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 |
|---|---|---|---|---|---|---|
What is the product of the numerator and the denominator when $0.\overline{0012}$ is expressed as a fraction in lowest terms? | 13332 | 0.9375 | 3,150.5 | 2,814.4 | 8,192 | |
Find the number of solutions in natural numbers for the equation \(\left\lfloor \frac{x}{10} \right\rfloor = \left\lfloor \frac{x}{11} \right\rfloor + 1\). | 110 | 0 | 8,192 | -1 | 8,192 | |
If there are $1, $2, and $3 bills in the board game "Silly Bills" and let x be the number of $1 bills, then x+11, x-18, and x+11+(x-18) = 2x-7 are the respective number of $2 and $3 bills, determine the value of x when the total amount of money is $100. | 22 | 0.875 | 1,929.9375 | 2,054.142857 | 1,060.5 | |
What is the number of square units in the area of trapezoid ABCD with vertices A(0,0), B(0,-2), C(4,0), and D(4,6)? | 16 | 1 | 2,895.6875 | 2,895.6875 | -1 | |
Given an arithmetic sequence $\{a\_n\}$ with a common ratio $q > 1$, and it satisfies: $a\_2 + a\_3 + a\_4 = 28$, and $a\_3 + 2$ is the arithmetic mean of $a\_2$ and $a\_4$.
(1) Find the general term formula of the sequence $\{a\_n\}$;
(2) If $b\_n = a\_n \log\_{ \frac {1}{2}}a\_n$, $S\_n = b\_1 + b\_2 + … + b\_n$, fin... | n = 6 | 0.9375 | 4,425.6875 | 4,394.466667 | 4,894 | |
Given eight distinguishable rings, let $n$ be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order of rings on each finger is significant, but it is not required that each finger have a ring. Find the leftmost three nonzero digits of $n$. | 376 | There are $\binom{8}{5}$ ways to choose the rings, and there are $5!$ distinct arrangements to order the rings [we order them so that the first ring is the bottom-most on the first finger that actually has a ring, and so forth]. The number of ways to distribute the rings among the fingers is equivalent the number of wa... | 0.0625 | 7,928.0625 | 6,433 | 8,027.733333 |
What is the greatest number of consecutive integers whose sum is $45?$ | 90 | To find the greatest number of consecutive integers whose sum is $45$, we need to consider sequences of integers, both positive and negative.
1. **Understanding the sum of consecutive integers**:
The sum of $N$ consecutive integers starting from $a$ can be expressed as:
\[
a + (a+1) + (a+2) + \cdots + (a+N-1)... | 0.8125 | 6,121.8125 | 5,745.307692 | 7,753.333333 |
Given that children enter at a discounted rate, half that of an adult ticket, and the total cost for $6$ adult tickets and $5$ child tickets amounts to $32.50$, calculate the total cost for $10$ adult tickets and $8$ child tickets. | 53.50 | 0 | 3,181.125 | -1 | 3,181.125 | |
The sum of an infinite geometric series is $16$ times the series that results if the first two terms of the original series are removed. What is the value of the series' common ratio? | \frac{1}{4} | 0.375 | 6,335.375 | 6,653.833333 | 6,144.3 | |
What is the smallest four-digit number that is divisible by $35$? | 1015 | 0.9375 | 2,841.75 | 2,485.066667 | 8,192 | |
Given the ellipse $C:\dfrac{x^2}{b^2}+\dfrac{y^2}{a^2}=1 \left( a > b > 0 \right)$ has an eccentricity of $\dfrac{\sqrt{2}}{2}$, and point $A(1,\sqrt{2})$ is on the ellipse.
