \frac {2}{3} | 0.75 | 4,832.1875 | 3,712.25 | 8,192 | |
Suppose that all four of the numbers \[3 - 2\sqrt{2}, \; -3-2\sqrt{2}, \; 1+\sqrt{7}, \; 1-\sqrt{7}\]are roots of the same nonzero polynomial with rational coefficients. What is the smallest possible degree of the polynomial? | 6 | 0.4375 | 5,095.5 | 4,007.142857 | 5,942 | |
What is equal to $\frac{\frac{1}{3}-\frac{1}{4}}{\frac{1}{2}-\frac{1}{3}}$? | \frac{1}{2} | 1. **Identify the Least Common Multiple (LCM):**
The denominators in the fractions are 3, 4, 2, and 3. The LCM of these numbers is the smallest number that each of these numbers can divide without leaving a remainder. The LCM of 2, 3, and 4 is $12$ (since $12 = 3 \times 4$ and is also divisible by 2).
2. **Rewrit... | 1 | 2,301.0625 | 2,301.0625 | -1 |
Given that $x$ is a positive integer less than 100, how many solutions does the congruence $x + 13 \equiv 55 \pmod{34}$ have? | 3 | 1 | 2,776.25 | 2,776.25 | -1 | |
Find the sum of all positive integers $n$ such that when $1^3+2^3+3^3+\cdots +n^3$ is divided by $n+5$, the remainder is $17$. | 239 | The formula for the sum of cubes is as follows: \[1^3+2^3+3^3+...+n^3=(1+2+3+...+n)^2=\left(\frac{n(n+1)}{2}\right)^2\] So let's apply this to this problem.
Let $m=n+5$. Then we have \[1^3+2^3+3^3+\cdots+(m-5)^3\equiv 17 \mod m\] \[\left(\frac{(m-5)(m-4)}{2}\right)^2\equiv 17 \mod m\] \[\frac{400}{4}\equiv 17 \mod m\]... | 0 | 4,818.875 | -1 | 4,818.875 |
For each positive integer $n$ , let $g(n)$ be the sum of the digits when $n$ is written in binary. For how many positive integers $n$ , where $1\leq n\leq 2007$ , is $g(n)\geq 3$ ? | 1941 | 0.1875 | 7,830.1875 | 6,544.333333 | 8,126.923077 | |
A confectionery factory received 5 spools of ribbon, each 60 meters long, for packaging cakes. How many cuts are needed to obtain pieces of ribbon, each 1 meter 50 centimeters long? | 195 | 0.1875 | 3,861 | 5,479.333333 | 3,487.538462 | |
In isosceles $\triangle ABC$ where $AB = AC = 2$ and $BC = 1$, equilateral triangles $ABD$, $BCE$, and $CAF$ are constructed outside the triangle. Calculate the area of polygon $DEF$.
A) $3\sqrt{3} - \sqrt{3.75}$
B) $3\sqrt{3} + \sqrt{3.75}$
C) $2\sqrt{3} - \sqrt{3.75}$
D) $2\sqrt{3} + \sqrt{3.75}$ | 3\sqrt{3} - \sqrt{3.75} | 0 | 8,192 | -1 | 8,192 | |
From the numbers $1, 2, 3, 4, 5$, 3 numbers are randomly drawn (with replacement) to form a three-digit number. What is the probability that the sum of its digits equals 9? | $\frac{19}{125}$ | 0 | 5,625.25 | -1 | 5,625.25 | |
The lateral surface area of a circular truncated cone is given by the formula, find the value for the lateral surface area of the cone where the radii of the upper and lower bases are $r=1$ and $R=4$ and the height is $4$. | 25\pi | 0.9375 | 2,687.875 | 2,320.933333 | 8,192 | |
In square $ABCD$, $AD$ is 4 centimeters, and $M$ is the midpoint of $\overline{CD}$. Let $O$ be the intersection of $\overline{AC}$ and $\overline{BM}$. What is the ratio of $OC$ to $OA$? Express your answer as a common fraction.
