problem stringlengths 10 5.15k | answer stringlengths 0 1.22k | solution stringlengths 0 11.1k | reward float64 0 1 | length float64 172 8.19k | correct_length float64 -1 8.19k | incorrect_length float64 -1 8.19k |
|---|---|---|---|---|---|---|
The following construction is used for training astronauts:
A circle $C_2$ of radius $2R$ rolls along the inside of another, fixed circle $C_1$ of radius $nR$, where $n$ is an integer greater than $2$. The astronaut is fastened to a third circle $C_3$ of radius $R$ which rolls along the inside of circle $C_2$ in such a... | n - 1 |
Consider the problem of determining the number of revolutions an astronaut performs relative to the ground while being fastened to a circle \( C_3 \) of radius \( R \). This circle rolls inside another circle \( C_2 \) of radius \( 2R \), which in turn rolls along the inside of a fixed circle \( C_1 \) of radius \( nR... | 0 | 7,917.75 | -1 | 7,917.75 |
Alli rolls a standard 8-sided die twice. What is the probability of rolling integers that differ by 3 on her first two rolls? Express your answer as a common fraction. | \frac{5}{32} | 0.8125 | 4,841.875 | 4,249.230769 | 7,410 | |
Simplify first, then find the value of $\frac{{{a^2}-{b^2}}}{{{a^2}b-a{b^2}}}÷(1+\frac{{{a^2}+{b^2}}}{2ab})$, where $a=\sqrt{3}-\sqrt{11}$ and $b=\sqrt{3}+\sqrt{11}$. | \frac{\sqrt{3}}{3} | 0 | 2,961.9375 | -1 | 2,961.9375 | |
In regular octagon $ABCDEFGH$, $M$ and $N$ are midpoints of $\overline{BC}$ and $\overline{FG}$ respectively. Compute $[ABMO]/[EDCMO]$. ($[ABCD]$ denotes the area of polygon $ABCD$.) [asy]
pair A,B,C,D,E,F,G,H;
F=(0,0); E=(2,0); D=(2+sqrt(2),sqrt(2)); C=(2+sqrt(2),2+sqrt(2));
B=(2,2+2sqrt(2)); A=(0,2+2*sqrt(2)); H=(-s... | \frac{3}{5} | 0.1875 | 7,860.375 | 7,692 | 7,899.230769 | |
The real numbers $x, y, z, w$ satisfy $$\begin{aligned} & 2 x+y+z+w=1 \\ & x+3 y+z+w=2 \\ & x+y+4 z+w=3 \\ & x+y+z+5 w=25 \end{aligned}$$ Find the value of $w$. | 11/2 | Multiplying the four equations by $12,6,4,3$ respectively, we get $$\begin{aligned} 24 x+12 y+12 z+12 w & =12 \\ 6 x+18 y+6 z+6 w & =12 \\ 4 x+4 y+16 z+4 w & =12 \\ 3 x+3 y+3 z+15 w & =75 \end{aligned}$$ Adding these yields $37 x+37 y+37 z+37 w=111$, or $x+y+z+w=3$. Subtract this from the fourth given equation to obtai... | 0.8125 | 5,407.25 | 4,764.615385 | 8,192 |
A refrigerator is offered at sale at $250.00 less successive discounts of 20% and 15%. The sale price of the refrigerator is: | 77\% of 250.00 | 1. **Calculate the price after the first discount:**
The original price of the refrigerator is $250.00$. The first discount is $20\%$ of the original price. Therefore, the price after the first discount is calculated as follows:
\[
250.00 \times (1 - 0.20) = 250.00 \times 0.80 = 200.00
\]
2. **Calculate ... | 0 | 2,382.9375 | -1 | 2,382.9375 |
The sum of the real values of $x$ satisfying the equality $|x+2|=2|x-2|$ is: | 6\frac{2}{3} | To solve the equation $|x+2| = 2|x-2|$, we need to consider the different cases for the absolute values based on the sign of the expressions inside them.