$(1)$ Find the equation of ellipse $C$;
$(2)$ If a line $l$ with a slope of $\sqrt{2}$ intersects the ellipse $C$ at two distinct points $B$ and... | \sqrt{2} | 0.6875 | 7,527.6875 | 7,225.727273 | 8,192 | |
In an eight-digit number, each digit (except the last one) is greater than the following digit. How many such numbers are there? | 45 | 0.5625 | 5,900.9375 | 5,087.666667 | 6,946.571429 | |
As we enter the autumn and winter seasons, the air becomes dry. A certain appliance store is preparing to purchase a batch of humidifiers. The cost price of each unit is $80$ yuan. After market research, the selling price is set at $100$ yuan per unit. The store can sell $500$ units per day. For every $1$ yuan increase... | 12250 | 1 | 2,655.625 | 2,655.625 | -1 | |
The mean of the numbers 3, 7, 10, and 15 is twice the mean of $x$, 20, and 6. What is the value of $x$? | -12.875 | 0.25 | 3,425.5625 | 3,188.75 | 3,504.5 | |
Given that $α \in (0,π)$, if $\sin α + \cos α = \frac{\sqrt{3}}{3}$, find the value of $\cos^2 α - \sin^2 α$. | \frac{\sqrt{5}}{3} | 0 | 7,397 | -1 | 7,397 | |
Compute the surface integral
$$
\iint_{\Sigma}(-x+3 y+4 z) d \sigma
$$
where $\Sigma$ is the part of the plane
$$
x + 2y + 3z = 1
$$
located in the first octant (i.e., $x \geq 0, y \geq 0, z \geq 0$). | \frac{\sqrt{14}}{18} | 0 | 6,568.0625 | -1 | 6,568.0625 | |
Given that the function $f(x)$ is an odd function defined on $\mathbb{R}$, and $f(x) = \begin{cases} \log_{2}(x+1), & x \geqslant 0 \\ g(x), & x < 0 \end{cases}$, find $g[f(-7)]$. | -2 | 0.875 | 2,761.0625 | 2,460.928571 | 4,862 | |
Bob the bomb-defuser has stumbled upon an active bomb. He opens it up, and finds the red and green wires conveniently located for him to cut. Being a seasoned member of the bomb-squad, Bob quickly determines that it is the green wire that he should cut, and puts his wirecutters on the green wire. But just before he sta... | \frac{23}{30} | Suppose Bob makes $n$ independent decisions, with probabilities of switching $p_{1}, p_{2}, \ldots, p_{n}$. Then in the expansion of the product $$P(x)=\left(p_{1}+\left(1-p_{1}\right) x\right)\left(p_{2}+\left(1-p_{2}\right) x\right) \cdots\left(p_{n}+\left(1-p_{n}\right) x\right)$$ the sum of the coefficients of even... | 0 | 8,192 | -1 | 8,192 |
Angela and Barry share a piece of land. The ratio of the area of Angela's portion to the area of Barry's portion is $3: 2$. They each grow corn and peas on their piece of land. The entire piece of land is covered by corn and peas in the ratio $7: 3$. On Angela's portion of the land, the ratio of corn to peas is $4: 1$.... | $11: 9$ | 0 | 4,974.625 | -1 | 4,974.625 | |
Let $n$ be a positive integer. Given that $n^{n}$ has 861 positive divisors, find $n$. | 20 | If $n=p_{1}^{\alpha_{1}} p_{2}^{\alpha_{2}} \ldots p_{k}^{\alpha_{k}}$, we must have $\left(n \alpha_{1}+1\right)\left(n \alpha_{2}+1\right) \ldots\left(n \alpha_{k}+1\right)=861=3 \cdot 7 \cdot 41$. If $k=1$, we have $n \mid 860$, and the only prime powers dividing 860 are $2,2^{2}, 5$, and 43 , which are not solution... | 0.25 | 7,971.3125 | 7,309.25 | 8,192 |
Equilateral triangles $ACB'$ and $BDC'$ are drawn on the diagonals of a convex quadrilateral $ABCD$ so that $B$ and $B'$ are on the same side of $AC$, and $C$ and $C'$ are on the same sides of $BD$. Find $\angle BAD + \angle CDA$ if $B'C' = AB+CD$. | 120^\circ |
Consider the convex quadrilateral \(ABCD\), and let equilateral triangles \(ACB'\) and \(BDC'\) be drawn on its diagonals such that points \(B'\) and \(C'\) are on specified sides of the lines, maintaining convexity. We are given that \(B'C' = AB + CD\).
Our objective is to find \(\angle BAD + \angle CDA\).
To solve... | 0.0625 | 8,112.75 | 6,924 | 8,192 |
The distance from the center \( O \) of a sphere with radius 12, which is circumscribed around a regular quadrangular pyramid, to a lateral edge is \( 4 \sqrt{2} \). Find:
1) the height of the pyramid;
2) the distance from point \( O \) to the lateral face of the pyramid;
3) the radius of the sphere inscribed in the p... | \frac{8}{3}\left(2 \sqrt{2} - 1\right) | 0 | 8,168.5625 | -1 | 8,168.5625 | |
Let the function $f(x) = \frac{bx}{\ln x} - ax$, where $e$ is the base of the natural logarithm.
(I) If the tangent line to the graph of the function $f(x)$ at the point $(e^2, f(e^2))$ is $3x + 4y - e^2 = 0$, find the values of the real numbers $a$ and $b$.