[asy]
size (3cm,3cm);
pair A,B,C,D,M;
D=(0,0);
C=(1,0);
B=(1,1);
A=(0,1);
draw(A--B--... | \frac{1}{2} | 0.8125 | 5,832.3125 | 5,287.769231 | 8,192 | |
Given the function $f(x)=x^3+ax^2+bx+16$ has an extremum of $10$ at $x=1$;
(1) Find the values of $a$ and $b$;
(2) Find the maximum and minimum values of $f(x)$ on the interval $[0,2]$. | 10 | 1 | 2,906.625 | 2,906.625 | -1 | |
The probability that a set of three distinct vertices chosen at random from among the vertices of a regular n-gon determine an obtuse triangle is $\frac{93}{125}$ . Find the sum of all possible values of $n$. | 503 | Inscribe the regular polygon inside a circle. A triangle inside this circle will be obtuse if and only if its three vertices lie on one side of a diameter of the circle. (This is because if an inscribed angle on a circle is obtuse, the arc it spans must be 180 degrees or greater).
Break up the problem into two cases: ... | 0 | 8,192 | -1 | 8,192 |
In the Cartesian coordinate system xOy, the parametric equations of line l are given by: $$\begin{cases}x=1+t\cos α \\ y=2+t\sin α\end{cases}$$ (t is a parameter, 0≤a<π). Establish a polar coordinate system with O as the pole and the positive semi-axis of the x-axis as the polar axis. The polar equation of curve C is ρ... | 2\sqrt{7} | 0.75 | 5,930.75 | 5,484.166667 | 7,270.5 | |
Given a function $f(x)$ that always satisfies the following conditions on its domain $\mathbb{R}$:
① $f(x) = f(-x)$, ② $f(2+x) = f(2-x)$, when $x \in [0, 4)$, $f(x) = -x^2 + 4x$.
(1) Find $f(8)$.
(2) Find the number of zeros of $f(x)$ in $[0, 2015]$. | 504 | 0.75 | 7,161.875 | 6,818.5 | 8,192 | |
On graph paper (1 cell = 1 cm), two equal triangles ABC and BDE are depicted.
Find the area of their common part. | 0.8 | 0 | 7,656.875 | -1 | 7,656.875 | |
Find the smallest natural number, which divides $2^{n}+15$ for some natural number $n$ and can be expressed in the form $3x^2-4xy+3y^2$ for some integers $x$ and $y$ . | 23 | 0.0625 | 8,192 | 8,192 | 8,192 | |
In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\left(\sin A+\sin B\right)\left(a-b\right)=c\left(\sin C-\sin B\right)$, and $D$ is a point on side $BC$ such that $AD$ bisects angle $BAC$ and $AD=2$. Find:<br/>
$(1)$ The measure of angle $A$;<br/>... | \frac{4\sqrt{3}}{3} | 0 | 5,956.125 | -1 | 5,956.125 | |
The increasing sequence $1,3,4,9,10,12,13\cdots$ consists of all those positive integers which are powers of 3 or sums of distinct powers of 3. Determine the $150^{\mbox{th}}$ term of this sequence. | 2280 | 0.75 | 5,663.9375 | 5,079.666667 | 7,416.75 | |
Let $T$ be the set of all positive integers that have five digits in base $2$. What is the sum of all the elements in $T$, when expressed in base $10$? | 376 | 0.625 | 5,519.875 | 3,916.6 | 8,192 | |
Triangle $ABC$ has vertices $A = (3,0)$, $B = (0,3)$, and $C$, where $C$ is on the line $x + y = 7$. What is the area of $\triangle ABC$? | 6 |
#### Step 1: Understanding the problem
We are given a triangle $ABC$ with vertices $A = (3,0)$, $B = (0,3)$, and $C$ on the line $x + y = 7$. We need to find the area of $\triangle ABC$.