#### Case 1: $x + 2 \geq 0$ and $x - 2 \geq 0$
This implies $x \geq 2$. The equation becomes:
\[ x+2 = 2(x-2) \]
\[ x+2 = 2x - 4 \]
\[ 2 + 4 = 2x - x \]
\[ 6 = x \]
... | 0 | 2,313.4375 | -1 | 2,313.4375 |
Four new students are to be assigned to three classes: A, B, and C, with at least one student in each class. Student A cannot be assigned to class A. How many different assignment plans are there? | 24 | 0.125 | 7,710.5625 | 4,340.5 | 8,192 | |
The radius of a cylinder is doubled and its height is tripled. If its original volume was 10 cubic feet, what is its volume now, in cubic feet? | 120 | 1 | 1,251.125 | 1,251.125 | -1 | |
From point $A$ to point $B$, three cars depart at equal time intervals. They all arrive at $B$ simultaneously, and then they proceed to point $C$, which is 120 km away from $B$. The first car arrives at point $C$ one hour after the second car. After reaching point $C$, the third car immediately turns back and meets th... | 30 | 0 | 8,192 | -1 | 8,192 | |
Given $\sin \left(\frac{3\pi }{2}+\theta \right)=\frac{1}{4}$,
find the value of $\frac{\cos (\pi +\theta )}{\cos \theta [\cos (\pi +\theta )-1]}+\frac{\cos (\theta -2\pi )}{\cos (\theta +2\pi )\cos (\theta +\pi )+\cos (-\theta )}$. | \frac{32}{15} | 0.9375 | 3,580.3125 | 3,272.866667 | 8,192 | |
Integers $1, 2, 3, ... ,n$ , where $n > 2$ , are written on a board. Two numbers $m, k$ such that $1 < m < n, 1 < k < n$ are removed and the average of the remaining numbers is found to be $17$ . What is the maximum sum of the two removed numbers? | 51 | 0.5 | 7,467.375 | 6,742.75 | 8,192 | |
Cameron writes down the smallest positive multiple of 30 that is a perfect square, the smallest positive multiple of 30 that is a perfect cube, and all the multiples of 30 between them. How many integers are in Cameron's list? | 871 | 0.8125 | 4,379.875 | 3,500.153846 | 8,192 | |
Sophia's age is $S$ years, which is thrice the combined ages of her two children. Her age $M$ years ago was four times the sum of their ages at that time. Find the value of $S/M$. | 21 | 0.875 | 1,594.6875 | 1,631.428571 | 1,337.5 | |
Define a regular \(n\)-pointed star as described in the original problem, but with a modification: the vertex connection rule skips by \(m\) steps where \(m\) is coprime with \(n\) and \(m\) is not a multiple of \(3\). How many non-similar regular 120-pointed stars adhere to this new rule? | 15 | 0 | 6,833.9375 | -1 | 6,833.9375 | |
The numbers \(a, b, c, d\) belong to the interval \([-4.5, 4.5]\). Find the maximum value of the expression \(a + 2b + c + 2d - ab - bc - cd - da\). | 90 | 0.0625 | 7,910.5625 | 8,112 | 7,897.133333 | |
If $\frac{\frac{x}{4}}{2}=\frac{4}{\frac{x}{2}}$, then $x=$ | \pm 8 | 1. Start by simplifying the given equation:
\[
\dfrac{\frac{x}{4}}{2} = \dfrac{4}{\frac{x}{2}}
\]
Simplify both sides:
\[
\frac{x}{4 \times 2} = \frac{4 \times 2}{x}
\]
\[
\frac{x}{8} = \frac{8}{x}
\]
2. Cross-multiply to eliminate the fractions:
\[
x \cdot x = 8 \cdot 8
\]
\[... | 1 | 3,131 | 3,131 | -1 |
Identical matches of length 1 are used to arrange the following pattern. If \( c \) denotes the total length of matches used, find \( c \). | 700 | 0 | 6,118.5 | -1 | 6,118.5 | |
Calculate using a simple method:<br/>$(1)100.2\times 99.8$;<br/>$(2)103^{2}$. | 10609 | 0.8125 | 610.5625 | 587.076923 | 712.333333 | |
Calculate the value of $\frac{1}{2 + \frac{1}{3 + \frac{1}{4}}}$. | \frac{13}{30} | 1 | 1,981.375 | 1,981.375 | -1 | |
1. Simplify and evaluate the expression: $\log_{\frac{1}{3}} \sqrt{27} + \lg 25 + \lg 4 + 7^{-\log_{7} 2} + (-0.98)^0$
2. Given a point $P(\sqrt{2}, -\sqrt{6})$ on the terminal side of angle $\alpha$, evaluate: $\frac{\cos \left( \frac{\pi}{2} + \alpha \right) \cos \left( 2\pi - \alpha \right) + \sin \left( -\alpha - \... | \frac{-\sqrt{3} - 1}{3} | 0 | 4,562.875 | -1 | 4,562.875 | |
Given that the midpoint of side $BC$ of triangle $\triangle ABC$ is $D$, point $E$ lies in the plane of $\triangle ABC$, and $\overrightarrow{CD}=3\overrightarrow{CE}-2\overrightarrow{CA}$, if $\overrightarrow{AC}=x\overrightarrow{AB}+y\overrightarrow{BE}$, then determine the value of $x+y$. | 11 | 0.8125 | 4,674.1875 | 4,068 | 7,301 | |