(II) When $b = 1$, if there exist $x_1, x_2 \in [e, e^2]$ s... | \frac{1}{2} - \frac{1}{4e^2} | 0 | 8,085.4375 | -1 | 8,085.4375 | |
What value of $x$ makes the equation below true: $$2x + 4 = |{-17 + 3}|$$ | 5 | 1 | 1,275.3125 | 1,275.3125 | -1 | |
Simplify \[\frac{1}{\dfrac{2}{\sqrt{5}+2} + \dfrac{3}{\sqrt{7}-2}}.\] | \frac{2\sqrt{5} + \sqrt{7} + 2}{23 + 4\sqrt{35}} | 0 | 7,470 | -1 | 7,470 | |
Let point $O$ be the origin of a two-dimensional coordinate system, and let points $A$ and $B$ be located on positive $x$ and $y$ axes, respectively. If $OA = \sqrt[3]{54}$ and $\angle AOB = 45^\circ,$ compute the length of the line segment $AB.$ | 54^{1/3} \sqrt{2} | 0 | 8,091.8125 | -1 | 8,091.8125 | |
Let $a, b \in \mathbb{R}^+$, and $a+b=1$. Find the minimum value of $\sqrt{a^2+1} + \sqrt{b^2+4}$. | \sqrt{10} | 0.9375 | 5,169.1875 | 4,967.666667 | 8,192 | |
Let $ (x_1,x_2,\cdots)$ be a sequence of positive numbers such that $ (8x_2 \minus{} 7x_1)x_1^7 \equal{} 8$ and
\[ x_{k \plus{} 1}x_{k \minus{} 1} \minus{} x_k^2 \equal{} \frac {x_{k \minus{} 1}^8 \minus{} x_k^8}{x_k^7x_{k \minus{} 1}^7} \text{ for }k \equal{} 2,3,\ldots
\]
Determine real number $ a$ such th... | 8^{1/8} | 0.0625 | 8,192 | 8,192 | 8,192 | |
In a certain circle, the chord of a $d$-degree arc is $22$ centimeters long, and the chord of a $2d$-degree arc is $20$ centimeters longer than the chord of a $3d$-degree arc, where $d < 120.$ The length of the chord of a $3d$-degree arc is $- m + \sqrt {n}$ centimeters, where $m$ and $n$ are positive integers. Find $m... | 174 | Let $z=\frac{d}{2}$, $R$ be the circumradius, and $a$ be the length of 3d degree chord. Using the extended sine law, we obtain: \[22=2R\sin(z)\] \[20+a=2R\sin(2z)\] \[a=2R\sin(3z)\] Dividing the second from the first we get $\cos(z)=\frac{20+a}{44}$ By the triple angle formula we can manipulate the third equation as fo... | 0.0625 | 8,155.0625 | 7,647 | 8,188.933333 |
What are the rightmost three digits of $5^{1993}$? | 125 | 0.9375 | 4,068 | 3,793.066667 | 8,192 | |
Find the sum of the digits of the greatest prime number that is a divisor of $16,385$. | 13 | 0 | 3,816.5625 | -1 | 3,816.5625 | |
In a corridor that is 100 meters long, there are 20 rugs with a total length of 1 kilometer. Each rug is as wide as the corridor. What is the maximum possible total length of the sections of the corridor that are not covered by the rugs? | 50 | 0.0625 | 7,943.5625 | 7,731 | 7,957.733333 | |
Given that the scores of a math exam follow a normal distribution N(102, 4²), the percentage of scores 114 and above is _______ (Note: P(μ-σ<X≤μ+σ)=0.6826, P(μ-2σ<X≤μ+2σ)=0.9544, P(μ-3σ<X≤μ+3σ)=0.9974). | 0.13\% | 0.625 | 5,255.1875 | 3,493.1 | 8,192 | |
A number $x$ is randomly selected from the interval $\left[ -\frac{\pi}{6}, \frac{\pi}{2} \right]$. Calculate the probability that $\sin x + \cos x \in [1, \sqrt{2}]$. | \frac{3}{4} | 0.5625 | 7,191.5 | 6,413.333333 | 8,192 | |
How many positive integers smaller than $1{,}000{,}000$ are powers of $2$, but are not powers of $8$? You may find it useful to consider that $2^{10}=1024$. | 13 | 0.5 | 6,685.3125 | 5,382.25 | 7,988.375 | |
Given two complex numbers $z_1$ and $z_2$ whose corresponding points in the complex plane are symmetrical about the imaginary axis, and $z_1 = 1 + 2i$ where $i$ is the imaginary unit, calculate the product of $z_1$ and $z_2$. | -5 | 0.9375 | 2,557.1875 | 2,594.733333 | 1,994 | |
A Moskvich car was sent to transport mail from the post office to the airfield. The plane carrying the mail landed earlier than expected, and the delivered mail was sent to the post office by a passing truck. After driving for 30 minutes, the truck met the Moskvich on the road, which received the mail and turned back i... | 40 | 0 | 7,933 | -1 | 7,933 | |
Two distinct numbers are selected from the set $\{1,2,3,4,\dots,38\}$ so that the sum of the remaining $36$ numbers equals the product of these two selected numbers plus one. Find the difference of these two numbers. | 20 | 1 | 2,554.0625 | 2,554.0625 | -1 | |
Let $A$ be the set $\{k^{19}-k: 1<k<20, k\in N\}$ . Let $G$ be the GCD of all elements of $A$ .
Then the value of $G$ is? | 798 | 0.1875 | 7,235.0625 | 4,985 | 7,754.307692 | |
If $2.4 \times 10^{8}$ is doubled, what is the result? | 4.8 \times 10^{8} | When $2.4 \times 10^{8}$ is doubled, the result is $2 \times 2.4 \times 10^{8}=4.8 \times 10^{8}$. | 0.625 | 303.6875 | 303.6 | 303.833333 |
Given the ellipse $$E: \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$$ has an eccentricity of $$\frac{\sqrt{2}}{2}$$, and the point $$A(1, \sqrt{2})$$ is on the ellipse.