#### Step 2: Analyzing the line $x + y = 7$
The line $x + y = 7$ is parallel to the line connecting $A$ and $B$, because the slope ... | 1 | 2,334.6875 | 2,334.6875 | -1 |
A shooter's probabilities of hitting the 10, 9, 8, 7 rings, and below 7 rings in a shooting are 0.24, 0.28, 0.19, 0.16, and 0.13, respectively. Calculate the probability that the shooter in a single shot:
(1) Hits the 10 or 9 rings,
(2) Hits at least the 7 ring,
(3) Hits less than 8 rings. | 0.29 | 0.6875 | 3,348.5 | 2,723.454545 | 4,723.6 | |
There are five positive integers that are divisors of each number in the list $$60, 120, -30, 180, 240$$. Find the sum of these five positive integers. | 17 | 0.25 | 7,912.3125 | 7,218.5 | 8,143.583333 | |
A sequence of 11 positive real numbers, $a_{1}, a_{2}, a_{3}, \ldots, a_{11}$, satisfies $a_{1}=4$ and $a_{11}=1024$ and $a_{n}+a_{n-1}=\frac{5}{2} \sqrt{a_{n} \cdot a_{n-1}}$ for every integer $n$ with $2 \leq n \leq 11$. For example when $n=7, a_{7}+a_{6}=\frac{5}{2} \sqrt{a_{7} \cdot a_{6}}$. There are $S$ such sequ... | 20 | Suppose that, for some integer $n \geq 2$, we have $a_{n}=x$ and $a_{n-1}=y$. The equation $a_{n}+a_{n-1}=\frac{5}{2} \sqrt{a_{n} \cdot a_{n-1}}$ can be re-written as $x+y=\frac{5}{2} \sqrt{x y}$. Since $x>0$ and $y>0$, squaring both sides of the equation gives an equivalent equation which is $(x+y)^{2}=\frac{25}{4} x ... | 0.75 | 5,947.9375 | 5,292.833333 | 7,913.25 |
Given that $0 \leqslant x \leqslant 2$, find the minimum and maximum values of the function $f(x) = 4^{x - \frac{1}{2}} - 3 \cdot 2^x + 5$. | \frac{5}{2} | 0.875 | 3,453.5 | 3,362.928571 | 4,087.5 | |
Given the function \( f(x) = \sin^4 x \),
1. Let \( g(x) = f(x) + f\left(\frac{\pi}{2} - x\right) \). Find the maximum and minimum values of \( g(x) \) in the interval \(\left[\frac{\pi}{6}, \frac{3\pi}{8}\right]\).
2. Find the value of \(\sum_{k=1}^{89} f\left(\frac{k\pi}{180}\right)\). | \frac{133}{4} | 0.0625 | 8,054.375 | 7,088 | 8,118.8 | |
In the convex quadrilateral \( MNLQ \), the angles at vertices \( N \) and \( L \) are right angles, and \(\operatorname{tg} \angle QMN = \frac{2}{3}\). Find the diagonal \( NQ \), given that the side \( LQ \) is half the length of side \( MN \) and is 2 units longer than side \( LN \). | 2\sqrt{13} | 0.25 | 6,980.9375 | 3,347.75 | 8,192 | |
Machines A, B, and C operate independently and are supervised by a single worker, who cannot attend to two or more machines simultaneously. Given that the probabilities of these machines operating without needing supervision are 0.9, 0.8, and 0.85 respectively, calculate the probability that during a certain period,
... | 0.059 | 0.25 | 4,777.375 | 3,699.25 | 5,136.75 | |
A rectangle has a perimeter of 80 inches and each side has an integer length. Additionally, one dimension must be at least twice as long as the other. How many non-congruent rectangles meet these criteria? | 13 | 0.6875 | 6,859.375 | 6,253.636364 | 8,192 | |
I want to choose a license plate which is 3 characters long, where the first character is a letter, the last character is a digit, and the middle is either a letter or a digit. I also want there to be two characters on my license plate which are the same. How many ways are there for me to choose a license plate with th... | 520 | 0.0625 | 7,910.875 | 3,694 | 8,192 | |
Consider a sequence of complex numbers $\left\{z_{n}\right\}$ defined as "interesting" if $\left|z_{1}\right|=1$ and for every positive integer $n$, the following holds:
$$
4 z_{n+1}^{2}+2 z_{n} z_{n+1}+z_{n}^{2}=0.