Given that $\frac{5+7+9}{3} = \frac{4020+4021+4022}{M}$, find $M$. | 1723 | 0 | 7,525.375 | -1 | 7,525.375 | |
A target consists of five zones: the center circle (bullseye) and four colored rings. The width of each ring is equal to the radius of the bullseye. It is known that the score for hitting each zone is inversely proportional to the probability of hitting that zone, and hitting the bullseye is worth 315 points. How many ... | 45 | 0.5625 | 5,990 | 4,360.888889 | 8,084.571429 | |
Patrick tosses four four-sided dice, each numbered $1$ through $4$ . What's the probability their product is a multiple of four? | \frac{13}{16} | 0.6875 | 6,194.875 | 5,287.090909 | 8,192 | |
Triangle $ABC$ has side lengths $AB=5$, $BC=6$, and $AC=7$. Two bugs start simultaneously from $A$ and crawl along the perimeter of the triangle in opposite directions at the same speed. They meet at point $D$. What is $BD$? | 4 | 0.9375 | 4,479.5625 | 4,232.066667 | 8,192 | |
A blackboard contains 68 pairs of nonzero integers. Suppose that for each positive integer $k$ at most one of the pairs $(k, k)$ and $(-k, -k)$ is written on the blackboard. A student erases some of the 136 integers, subject to the condition that no two erased integers may add to 0. The student then scores one point... | \[
43
\] | Answer: 43
Attainability: Consider 8 distinct positive numbers. Let there be 5 pairs for each of the numbers including 2 clones of that number. Let there also be 28 pairs that include the negatives of those numbers such that each negative associates with another negative once and exactly once (in graph theoretic term... | 0 | 8,192 | -1 | 8,192 |
Ali Baba and the 40 thieves decided to divide a treasure of 1987 gold coins in the following manner: the first thief divides the entire treasure into two parts, then the second thief divides one of the parts into two parts, and so on. After the 40th division, the first thief picks the largest part, the second thief pi... | 49 | 0.0625 | 8,189.8125 | 8,157 | 8,192 | |
Knights, who always tell the truth, and liars, who always lie, live on an island. One day, 30 inhabitants of this island sat around a round table. Each of them said one of two phrases: "My neighbor on the left is a liar" or "My neighbor on the right is a liar." What is the minimum number of knights that can be at the t... | 10 | 0.0625 | 8,104.375 | 7,835 | 8,122.333333 | |
A sequence of numbers is written on the blackboard: \(1, 2, 3, \cdots, 50\). Each time, the first 4 numbers are erased, and the sum of these 4 erased numbers is written at the end of the sequence, creating a new sequence. This operation is repeated until there are fewer than 4 numbers remaining on the blackboard. Deter... | 755 | 0 | 7,952.3125 | -1 | 7,952.3125 | |
Consider the set $\{0.34,0.304,0.034,0.43\}$. The sum of the smallest and largest numbers in the set is | 0.464 | 1 | 2,491.1875 | 2,491.1875 | -1 | |
In the diagram, the numbers 1 to 10 are placed around a circle. Sandy crosses out 1, then 4, and then 7. Continuing in a clockwise direction, she crosses out every third number of those remaining, until only two numbers are left. The sum of these two numbers is: | 10 | 0.375 | 5,815.9375 | 5,961.333333 | 5,728.7 | |
Given the sequence \( a_{1}, a_{2}, \cdots, a_{n}, \cdots \) that satisfies \( a_{1}=a_{2}=1, a_{3}=2 \), and for any natural number \( n \), \( a_{n} a_{n+1} a_{n+2} \neq 1 \). Furthermore, it is given that \( a_{n} a_{n+1} a_{n+2} a_{n+3} = a_{1} + a_{n+1} + a_{n+2} + a_{n+3} \). Find the value of \( a_{1} + a_{2} + ... | 200 | 0.125 | 7,789.4375 | 4,971.5 | 8,192 | |
Suppose one of the eight lettered identical squares is included with the four squares in the T-shaped figure outlined. How many of the resulting figures can be folded into a topless cubical box? | 6 | To solve this problem, we need to determine how many of the eight lettered squares can be added to the T-shaped figure to create a figure that can be folded into a topless cubical box. The T-shaped figure already consists of four squares that can be thought of as forming four sides of a cube.