(1) Find the equation of ellipse E;
(2) If a line $l$ with a slope of $$\sqrt{2}$$ intersects the ellipse E at two distinct points B a... | \sqrt{2} | 0 | 7,215.9375 | -1 | 7,215.9375 | |
How many elements are in the set obtained by transforming $\{(0,0),(2,0)\} 14$ times? | 477 | Transforming it $k \geq 1$ times yields the diamond $\{(n, m):|n-1|+|m| \leq k+1\}$ with the points $(1, k),(1, k+1),(1,-k),(1,-k-1)$ removed (this can be seen inductively). So we get $(k+1)^{2}+k^{2}-4$ lattice points, making the answer 477. | 0 | 7,381.5625 | -1 | 7,381.5625 |
Find the number of ordered pairs of positive integers $(x, y)$ with $x, y \leq 2020$ such that $3 x^{2}+10 x y+3 y^{2}$ is the power of some prime. | 29 | We can factor as $(3 x+y)(x+3 y)$. If $x \geq y$, we need $\frac{3 x+y}{x+3 y} \in\{1,2\}$ to be an integer. So we get the case where $x=y$, in which we need both to be a power of 2, or the case $x=5 y$, in which case we need $y$ to be a power of 2. This gives us $11+9+9=29$ solutions, where we account for $y=5 x$ as w... | 0 | 8,070.6875 | -1 | 8,070.6875 |
Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth. | 0.294 | 0 | 7,585.9375 | -1 | 7,585.9375 | |
A circular garden is enlarged so that the new diameter is twice the old diameter. What is the ratio of the original area to the enlarged area? Express your answer as a common fraction. | \frac{1}{4} | 1 | 1,035.4375 | 1,035.4375 | -1 | |
Given $a^2 = 16$, $|b| = 3$, $ab < 0$, find the value of $(a - b)^2 + ab^2$. | 13 | 0.125 | 8,192 | 8,192 | 8,192 | |
Find all solutions to the equation $\displaystyle\sqrt[3]{3 - x} = -\frac{3}{2}$. | \frac{51}{8} | 1 | 1,900.125 | 1,900.125 | -1 | |
Let $\sigma (n)$ denote the sum and $\tau (n)$ denote the amount of natural divisors of number $n$ (including $1$ and $n$ ). Find the greatest real number $a$ such that for all $n>1$ the following inequality is true: $$ \frac{\sigma (n)}{\tau (n)}\geq a\sqrt{n} $$ | \frac{3 \sqrt{2}}{4} | 0 | 8,192 | -1 | 8,192 | |
Two triangles are similar. The ratio of their areas is 1:4. If the height of the smaller triangle is 3 cm, how long is the corresponding height of the larger triangle, in centimeters? | 6 | 1 | 1,187.75 | 1,187.75 | -1 | |
Let \(a, b, c, d\) be distinct positive odd numbers. What is the minimum value of
\[
2abcd - (abc + abd + acd + bcd)
\] | 34 | 0.25 | 7,855.6875 | 6,846.75 | 8,192 | |
How many lattice points are enclosed by the triangle with vertices $(0,99),(5,100)$, and $(2003,500) ?$ Don't count boundary points. | 0 | Using the determinant formula, we get that the area of the triangle is $$\left|\begin{array}{cc} 5 & 1 \\ 2003 & 401 \end{array}\right| / 2=1$$ There are 4 lattice points on the boundary of the triangle (the three vertices and $(1004,300)$ ), so it follows from Pick's Theorem that there are 0 in the interior. | 0.5625 | 6,530.5625 | 5,238.333333 | 8,192 |
Three musicians, Janek, Mikeš, and Vávra usually divide their shared fee in the ratio $4: 5: 6$, with Janek receiving the least and Vávra the most. This time, Vávra did not perform well, so he gave up his portion. Janek suggested that Vávra's share should be divided equally between him and Mikeš. However, Mikeš insiste... | 1800 | 0.125 | 5,749.5625 | 5,398.5 | 5,799.714286 | |
Let $\alpha$, $\beta$, $\gamma$ represent three different planes, and $a$, $b$, $c$ represent three different lines. Consider the following five propositions:
(1) If $a \parallel \alpha$, $b \parallel \beta$, and $a \parallel b$, then $\alpha \parallel \beta$;
(2) If $a \parallel \alpha$, $b \parallel \alpha$, $\be... | (2) | 0 | 3,944.8125 | -1 | 3,944.8125 | |
How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots? | 100 | 0.3125 | 6,456.3125 | 4,651.2 | 7,276.818182 | |
In triangle \(A B C\), side \(B C\) equals 4, and the median drawn to this side equals 3. Find the length of the common chord of two circles, each of which passes through point \(A\) and is tangent to \(B C\), with one tangent at point \(B\) and the other at point \(C\). | \frac{5}{3} | 0.3125 | 7,359.1875 | 5,527 | 8,192 | |