$$
Find the largest constant $C$ such that for any "interesting" sequence $\left\{z_{n}\right\}$ and an... | \frac{\sqrt{3}}{3} | 0 | 8,192 | -1 | 8,192 | |
Given that $a\in\{0,1,2\}$ and $b\in\{-1,1,3,5\}$, find the probability that the function $f(x)=ax^{2}-2bx$ is increasing on the interval $(1,+\infty)$. | \frac{5}{12} | 0.3125 | 5,535.375 | 5,854 | 5,390.545455 | |
Compute \[\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor + \cdots + \lfloor \sqrt{25} \rfloor.\] | 75 | 0.75 | 5,678 | 4,840 | 8,192 | |
What percent of the positive integers less than or equal to $150$ have no remainders when divided by $6$? | 16.67\% | 0.375 | 4,523.1875 | 2,397.833333 | 5,798.4 | |
In a similar tournament setup, the top 6 bowlers have a playoff. First #6 bowls #5, and the loser gets the 6th prize. The winner then bowls #4, and the loser of this match gets the 5th prize. The process continues with the previous winner bowling the next highest ranked bowler until the final match, where the winner of... | 32 | 0.4375 | 7,206.8125 | 5,940.142857 | 8,192 | |
Let $f(x) = \sqrt{-x^2 + 5x + 6}$.
$(1)$ Find the domain of $f(x)$.
$(2)$ Determine the intervals where $f(x)$ is increasing or decreasing.
$(3)$ Find the maximum and minimum values of $f(x)$ on the interval $[1,5]$. | \sqrt{6} | 0.9375 | 3,296 | 3,286.133333 | 3,444 | |
A pedestrian is moving in a straight line towards a crosswalk at a constant speed of 3.6 km/h. Initially, the distance from the pedestrian to the crosswalk is 40 meters. The length of the crosswalk is 6 meters. What distance from the crosswalk will the pedestrian be after two minutes? | 74 | 0.0625 | 7,345.6875 | 2,766 | 7,651 | |
Compute the triple integral \( I = \iiint_{G} \frac{d x d y}{1-x-y} \), where the region \( G \) is bounded by the planes:
1) \( x + y + z = 1 \), \( x = 0 \), \( y = 0 \), \( z = 0 \)
2) \( x = 0 \), \( x = 1 \), \( y = 2 \), \( y = 5 \), \( z = 2 \), \( z = 4 \). | 1/2 | 0.1875 | 7,905.625 | 6,672 | 8,190.307692 | |
In the first week after his birthday, Bill's uncle gave him some money for his piggy bank. Every week after that Bill put $2 into his piggy bank. At the end of the ninth week after his birthday, Bill had trebled the amount he started with. How much did he have in total at the end of the ninth week? | 24 | 0.75 | 663.1875 | 672.833333 | 634.25 | |
Construct a square on one side of an equilateral triangle. On one non-adjacent side of the square, construct a regular pentagon, as shown. On a non-adjacent side of the pentagon, construct a hexagon. Continue to construct regular polygons in the same way, until you construct an octagon. How many sides does the resultin... | 23 | 1. **Identify the shapes and their sides**: We are given an equilateral triangle (3 sides), a square (4 sides), a regular pentagon (5 sides), a regular hexagon (6 sides), a regular heptagon (7 sides), and a regular octagon (8 sides).
2. **Determine the adjacency of the shapes**:
- The equilateral triangle and the ... | 0.1875 | 7,059.9375 | 6,436.666667 | 7,203.769231 |
Compute the definite integral:
$$
\int_{0}^{\pi}\left(x^{2}-3 x+2\right) \sin x \, dx
$$ | \pi^2 - 3\pi | 0.8125 | 5,103.25 | 4,390.461538 | 8,192 | |
What is the value of $\left(\left((3+2)^{-1}-1\right)^{-1}-1\right)^{-1}-1$? | -\frac{13}{9} | 0.75 | 3,412.1875 | 2,798.75 | 5,252.5 | |
Find the maximum value of the expression \((\sqrt{36-4 \sqrt{5}} \sin x-\sqrt{2(1+\cos 2 x)}-2) \cdot (3+2 \sqrt{10-\sqrt{5}} \cos y-\cos 2 y)\). If the answer is not an integer, round it to the nearest whole number. | 27 | 0.25 | 7,751.25 | 6,632.5 | 8,124.166667 | |
If $2x-3=10$, what is the value of $4x$? | 26 | Since $2x-3=10$, then $2x=13$ and so $4x=2(2x)=2(13)=26$.