#### Step-by-step Analysi... | 0 | 7,781.5 | -1 | 7,781.5 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $b= \sqrt {2}a$, $\sqrt {3}\cos B= \sqrt {2}\cos A$, $c= \sqrt {3}+1$. Find the area of $\triangle ABC$. | \frac { \sqrt {3}+1}{2} | 0 | 7,022.1875 | -1 | 7,022.1875 | |
Calculate the lengths of the arcs of curves defined by the equations in polar coordinates.
$$
\rho=5(1-\cos \varphi),-\frac{\pi}{3} \leq \varphi \leq 0
$$ | 20 \left(1 - \frac{\sqrt{3}}{2}\right) | 0 | 7,327.375 | -1 | 7,327.375 | |
Equilateral triangle $ABC$ has been creased and folded so that vertex $A$ now rests at $A'$ on $\overline{BC}$ as shown. If $BA' = 1$ and $A'C = 2,$ then find the length of crease $\overline{PQ}.$
[asy]
unitsize(1 cm);
pair A, Ap, B, C, P, Q;
A = 3*dir(60);
B = (0,0);
C = (3,0);
Ap = (1,0);
P = 8/5*dir(60);
Q = C +... | \frac{7 \sqrt{21}}{20} | 0 | 6,361.3125 | -1 | 6,361.3125 | |
What is $\frac{2468_{10}}{123_{5}} \times 107_{8} + 4321_{9}$? Express your answer in base 10. | 7789 | 0 | 7,658.5625 | -1 | 7,658.5625 | |
Let $f(x) = \sin{x} + 2\cos{x} + 3\tan{x}$, using radian measure for the variable $x$. Let $r$ be the smallest positive value of $x$ for which $f(x) = 0$. Find $\lfloor r \rfloor.$ | 3 | 0 | 8,192 | -1 | 8,192 | |
Carl wrote a list of 10 distinct positive integers on a board. Each integer in the list, apart from the first, is a multiple of the previous integer. The last of the 10 integers is between 600 and 1000. What is this last integer? | 768 | 0 | 8,192 | -1 | 8,192 | |
How many distinct trees with exactly 7 vertices are there? Here, a tree in graph theory refers to a connected graph without cycles, which can be simply understood as connecting \(n\) vertices with \(n-1\) edges. | 11 | 0.0625 | 3,254.75 | 4,507 | 3,171.266667 | |
In $\Delta ABC$, it is known that $c^2-a^2=5b$ and $3\sin A\cos C=\cos A\sin C$. Find the value of $b$. | 10 | 0.625 | 6,410.6875 | 5,341.9 | 8,192 | |
Does there exist a natural number \( n \), greater than 1, such that the value of the expression \(\sqrt{n \sqrt{n \sqrt{n}}}\) is a natural number? | 256 | 0.9375 | 3,475.375 | 3,342.733333 | 5,465 | |
In the set of equations $z^x = y^{2x}$, $2^z = 2 \cdot 4^x$, $x + y + z = 16$, the integral roots in the order $x,y,z$ are: | 4,3,9 | Let's analyze and solve the given set of equations step by step.
#### Step 1: Simplify the equations
**Equation 1:**
\[ z^x = y^{2x} \]
Taking the $x$-th root on both sides (assuming $x \neq 0$),
\[ z = y^2 \]
**Equation 2:**
\[ 2^z = 2 \cdot 4^x \]
\[ 2^z = 2 \cdot (2^2)^x \]
\[ 2^z = 2 \cdot 2^{2x} \]
\[ 2^z = 2^{... | 0 | 3,619.9375 | -1 | 3,619.9375 |
If three different numbers are selected from 2, 3, 4, 5, 6 to be $a$, $b$, $c$ such that $N = abc + ab + bc + a - b - c$ reaches its maximum value, then this maximum value is. | 167 | 0 | 7,762.0625 | -1 | 7,762.0625 | |
In parallelogram $EFGH$, point $Q$ is on $\overline{EF}$ such that $\frac {EQ}{EF} = \frac {23}{1005}$, and point $R$ is on $\overline{EH}$ such that $\frac {ER}{EH} = \frac {23}{2011}$. Let $S$ be the point of intersection of $\overline{EG}$ and $\overline{QR}$. Find $\frac {EG}{ES}$. | 131 | 0.0625 | 7,315 | 3,292 | 7,583.2 | |
Given a triangle $ABC$ with sides $a$, $b$, $c$, and area $S$ satisfying $S=a^{2}-(b-c)^{2}$, and $b+c=8$.