You have a length of string and 7 beads in the 7 colors of the rainbow. You place the beads on the string as follows - you randomly pick a bead that you haven't used yet, then randomly add it to either the left end or the right end of the string. What is the probability that, at the end, the colors of the beads are the... | \frac{1}{5040} | The threading method does not depend on the colors of the beads, so at the end all configurations are equally likely. Since there are $7!=5040$ configurations in total, the probability of any particular configuration is $\frac{1}{5040}$. | 0.125 | 7,651.875 | 5,264.5 | 7,992.928571 |
Point $E$ is the midpoint of side $\overline{CD}$ in square $ABCD,$ and $\overline{BE}$ meets diagonal $\overline{AC}$ at $F.$ The area of quadrilateral $AFED$ is $45.$ What is the area of $ABCD?$ | 108 | 1. **Identify the Coordinates of Points in the Square:**
We place square $ABCD$ in the Cartesian coordinate system with $D$ as the origin. Thus, we have:
- $D = (0, 0)$
- $C = (1, 0)$
- $B = (1, 1)$
- $A = (0, 1)$
2. **Determine the Coordinates of Point $E$:**
Since $E$ is the midpoint of $\overline{... | 0.8125 | 5,681.625 | 5,102.307692 | 8,192 |
For any positive integer $x$, define $\operatorname{Accident}(x)$ to be the set of ordered pairs $(s, t)$ with $s \in \{0,2,4,5,7,9,11\}$ and $t \in\{1,3,6,8,10\}$ such that $x+s-t$ is divisible by 12. For any nonnegative integer $i$, let $a_{i}$ denote the number of $x \in\{0,1, \ldots, 11\}$ for which $|\operatorname... | 26 | Modulo twelve, the first set turns out to be $\{-1 \cdot 7,0 \cdot 7, \ldots, 5 \cdot 7\}$ and the second set turns out to be be $\{6 \cdot 7, \ldots, 10 \cdot 7\}$. We can eliminate the factor of 7 and shift to reduce the problem to $s \in\{0,1, \ldots, 6\}$ and $t \in\{7, \ldots, 11\}$. With this we can easily comput... | 0 | 7,349.75 | -1 | 7,349.75 |
The equation $x^2-kx-12=0$ has only integer solutions for certain positive integers $k$. What is the sum of all such values of $k$? | 16 | 1 | 3,136.1875 | 3,136.1875 | -1 | |
Losyash is walking to Sovunya's house along the river at a speed of 4 km/h. Every half-hour, he launches paper boats that travel to Sovunya at a speed of 10 km/h. What is the time interval at which the boats arrive at Sovunya's house? | 18 | 0.0625 | 6,399.625 | 7,438 | 6,330.4 | |
Given vectors $\mathbf{a}$ and $\mathbf{b},$ let $\mathbf{p}$ be a vector such that
\[\|\mathbf{p} - \mathbf{b}\| = 2 \|\mathbf{p} - \mathbf{a}\|.\]Among all such vectors $\mathbf{p},$ there exists constants $t$ and $u$ such that $\mathbf{p}$ is at a fixed distance from $t \mathbf{a} + u \mathbf{b}.$ Enter the ordered... | \left( \frac{4}{3}, -\frac{1}{3} \right) | 0.5 | 6,352.5 | 5,418 | 7,287 | |
Let $f(x, y)=x^{2}+2 x+y^{2}+4 y$. Let \(x_{1}, y_{1}\), \(x_{2}, y_{2}\), \(x_{3}, y_{3}\), and \(x_{4}, y_{4}\) be the vertices of a square with side length one and sides parallel to the coordinate axes. What is the minimum value of \(f\left(x_{1}, y_{1}\right)+f\left(x_{2}, y_{2}\right)+f\left(x_{3}, y_{3}\right)+f\... | -18 | The square's corners must be at $(x, y),(x+1, y),(x+1, y+1)$, and $(x, y+1)$ for some $x$ and $y$. So, $$\begin{aligned} f\left(x_{1}, y_{1}\right) & +f\left(x_{2}, y_{2}\right)+f\left(x_{3}, y_{3}\right)+f\left(x_{4}, y_{4}\right) \\ & =2\left(x^{2}+2 x\right)+2\left((x+1)^{2}+2(x+1)\right)+2\left(y^{2}+4 y\right)+2\l... | 0.5 | 7,031.1875 | 6,446 | 7,616.375 |
How many sequences of integers $(a_{1}, \ldots, a_{7})$ are there for which $-1 \leq a_{i} \leq 1$ for every $i$, and $a_{1} a_{2}+a_{2} a_{3}+a_{3} a_{4}+a_{4} a_{5}+a_{5} a_{6}+a_{6} a_{7}=4$? | 38 | For $i=1,2, \ldots, 6$, let $b_{i}=a_{i} a_{i+1}$. From the problem condition each of $b_{1}, b_{2}, \ldots, b_{6}$ can only be $-1,0$, or 1 . Since the sum of these six numbers is 4 , either there are five 1 s and a -1 or there are four 1 s and two 0s. In the first case, there are 6 ways to choose $i$ such that $b_{i}... | 0 | 8,192 | -1 | 8,192 |
The area of triangle \(ABC\) is 1. Let \(A_1\), \(B_1\), and \(C_1\) be the midpoints of the sides \(BC\), \(CA\), and \(AB\) respectively. Points \(K\), \(L\), and \(M\) are taken on segments \(AB_1\), \(CA_1\), and \(BC_1\) respectively. What is the minimum area of the common part of triangles \(KLM\) and \(A_1B_1C_1... | 1/8 | 0.1875 | 8,162.625 | 8,035.333333 | 8,192 | |