(We did not have to determine the value of $x$.) | 1 | 1,233 | 1,233 | -1 |
Three fair dice are tossed at random (i.e., all faces have the same probability of coming up). What is the probability that the three numbers turned up can be arranged to form an arithmetic progression with common difference one? | \frac{1}{9} | 1. **Total Outcomes**: When three fair dice are tossed, each die has 6 faces, and each face is equally likely to come up. Therefore, the total number of outcomes when three dice are tossed is $6 \times 6 \times 6 = 6^3 = 216$.
2. **Favorable Outcomes**: We need to find the number of outcomes where the numbers on the t... | 0.875 | 4,850.5625 | 4,453.714286 | 7,628.5 |
In $\triangle ABC$, point $D$ is the midpoint of side $BC$. Point $E$ is on $AC$ such that $AE:EC =1:2$. Point $F$ is on $AD$ such that $AF:FD=3:1$. If the area of $\triangle DEF$ is 17, determine the area of $\triangle ABC$. [asy]
size(6cm);defaultpen(fontsize(11));
pair b =(0,0);pair c = (10, 0);pair a=(4, 6);
pair... | 408 | 0.625 | 6,740.125 | 5,869 | 8,192 | |
If the ratio of the legs of a right triangle is $1:3$, then the ratio of the corresponding segments of the hypotenuse made by a perpendicular upon it from the vertex is:
A) $1:3$
B) $1:9$
C) $3:1$
D) $9:1$ | 9:1 | 0 | 4,053.4375 | -1 | 4,053.4375 | |
A number in the set $\{50, 51, 52, 53, ... , 500\}$ is randomly selected. What is the probability that it is a two-digit number divisible by 3? Express your answer as a common fraction. | \frac{17}{451} | 0.9375 | 4,529.4375 | 4,285.266667 | 8,192 | |
Consider the set $$ \mathcal{S}=\{(a, b, c, d, e): 0<a<b<c<d<e<100\} $$ where $a, b, c, d, e$ are integers. If $D$ is the average value of the fourth element of such a tuple in the set, taken over all the elements of $\mathcal{S}$ , find the largest integer less than or equal to $D$ . | 66 | 0.0625 | 8,008.4375 | 5,255 | 8,192 | |
A wooden block is 5 inches long, 5 inches wide, and 1 inch high. The block is painted red on all six sides and then cut into twenty-five 1-inch cubes. How many of the resulting cubes each have a total number of red faces that is an odd number? | 13 | 0 | 8,058.1875 | -1 | 8,058.1875 | |
The least common multiple of two integers is 36 and 6 is their greatest common divisor. What is the product of the two numbers? | 216 | 1 | 1,919.4375 | 1,919.4375 | -1 | |
In an opaque bag, there are four small balls labeled with the Chinese characters "阳", "过", "阳", and "康" respectively. Apart from the characters, the balls are indistinguishable. Before each draw, the balls are thoroughly mixed.<br/>$(1)$ If one ball is randomly drawn from the bag, the probability that the character on ... | \frac{1}{3} | 0.4375 | 5,441.25 | 4,105.571429 | 6,480.111111 | |
$E$ is the midpoint of side $BC$ of parallelogram $ABCD$. Line $AE$ intersects the diagonal $BD$ at point $G$. If the area of triangle $\triangle BEG$ is 1, find the area of parallelogram $ABCD$. | 12 | 0.875 | 5,305.25 | 5,048.071429 | 7,105.5 | |
Let \( A = \{1, 2, \cdots, 2004\} \) and \( f: A \rightarrow A \) be a bijection satisfying \( f^{[2004]}(x) = f(x) \), where \( f^{[2004]}(x) \) denotes applying \( f \) 2004 times to \( x \). How many such functions \( f \) are there? | 1 + 2004! | 0 | 7,652.4375 | -1 | 7,652.4375 | |
Let $\mathbf{A} =\begin{pmatrix} -1 & 2 \\ 3 & 4 \end{pmatrix}.$ Then there exist scalars $p$ and $q$ such that
\[\mathbf{A}^6 = p \mathbf{A} + q \mathbf{I}.\]Enter the ordered pair $(p,q).$ | (2223,4510) | 0.5625 | 6,561.875 | 5,294 | 8,192 | |
Some boys and girls are having a car wash to raise money for a class trip to China. Initially $40\%$ of the group are girls. Shortly thereafter two girls leave and two boys arrive, and then $30\%$ of the group are girls. How many girls were initially in the group? | 8 | 1. **Define Variables:**
Let $p$ be the total number of people initially in the group. Since $40\%$ of the group are girls, the number of girls initially is $0.4p$.