$(1)$ Find $\cos A$;
$(2)$ Find the maximum value of $S$. | \frac{64}{17} | 0.875 | 4,884.75 | 4,412.285714 | 8,192 | |
6 * cos(18°) + 2 * cos(36°) + 4 * cos(54°) + ... + 20 * cos(360°) = ? | 10 | 0 | 8,192 | -1 | 8,192 | |
An ant starts at the point \((1,0)\). Each minute, it walks from its current position to one of the four adjacent lattice points until it reaches a point \((x, y)\) with \(|x|+|y| \geq 2\). What is the probability that the ant ends at the point \((1,1)\)? | 7/24 | 0 | 8,012.875 | -1 | 8,012.875 | |
The regular tetrahedron, octahedron, and icosahedron have equal surface areas. How are their edges related? | 2 \sqrt{10} : \sqrt{10} : 2 | 0 | 5,934.75 | -1 | 5,934.75 | |
Let $a,$ $b,$ $c$ be distinct, nonzero real numbers such that
\[a + \frac{1}{b} = b + \frac{1}{c} = c + \frac{1}{a}.\]Find $|abc|.$
Note: Intermediate Algebra writing problem, week 12. | 1 | 0.6875 | 6,158.9375 | 5,234.818182 | 8,192 | |
A line segment begins at $(2, 5)$. It is 10 units long and ends at the point $(-6, y)$ where $y > 0$. What is the value of $y$? | 11 | 1 | 1,379.3125 | 1,379.3125 | -1 | |
The function $g$ is defined on positive integers as follows:
\[g(n) = \left\{
\begin{array}{cl}
n + 12 & \text{if $n < 12$}, \\
g(n - 7) & \text{if $n \ge 12$}.
\end{array}
\right.\]
Find the maximum value of the function. | 23 | 0.5 | 6,059.75 | 3,927.5 | 8,192 | |
A floor 9 feet by 12 feet, is to be tiled with 4-inch-by-6-inch tiles. How many tiles are needed to cover the floor? | 648 | 1 | 2,804.375 | 2,804.375 | -1 | |
Compute $\cos 0^\circ$. | 1 | 1 | 1,816.125 | 1,816.125 | -1 | |
A red ball and a green ball are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin $k$ is $2^{-k}$ for $k = 1,2,3....$ What is the probability that the red ball is tossed into a higher-numbered bin than the green ball? | \frac{1}{3} |
We are given that the probability that a ball is tossed into bin $k$ is $2^{-k}$ for $k = 1, 2, 3, \ldots$. We need to find the probability that the red ball is tossed into a higher-numbered bin than the green ball.
#### Step-by-step Analysis:
1. **Probability of Landing in the Same Bin:**
Let's first calculate t... | 1 | 4,633.3125 | 4,633.3125 | -1 |
Solve \[\frac{5x+1}{2x^2+5x-3}=\frac{2x}{2x-1}\]for $x$. | -1 | 0.9375 | 2,839.625 | 2,876.133333 | 2,292 | |
Let $n>5$ be an integer. There are $n$ points in the plane, no three of them collinear. Each day, Tom erases one of the points, until there are three points left. On the $i$-th day, for $1<i<n-3$, before erasing that day's point, Tom writes down the positive integer $v(i)$ such that the convex hull of the points at tha... | 2n - 8 |
Given an integer \( n > 5 \), there are \( n \) points in the plane with no three collinear. Tom sequentially erases a point each day until only three points remain. On the \( i \)-th day (\( 1 < i < n-3 \)), he notes a positive integer \( v(i) \) representing the number of vertices in the current convex hull. Finally... | 0 | 7,772.4375 | -1 | 7,772.4375 |
One hundred students at Century High School participated in the AHSME last year, and their mean score was 100. The number of non-seniors taking the AHSME was $50\%$ more than the number of seniors, and the mean score of the seniors was $50\%$ higher than that of the non-seniors. What was the mean score of the seniors? | 125 | 1. **Define Variables:**
Let $s$ be the number of seniors and $n$ be the number of non-seniors. According to the problem, the number of non-seniors is $50\%$ more than the number of seniors, so we can write:
\[
n = s + 0.5s = 1.5s
\]
2. **Total Students:**
The total number of students is given as 100. T... | 1 | 2,347.25 | 2,347.25 | -1 |
There is a basket of apples. If Class A shares the apples such that each person gets 3 apples, 10 apples remain. If Class B shares the apples such that each person gets 4 apples, 11 apples remain. If Class C shares the apples such that each person gets 5 apples, 12 apples remain. How many apples are there in the basket... | 67 | 0.3125 | 7,685.6875 | 7,118.8 | 7,943.363636 | |
Real numbers $x, y, z$ satisfy $$x+x y+x y z=1, \quad y+y z+x y z=2, \quad z+x z+x y z=4$$ The largest possible value of $x y z$ is $\frac{a+b \sqrt{c}}{d}$, where $a, b, c, d$ are integers, $d$ is positive, $c$ is square-free, and $\operatorname{gcd}(a, b, d)=1$. Find $1000 a+100 b+10 c+d$. | 5272 | Solution 1: Let $p=x y z$ and $q=(x+1)(y+1)(z+1)$. Then, we get $$p q=[x(1+y)] \cdot[y(1+z)] \cdot[z(1+x)]=(1-p)(2-p)(4-p)$$ Additionally, note that $$q-p=x y+y z+z x+x+y+z+1=(x+x y)+(y+y z)+(z+x z)+1=8-3 p$$ Therefore, we have $q=8-2 p$. Substituting this into our earlier equation gives us $$p(8-2 p)=(1-p)(2-p)(4-p)$$... | 0 | 8,192 | -1 | 8,192 |
Externally tangent circles with centers at points $A$ and $B$ have radii of lengths $5$ and $3$, respectively. A line externally tangent to both circles intersects ray $AB$ at point $C$. What is $BC$? | 12 | 1. **Identify the Configuration**:
Let $A$ and $B$ be the centers of two externally tangent circles with radii $5$ and $3$, respectively. The line tangent to both circles intersects ray $AB$ at point $C$. Let $D$ and $E$ be the points of tangency on circles centered at $A$ and $B$, respectively.