Shuai Shuai memorized more than one hundred words in seven days. The number of words memorized in the first three days is $20\%$ less than the number of words memorized in the last four days, and the number of words memorized in the first four days is $20\%$ more than the number of words memorized in the last three day... | 198 | 0.3125 | 6,792.4375 | 6,257.2 | 7,035.727273 | |
For how many integer values of $a$ does the equation $$x^2 + ax + 8a = 0$$ have integer solutions for $x$? | 8 | 0.625 | 6,461.375 | 5,423 | 8,192 | |
Simplify the following expression: $(x^5+x^4+x+10)-(x^5+2x^4-x^3+12).$ Express your answer as a polynomial with the degrees of the terms in decreasing order. | -x^4+x^3+x-2 | 0.875 | 2,655.25 | 2,334.785714 | 4,898.5 | |
Positive numbers \( \mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d} \) satisfy \( a+b+c+d=100 \) and \( \frac{a}{b+c+d}+\frac{b}{a+c+d}+\frac{c}{a+b+d}+\frac{d}{a+b+c}=95 \). Then, \( \frac{1}{b+c+d}+\frac{1}{a+c+d}+\frac{1}{a+b+d}+\frac{1}{a+b+c} = \quad \) | \frac{99}{100} | 0.875 | 4,587.3125 | 4,072.357143 | 8,192 | |
Find the smallest positive integer $n\neq 2004$ for which there exists a polynomial $f\in\mathbb{Z}[x]$ such that the equation $f(x)=2004$ has at least one, and the equation $f(x)=n$ has at least $2004$ different integer solutions. | (1002!)^2 + 2004 | 0 | 8,192 | -1 | 8,192 | |
In how many ways can one fill a \(4 \times 4\) grid with a 0 or 1 in each square such that the sum of the entries in each row, column, and long diagonal is even? | 256 | First we name the elements of the square as follows: \(a_{11}, a_{12}, a_{13}, a_{14}, a_{21}, a_{22}, a_{23}, a_{24}, a_{31}, a_{32}, a_{33}, a_{34}, a_{41}, a_{42}, a_{43}, a_{44}\). We claim that for any given values of \(a_{11}, a_{12}, a_{13}, a_{21}, a_{22}, a_{23}, a_{32}\), and \(a_{33}\) (the + signs in the di... | 0.125 | 7,511.625 | 6,409 | 7,669.142857 |
Xiaoming shot a total of 80 times in 3 minutes and scored 50 goals. Calculate the frequency of Xiaoming's scoring. | 0.625 | 0.8125 | 306.5 | 299.384615 | 337.333333 | |
A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that... | \frac{1}{729} | 1. **Understanding the Problem**: A bug starts at one vertex of a cube and moves along the edges. At each vertex, it can choose any of the three connected edges with equal probability. We need to find the probability that after seven moves, the bug visits every vertex exactly once.
2. **Visualizing the Cube**: Conside... | 0 | 7,830.125 | -1 | 7,830.125 |
Let $A B C D$ be a quadrilateral inscribed in a unit circle with center $O$. Suppose that $\angle A O B=\angle C O D=135^{\circ}, B C=1$. Let $B^{\prime}$ and $C^{\prime}$ be the reflections of $A$ across $B O$ and $C O$ respectively. Let $H_{1}$ and $H_{2}$ be the orthocenters of $A B^{\prime} C^{\prime}$ and $B C D$,... | \frac{1}{4}(8-\sqrt{6}-3 \sqrt{2}) | Put the diagram on the complex plane with $O$ at the origin and $A$ at 1. Let $B$ have coordinate $b$ and $C$ have coordinate $c$. We obtain easily that $B^{\prime}$ is $b^{2}, C^{\prime}$ is $c^{2}$, and $D$ is $b c$. Therefore, $H_{1}$ is $1+b^{2}+c^{2}$ and $H_{2}$ is $b+c+b c$ (we have used the fact that for triang... | 0 | 8,192 | -1 | 8,192 |
The numbers \( x \) and \( y \) are such that the equalities \( \operatorname{ctg} x - \operatorname{ctg} y = 2 \) and \( 5 \sin (2x - 2y) = \sin 2x \sin 2y \) hold. Find \( \operatorname{tg} x \operatorname{tg} y \). | -\frac{6}{5} | 0.3125 | 7,144.1875 | 5,805.8 | 7,752.545455 | |
Find the area of the region between a circle of radius 100 and a circle of radius 99. | 199 \pi | The area of a circle of radius 100 is $100^{2} \pi$, and the area of a circle of radius 99 is $99^{2} \pi$. Therefore, the area of the region between them is $(100^{2}-99^{2}) \pi=(100+99)(100-99) \pi=199 \pi$. | 1 | 1,654.4375 | 1,654.4375 | -1 |
Given a set of points in space, a *jump* consists of taking two points, $P$ and $Q,$ and replacing $P$ with the reflection of $P$ over $Q$ . Find the smallest number $n$ such that for any set of $n$ lattice points in $10$ -dimensional-space, it is possible to perform a finite number of jumps so that some ... | 1025 | 0.3125 | 7,924.8125 | 7,337 | 8,192 | |