2. **Change in Group Composition:**
After two girls leave and two boys arrive, the total number of people remains the same, $p$. However, the number... | 1 | 2,278.1875 | 2,278.1875 | -1 |
Find the minimum value of $\frac{9x^2\sin^2 x + 4}{x\sin x}$ for $0 < x < \pi$. | 12 | We can rewrite the numerator to be a perfect square by adding $-\dfrac{12x \sin x}{x \sin x}$. Thus, we must also add back $12$.
This results in $\dfrac{(3x \sin x-2)^2}{x \sin x}+12$.
Thus, if $3x \sin x-2=0$, then the minimum is obviously $12$. We show this possible with the same methods in Solution 1; thus the ans... | 0.875 | 4,993.125 | 4,536.142857 | 8,192 |
Given the equation $x^{2}-px+q=0$ ($p > 0, q > 0$) with two distinct roots $x_{1}$, $x_{2}$, and the fact that $x_{1}$, $x_{2}$, and $-2$ can be appropriately sorted to form an arithmetic sequence as well as a geometric sequence, find the value of $p \times q$. | 20 | 0.75 | 6,765.3125 | 6,289.75 | 8,192 | |
In the Cartesian coordinate system $(xOy)$, the sum of the distances from point $P$ to two points $(0,-\sqrt{3})$ and $(0,\sqrt{3})$ is equal to $4$. Let the trajectory of point $P$ be $C$.
(I) Write the equation of $C$;
(II) Given that the line $y=kx+1$ intersects $C$ at points $A$ and $B$, for what value of $k$ is $\... | \frac{4\sqrt{65}}{17} | 0 | 5,297.5625 | -1 | 5,297.5625 | |
The equation \(2008=1111+444+222+99+77+55\) is an example of decomposing the number 2008 as a sum of distinct numbers with more than one digit, where each number's representation (in the decimal system) uses only one digit.
i) Find a similar decomposition for the number 2009.
ii) Determine all possible such decomposi... | 1111 + 777 + 66 + 55 | 0 | 8,192 | -1 | 8,192 | |
You are given a sequence of $58$ terms; each term has the form $P+n$ where $P$ stands for the product $2 \times 3 \times 5 \times\ldots \times 61$ of all prime numbers less than or equal to $61$, and $n$ takes, successively, the values $2, 3, 4,\ldots, 59$. Let $N$ be the number of primes appearing in this sequence. Th... | 0 | 3,432.125 | -1 | 3,432.125 | ||
A triangular wire frame with side lengths of $13, 14, 15$ is fitted over a sphere with a radius of 10. Find the distance between the plane containing the triangle and the center of the sphere. | 2\sqrt{21} | 0 | 4,240.8125 | -1 | 4,240.8125 | |
Among the scalene triangles with natural number side lengths, a perimeter not exceeding 30, and the sum of the longest and shortest sides exactly equal to twice the third side, there are ____ distinct triangles. | 20 | 0.25 | 8,018 | 7,556.25 | 8,171.916667 | |
Given a checkerboard with 31 rows and 29 columns, where each corner square is black and the squares alternate between red and black, determine the number of black squares on this checkerboard. | 465 | 0 | 6,075.4375 | -1 | 6,075.4375 | |
Find the distance between the points $(1,1)$ and $(4,7)$. Express your answer in simplest radical form. | 3\sqrt{5} | 1 | 1,320.875 | 1,320.875 | -1 | |
Calculate \( t(0) - t(\pi / 5) + t\left((\pi / 5) - t(3 \pi / 5) + \ldots + t\left(\frac{8 \pi}{5}\right) - t(9 \pi / 5) \right) \), where \( t(x) = \cos 5x + * \cos 4x + * \cos 3x + * \cos 2x + * \cos x + * \). A math student mentioned that he could compute this sum without knowing the coefficients (denoted by *). Is ... | 10 | 0 | 8,071.125 | -1 | 8,071.125 | |