2. **Distance Betw... | 0.9375 | 4,358.1875 | 4,102.6 | 8,192 |
Find the smallest six-digit number that is divisible by 3, 7, and 13 without a remainder. | 100191 | 0.5625 | 5,202.875 | 3,752.666667 | 7,067.428571 | |
Given the function f(x) = sinωx + cosωx, if there exists a real number x₁ such that for any real number x, f(x₁) ≤ f(x) ≤ f(x₁ + 2018) holds true, find the minimum positive value of ω. | \frac{\pi}{2018} | 0.5 | 6,114.6875 | 4,479.875 | 7,749.5 | |
Ramon sells two enchiladas and three tacos for $\$$2.50 and he sells three enchiladas and two tacos for $\$$2.70. Assuming a fixed price per item, what is the cost, in dollars, of three enchiladas and four tacos? Express your answer as a decimal to the nearest hundredth. | \$3.54 | 1 | 2,292.5 | 2,292.5 | -1 | |
Find the greatest positive integer $N$ with the following property: there exist integers $x_1, . . . , x_N$ such that $x^2_i - x_ix_j$ is not divisible by $1111$ for any $i\ne j.$ | 1000 | 0 | 7,639.125 | -1 | 7,639.125 | |
Determine the share of the Chinese yuan in the currency structure of the National Wealth Fund (NWF) as of 01.07.2021 by one of the following methods:
First method:
a) Find the total amount of NWF funds placed in Chinese yuan as of 01.07.2021:
\[ CNY_{22} = 1213.76 - 3.36 - 38.4 - 4.25 - 600.3 - 340.56 - 0.29 = 226.6 \... | 1.2 | 0.375 | 4,024.5 | 2,909.333333 | 4,693.6 | |
In triangle $ABC$, where $AB = 50$, $BC = 36$, and $AC = 42$. A line $CX$ from $C$ is perpendicular to $AB$ and intersects $AB$ at point $X$. Find the ratio of the area of $\triangle BCX$ to the area of $\triangle ACX$. Express your answer as a simplified common fraction. | \frac{6}{7} | 0.125 | 6,746.4375 | 3,641 | 7,190.071429 | |
Let $p(x)$ be a monic polynomial of degree 7 such that $p(0) = 0,$ $p(1) = 1,$ $p(2) = 2,$ $p(3) = 3,$ $p(4) = 4,$ $p(5) = 5,$ $p(6) = 6,$ and $p(7) = 7.$ Find $p(8).$ | 40328 | 0 | 7,950.625 | -1 | 7,950.625 | |
In the Cartesian coordinate system, with the origin as the pole and the positive x-axis as the polar axis, the polar equation of line $l$ is $$ρ\cos(θ+ \frac {π}{4})= \frac { \sqrt {2}}{2}$$, and the parametric equation of curve $C$ is $$\begin{cases} x=5+\cos\theta \\ y=\sin\theta \end{cases}$$, (where $θ$ is the para... | 2+ \sqrt {34} | 0 | 7,709.8125 | -1 | 7,709.8125 | |
Let \( a \) be an integer such that \( |a| \leq 2005 \). Find the number of values of \( a \) for which the system of equations
\[
\begin{cases}
x^2 = y + a, \\
y^2 = x + a
\end{cases}
\]
has integer solutions. | 90 | 0 | 8,192 | -1 | 8,192 | |
In square ABCD, point E is on AB and point F is on BC such that AE=3EB and BF=FC. Find the ratio of the area of triangle DEF to the area of square ABCD. | \frac{5}{16} | 1 | 3,640.125 | 3,640.125 | -1 | |
A square with an area of 40 is inscribed in a semicircle. If another square is inscribed in a full circle with the same radius, what is the area of this square? | 80 | 0 | 4,596.25 | -1 | 4,596.25 | |
Given that the sequence $\left\{\frac{1}{b_{n}}\right\}$ is a "dream sequence" defined by $\frac{1}{a_{n+1}}- \frac{2}{a_{n}}=0$, and that $b_1+b_2+b_3=2$, find the value of $b_6+b_7+b_8$. | 64 | 0.5625 | 3,702.6875 | 3,980.222222 | 3,345.857143 | |