I have 7 books, two of which are identical copies of a science book and another two identical copies of a math book, while the rest of the books are all different. In how many ways can I arrange them on a shelf, and additionally, how many of these arrangements can be made if I decide to highlight exactly two books (not... | 26460 | 0.3125 | 6,547.5 | 5,070.8 | 7,218.727273 | |
Arrange all the four-digit numbers formed using $1, 2, 3,$ and $4$, each used exactly once, in ascending order. What is the difference between the 23rd number and the 21st number? | 99 | 0.75 | 5,010.0625 | 4,274.25 | 7,217.5 | |
The positive number $a$ is chosen such that the terms $25, a, b, \frac{1}{25}$ are the first, second, third, and fourth terms, respectively, of a geometric sequence. What is the value of $a$ and $b$? | 25^{-1/3} | 0 | 5,634.25 | -1 | 5,634.25 | |
Let $k$ be a positive integer, and the coefficient of the fourth term in the expansion of $(1+ \frac{x}{k})^{k}$ is $\frac{1}{16}$. Consider the functions $y= \sqrt{8x-x^{2}}$ and $y= \frac{1}{4}kx$, and let $S$ be the shaded region enclosed by their graphs. Calculate the probability that the point $(x,y)$ lies within ... | \frac{\pi}{4} - \frac{1}{2} | 0 | 4,613.4375 | -1 | 4,613.4375 | |
What is the sum of the digits of the base $7$ representation of $777_{10}$? | 9 | 1 | 2,895.3125 | 2,895.3125 | -1 | |
Square $ABCD$ has center $O,\ AB=900,\ E$ and $F$ are on $AB$ with $AE<BF$ and $E$ between $A$ and $F, m\angle EOF =45^\circ,$ and $EF=400.$ Given that $BF=p+q\sqrt{r},$ where $p,q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p+q+r.$ | 307 | Draw AO, OB, and extend OB to D. Let $\angle{FOB} = \alpha.$ Then, after angle chasing, we find that \[\angle{AEB} = 90 + \alpha\]. Using this, we draw a line perpendicular to $AB$ at $E$ to meet $BD$ at $M$. Since $\angle{MEO} = \alpha$ and $\angle{EMO} = 45$, we have that \[\triangle{EMO} \sim \triangle{OBF}\] Let $F... | 0.0625 | 8,168.1875 | 7,811 | 8,192 |
When two fair 12-sided dice are tossed, the numbers $a$ and $b$ are obtained. What is the probability that both the two-digit number $ab$ (where $a$ and $b$ are digits) and each of $a$ and $b$ individually are divisible by 4? | \frac{1}{16} | 0.25 | 7,248.625 | 4,738 | 8,085.5 | |
Regular polygons with $5, 6, 7,$ and $8$ sides are inscribed in the same circle. No two of the polygons share a vertex, and no three of their sides intersect at a common point. At how many points inside the circle do two of their sides intersect? | 68 | To solve this problem, we need to calculate the number of intersection points inside the circle where two sides from different polygons intersect. We consider pairs of polygons and count the intersections for each pair.
1. **Understanding the Intersection Rule**:
- When two polygons with $m$ and $n$ sides ($m > n$)... | 0 | 8,192 | -1 | 8,192 |
Let $S_{n}$ and $T_{n}$ represent the sum of the first $n$ terms of the arithmetic sequences ${a_{n}}$ and ${b_{n}}$, respectively. Given that $\frac{S_{n}}{T_{n}} = \frac{2n+1}{4n-2}$ for all positive integers $n$, find the value of $\frac{a_{10}}{b_{3}+b_{18}} + \frac{a_{11}}{b_{6}+b_{15}}$. | \frac{41}{78} | 0.75 | 5,271.75 | 5,017.75 | 6,033.75 | |
Each row of a $24 \times 8$ table contains some permutation of the numbers $1, 2, \cdots , 8.$ In each column the numbers are multiplied. What is the minimum possible sum of all the products?
*(C. Wu)* | 8 * (8!)^3 | 0 | 7,379.5625 | -1 | 7,379.5625 | |
How many natural numbers greater than one have a product with their smallest prime divisor that is not greater than 100? | 33 | 0 | 8,122.75 | -1 | 8,122.75 | |
Mr. Garcia asked the members of his health class how many days last week they exercised for at least 30 minutes. The results are summarized in the following bar graph, where the heights of the bars represent the number of students.
What was the mean number of days of exercise last week, rounded to the nearest hundred... | 4.36 | To find the mean number of days of exercise reported by the students, we need to calculate the total number of days exercised by all students and then divide this by the total number of students.