If $\mathbf{a}$ and $\mathbf{b}$ are two unit vectors, with an angle of $\frac{\pi}{3}$ between them, then compute the volume of the parallelepiped generated by $\mathbf{a},$ $\mathbf{b} + \mathbf{b} \times \mathbf{a},$ and $\mathbf{b}.$ | \frac{3}{4} | 0.875 | 4,785.0625 | 4,298.357143 | 8,192 | |
Points $A=(6,13)$ and $B=(12,11)$ lie on circle $\omega$ in the plane. Suppose that the tangent lines to $\omega$ at $A$ and $B$ intersect at a point on the $x$-axis. What is the area of $\omega$? | \frac{85\pi}{8} | 1. **Identify the midpoint of segment $AB$**:
Given points $A=(6,13)$ and $B=(12,11)$, the midpoint $D$ of $AB$ is calculated as:
\[
D = \left(\frac{6+12}{2}, \frac{13+11}{2}\right) = (9, 12).
\]
2. **Determine the slope of line $AB$**:
The slope of $AB$ is given by:
\[
\text{slope of } AB = \frac... | 0.125 | 6,552.625 | 5,159 | 6,751.714286 |
A 6 m by 8 m rectangular field has a fence around it. There is a post at each of the four corners of the field. Starting at each corner, there is a post every 2 m along each side of the fence. How many posts are there? | 14 | A rectangle that is 6 m by 8 m has perimeter $2 \times(6 \mathrm{~m}+8 \mathrm{~m})=28 \mathrm{~m}$. If posts are put in every 2 m around the perimeter starting at a corner, then we would guess that it will take $\frac{28 \mathrm{~m}}{2 \mathrm{~m}}=14$ posts. | 0.875 | 5,311.875 | 4,900.428571 | 8,192 |
Given $(a+i)i=b+ai$, solve for $|a+bi|$. | \sqrt{2} | 0 | 5,061.4375 | -1 | 5,061.4375 | |
What is the smallest positive integer $n$ such that $5n \equiv 2024 \pmod{26}$? | 20 | 1 | 2,560.25 | 2,560.25 | -1 | |
Problem
Steve is piling $m\geq 1$ indistinguishable stones on the squares of an $n\times n$ grid. Each square can have an arbitrarily high pile of stones. After he finished piling his stones in some manner, he can then perform stone moves, defined as follows. Consider any four grid squares, which are corners of a recta... | \[
\binom{n+m-1}{m}^{2}
\] | Let the number of stones in row $i$ be $r_i$ and let the number of stones in column $i$ be $c_i$ . Since there are $m$ stones, we must have $\sum_{i=1}^n r_i=\sum_{i=1}^n c_i=m$
Lemma 1: If any $2$ pilings are equivalent, then $r_i$ and $c_i$ are the same in both pilings $\forall i$ .
Proof: We suppose the contrary. N... | 0 | 8,192 | -1 | 8,192 |
Petya wants to create an unusual die, which should have the shape of a cube, with dots drawn on the faces (different numbers of dots on different faces). Additionally, on each pair of adjacent faces, the number of dots must differ by at least two (it is allowed to have more than six dots on some faces). How many dots i... | 27 | 0 | 8,088.0625 | -1 | 8,088.0625 | |
In the arithmetic sequence $\{a_n\}$, $a_3+a_6+a_9=54$. Let the sum of the first $n$ terms of the sequence $\{a_n\}$ be $S_n$. Then, determine the value of $S_{11}$. | 99 | 0 | 2,123.6875 | -1 | 2,123.6875 | |
A palindrome between $1000$ and $10000$ is chosen at random. What is the probability that it is divisible by $7$? | \frac{1}{5} | 1. **Identify the form of the palindrome**: A four-digit palindrome can be expressed in the form $\overline{abba}$, where $a$ and $b$ are digits, and $a \neq 0$ to ensure it is a four-digit number.