A binary string of length $n$ is a sequence of $n$ digits, each of which is 0 or 1 . The distance between two binary strings of the same length is the number of positions in which they disagree; for example, the distance between the strings 01101011 and 00101110 is 3 since they differ in the second, sixth, and eighth p... | \begin{tabular}{ll} 00000000 & 00110101 \ 11001010 & 10011110 \ 11100001 & 01101011 \ 11010100 & 01100110 \ 10111001 & 10010011 \ 01111100 & 11001101 \ 00111010 & 10101100 \ 01010111 & 11110010 \ 00001111 & 01011001 \ 10100111 & 11111111 \ \end{tabular} | The maximum possible number of such strings is 20 . An example of a set attaining this bound is \begin{tabular}{ll} 00000000 & 00110101 \\ 11001010 & 10011110 \\ 11100001 & 01101011 \\ 11010100 & 01100110 \\ 10111001 & 10010011 \\ 01111100 & 11001101 \\ 00111010 & 10101100 \\ 01010111 & 11110010 \\ 00001111 & 01011001 ... | 0 | 6,609.875 | -1 | 6,609.875 |
Find the smallest period of the function \( y = \cos^{10} x + \sin^{10} x \). | \frac{\pi}{2} | 0.125 | 8,116 | 7,866.5 | 8,151.642857 | |
There are 1987 sets, each with 45 elements. The union of any two sets has 89 elements. How many elements are there in the union of all 1987 sets? | 87429 | 0 | 8,192 | -1 | 8,192 | |
Ten Cs are written in a row. Some Cs are upper-case and some are lower-case, and each is written in one of two colors, green and yellow. It is given that there is at least one lower-case C, at least one green C, and at least one C that is both upper-case and yellow. Furthermore, no lower-case C can be followed by an up... | 36 | By the conditions of the problem, we must pick some point in the line where the green Cs transition to yellow, and some point where the upper-case Cs transition to lower-case. We see that the first transition must occur before the second, and that they cannot occur on the same C. Hence, the answer is $\binom{9}{2}=36$. | 0 | 8,192 | -1 | 8,192 |
Let \( n \geq 1 \) be a positive integer. We say that an integer \( k \) is a fan of \( n \) if \( 0 \leq k \leq n-1 \) and there exist integers \( x, y, z \in \mathbb{Z} \) such that
\[
\begin{aligned}
x^2 + y^2 + z^2 &\equiv 0 \pmod{n}; \\
xyz &\equiv k \pmod{n}.
\end{aligned}
\]
Let \( f(n) \) be the number of fans... | 101 | 0 | 8,192 | -1 | 8,192 | |
Each face of a regular tetrahedron is labeled with one of the numbers $1, 2, 3, 4$. Four identical regular tetrahedrons are simultaneously rolled onto a table. What is the probability that the product of the four numbers on the faces touching the table is divisible by 4? | $\frac{13}{16}$ | 0 | 7,268.625 | -1 | 7,268.625 | |
The figure is constructed from $11$ line segments, each of which has length $2$. The area of pentagon $ABCDE$ can be written as $\sqrt{m} + \sqrt{n}$, where $m$ and $n$ are positive integers. What is $m + n ?$ | 23 | 1. **Identify Key Triangles and Midpoint**:
Let $M$ be the midpoint of $CD$. Given that the figure is constructed from line segments each of length $2$, and considering the symmetry and angles in the figure, we can identify that $\triangle AED$ and $\triangle ABC$ are $30^\circ-60^\circ-90^\circ$ triangles. This is... | 0 | 8,192 | -1 | 8,192 |
A square with a side length of one unit has one of its vertices separated from the other three by a line \( e \). The products of the distances of opposite vertex pairs from \( e \) are equal. What is the distance of the center of the square from \( e \)? | \frac{1}{2} | 0.0625 | 8,192 | 8,192 | 8,192 | |
A rectangular flag is divided into four triangles, labelled Left, Right, Top, and Bottom. Each triangle is to be colored one of red, white, blue, green, and purple such that no two triangles that share an edge are the same color. Determine the total number of different flags that can be made. | 260 | 0.625 | 6,609.9375 | 5,660.7 | 8,192 | |
Given the random variables $\xi + \eta = 8$, if $\xi \sim B(10, 0.6)$, then calculate $E\eta$ and $D\eta$. | 2.4 | 1 | 2,013.625 | 2,013.625 | -1 | |