1. **Calculate the total number of days exercised**:
- Each student who exercised for 1 day contributes 1 day to the tot... | 0 | 7,656.5 | -1 | 7,656.5 |
Given that the year 2010 corresponds to the Geng-Yin year, determine the year of the previous Geng-Yin year. | 1950 | 0.0625 | 3,321.1875 | 422 | 3,514.466667 | |
Let $S = \{1, 2,..., 8\}$ . How many ways are there to select two disjoint subsets of $S$ ? | 6561 | 0.3125 | 6,787.5 | 5,201 | 7,508.636364 | |
Let $n\geq 3$ be a fixed integer. Each side and each diagonal of a regular $n$-gon is labelled with a number from the set $\left\{1;\;2;\;...;\;r\right\}$ in a way such that the following two conditions are fulfilled:
[b]1.[/b] Each number from the set $\left\{1;\;2;\;...;\;r\right\}$ occurs at least once as a label.
... | {\frac{n!(n-1)!}{2^{n-1}}} |
To solve the given problem, we consider a regular \( n \)-gon with sides and diagonals labeled from a set \(\{1, 2, \ldots, r\}\). The goal is to find the maximal \( r \) such that the labeling satisfies the provided conditions.
### Part (a): Finding the maximal \( r \)
1. **Understanding Conditions**:
- Each nu... | 0 | 8,192 | -1 | 8,192 |
In the following two equations, the same Chinese character represents the same digit, and different Chinese characters represent different digits:
数字花园 + 探秘 = 2015,
探秘 + 1 + 2 + 3 + ... + 10 = 花园
So the four-digit 数字花园 = ______ | 1985 | 0 | 3,511.9375 | -1 | 3,511.9375 | |
A circle of radius $2$ is centered at $O$. Square $OABC$ has side length $1$. Sides $AB$ and $CB$ are extended past $B$ to meet the circle at $D$ and $E$, respectively. What is the area of the shaded region in the figure, which is bounded by $BD$, $BE$, and the minor arc connecting $D$ and $E$? | \frac{\pi}{3}+1-\sqrt{3} | 1. **Identify the Geometry and Key Points**: We have a circle centered at $O$ with radius $2$, and a square $OABC$ with side length $1$. The sides $AB$ and $CB$ are extended to meet the circle at points $D$ and $E$ respectively.
2. **Calculate $DA$ and $CE$ Using the Pythagorean Theorem**: Since $OA = OC = 1$ (radii o... | 0 | 8,112.4375 | -1 | 8,112.4375 |
How many different positive, four-digit integers can be formed using the digits 2, 2, 9 and 9? | 6 | 1 | 1,616 | 1,616 | -1 | |
All letters in the word $VUQAR$ are different and chosen from the set $\{1,2,3,4,5\}$. Find all solutions to the equation \[\frac{(V+U+Q+A+R)^2}{V-U-Q+A+R}=V^{{{U^Q}^A}^R}.\] | (5, 2, 1, 3, 4) \text{ and } (5, 2, 1, 4, 3) |
Let's consider the given problem: we need to find all solutions for the letters \( V, U, Q, A, R \) in the equation:
\[
\frac{(V+U+Q+A+R)^2}{V-U-Q+A+R} = V^{{{U^Q}^A}^R},
\]
where each letter is from the set \(\{1,2,3,4,5\}\) and all letters are different.
### Step-by-step Strategy:
1. **Analyze the Equation**:
... | 0 | 8,152.75 | -1 | 8,152.75 |
How many alphabetic sequences (that is, sequences containing only letters from $a\cdots z$ ) of length $2013$ have letters in alphabetic order? | \binom{2038}{25} | 0 | 3,666.25 | -1 | 3,666.25 | |
Calculate $7 \cdot 9\frac{2}{5}$. | 65\frac{4}{5} | 0.5 | 1,388.75 | 1,013.25 | 1,764.25 | |
In a class organizing a cultural evening, they plan to select 4 programs from 8 programs, with the requirement that at least one of the programs A or B must be selected, and when both A and B are selected, their performance order cannot be adjacent. Express the number of different performance orders as a value. | 1140 | 0.375 | 6,578.5625 | 4,643.5 | 7,739.6 | |
Determine the number of quadratic polynomials $P(x)=p_{1} x^{2}+p_{2} x-p_{3}$, where $p_{1}, p_{2}, p_{3}$ are not necessarily distinct (positive) prime numbers less than 50, whose roots are distinct rational numbers. | 31 | The existence of distinct rational roots means that the given quadratic splits into linear factors. Then, since $p_{1}, p_{3}$ are both prime, we get that the following are the only possible factorizations: - $(p_{1} x-p_{3})(x+1) \Rightarrow p_{2}=p_{1}-p_{3}$ - $(p_{1} x+p_{3})(x-1) \Rightarrow p_{2}=-p_{1}+p_{3}$ - ... | 0 | 8,192 | -1 | 8,192 |
Given the function
$$
f(x)=\left(1-x^{2}\right)\left(x^{2}+b x+c\right) \text{ for } x \in [-1, 1].
$$
Let $\mid f(x) \mid$ have a maximum value of $M(b, c)$. As $b$ and $c$ vary, find the minimum value of $M(b, c)$. | 3 - 2\sqrt{2} | 0 | 8,192 | -1 | 8,192 |
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