2. **Total number of palindromes**: Since $a$ can be any digit from 1 to 9 (9 choices) and $b$ can be any digit from 0 to... | 1 | 4,004.9375 | 4,004.9375 | -1 |
Five people are crowding into a booth against a wall at a noisy restaurant. If at most three can fit on one side, how many seating arrangements accommodate them all? | 240 | 0.3125 | 7,277.625 | 6,711.6 | 7,534.909091 | |
In the rectangular coordinate system $xOy$, with $O$ as the pole and the positive semi-axis of $x$ as the polar axis, the polar coordinate system is established. The polar coordinate equation of the curve $C$ is $\rho=2\sin\theta+2a\cos\theta$ ($a>0$); the parameter equation of the line $l$ is $$\begin{cases} x=-2+ \fr... | a=2 | 0.625 | 6,202.6875 | 5,186 | 7,897.166667 | |
Given a cube with a side length of \(4\), if a solid cube of side length \(1\) is removed from each corner, calculate the total number of edges of the resulting structure. | 36 | 0.375 | 7,890.625 | 7,855.833333 | 7,911.5 | |
Given $A=\{4, a^2\}$, $B=\{a-6, a+1, 9\}$, if $A \cap B = \{9\}$, find the value of $a$. | -3 | 0.9375 | 2,236.25 | 1,839.2 | 8,192 | |
How many distinct sets of 8 positive odd integers sum to 20 ? | 11 | This is the same as the number of ways 8 nonnegative even integers sum to 12 (we subtract 1 from each integer in the above sum). All 11 possibilities are (leaving out 0s): $12,10+2,8+4,8+2+2,6+6,6+4+2,6+2+2+2+2,4+4+4,4+4+2+2$, $4+2+2+2+2,2+2+2+2+2+2$. | 0.0625 | 6,647 | 7,036 | 6,621.066667 |
$\frac{2+4+6+\cdots + 34}{3+6+9+\cdots+51}=$ | $\frac{2}{3}$ | 1. **Identify the sequences and their properties:**
- The numerator is an arithmetic sequence with the first term $a = 2$, common difference $d = 2$, and last term $l = 34$.
- The denominator is an arithmetic sequence with the first term $a = 3$, common difference $d = 3$, and last term $l = 51$.
2. **Determine ... | 0 | 2,569.25 | -1 | 2,569.25 |
The sum of the base-$10$ logarithms of the divisors of $6^n$ is $540$. What is $n$?
A) 9
B) 10
C) 11
D) 12
E) 13 | 10 | 0 | 8,192 | -1 | 8,192 | |
Let $x_0$ be a zero of the function $f(x) = \sin \pi x$, and it satisfies $|x_{0}| + f(x_{0} + \frac {1}{2}) < 11$. Determine the number of such zeros. | 21 | 0.5625 | 6,268.875 | 4,773.111111 | 8,192 | |
During the National Day holiday, a fruit company organized 20 cars to transport three types of fruits, $A$, $B$, and $C$, totaling 120 tons for sale in other places. It is required that all 20 cars be fully loaded, each car can only transport the same type of fruit, and each type of fruit must be transported by at leas... | 198900 | 0.1875 | 3,489.4375 | 4,610 | 3,230.846154 | |
Let \( f(x) \) be a function from \( \mathbf{R} \) to \( \mathbf{R} \), and for any real numbers, it holds that
$$
f(x^{2}+x) + 2 f(x^{2}-3x+2) = 9x^{2} - 15x,
$$
then the value of \( f(50) \) is ( ). | 146 | 0.8125 | 5,639.625 | 5,050.615385 | 8,192 |
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