If there exist $n$ real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfying $x_{1}+2 x_{2}+\cdots+ n x_{n}=2009$ and $x_{1}+x_{2}+\cdots+x_{n}=0$, where each $x_{i}= \pm 7$ for $i=1,2, \cdots, n$, determine the minimum value of $n$. | 34 | 0 | 8,192 | -1 | 8,192 | |
Given that \([x]\) represents the largest integer not exceeding \( x \), if \([x+0.1] + [x+0.2] + \ldots + [x+0.9] = 104\), what is the minimum value of \( x \)? | 11.5 | 0 | 8,192 | -1 | 8,192 | |
There are 40 identical looking coins, among which 3 are counterfeit - they weigh the same and are lighter than the genuine coins (the genuine coins also weigh the same). How can you use three weighings on a balance scale without weights to select 16 genuine coins? | 16 | 0.125 | 8,031.4375 | 6,907.5 | 8,192 | |
Two dice are tossed. What is the probability that the sum is greater than three? | \frac{11}{12} | 0.625 | 4,966.4375 | 3,031.1 | 8,192 | |
For \( x \in \mathbb{R} \), the function \( f(x) \) satisfies \( f(x+4) + f(x-4) = f(x) \). Thus, it is a periodic function. The common minimum period of such functions is: | 24 | 0.4375 | 7,100.75 | 5,835.142857 | 8,085.111111 | |
Given the sequence $\{a\_n\}(n=1,2,3,...,2016)$, circle $C\_1$: $x^{2}+y^{2}-4x-4y=0$, circle $C\_2$: $x^{2}+y^{2}-2a_{n}x-2a_{2017-n}y=0$. If circle $C\_2$ bisects the circumference of circle $C\_1$, then the sum of all terms in the sequence $\{a\_n\}$ is $\_\_\_\_\_\_$. | 4032 | 0.4375 | 5,114 | 4,258.857143 | 5,779.111111 | |
Call a permutation $a_1, a_2, \ldots, a_n$ of the integers $1, 2, \ldots, n$ quasi-increasing if $a_k \leq a_{k+1} + 2$ for each $1 \leq k \leq n-1$. For example, 53421 and 14253 are quasi-increasing permutations of the integers $1, 2, 3, 4, 5$, but 45123 is not. Find the number of quasi-increasing permutations of the ... | 486 | The simple recurrence can be found.
When inserting an integer $n$ into a string with $n - 1$ integers, we notice that the integer $n$ has 3 spots where it can go: before $n - 1$, before $n - 2$, and at the very end.
EXAMPLE: Putting 4 into the string 123: 4 can go before the 2: 1423, Before the 3: 1243, And at the ve... | 0 | 8,192 | -1 | 8,192 |
Find the largest real number \(\lambda\) such that for the real coefficient polynomial \(f(x) = x^3 + ax^2 + bx + c\) with all non-negative real roots, it holds that \(f(x) \geqslant \lambda(x - a)^3\) for all \(x \geqslant 0\). Additionally, determine when the equality in the expression is achieved. | -1/27 | 0 | 8,009.5625 | -1 | 8,009.5625 | |
What is the area of the shaded figure shown below? | 6 | To find the area of the shaded figure with vertices given as $(1,0)$, $(3,2)$, $(5,0)$, and $(3,5)$, we can use the Shoelace Theorem. The Shoelace Theorem states that the area of a polygon with vertices $(x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)$ is given by:
\[
\text{Area} = \frac{1}{2} \left| \sum_{i=1}^{n-1} (x_i ... | 0 | 7,924.75 | -1 | 7,924.75 |
What is the first nonzero digit to the right of the decimal point of the fraction $\frac{1}{129}$? | 7 | 0.875 | 6,206.8125 | 5,923.214286 | 8,192 | |
The problem involves finding the value of the expressions $\lg 2 + \lg 5$ and $4(-100)^4$. | 400000000 | 0.9375 | 1,686.8125 | 1,670.733333 | 1,928 | |
Two long cylindrical tubes of the same length but different diameters lie parallel to each other on a flat surface. The larger tube has radius $72$ and rolls along the surface toward the smaller tube, which has radius $24$. It rolls over the smaller tube and continues rolling along the flat surface until it comes to re... | 179 | If it weren’t for the small tube, the larger tube would travel $144\pi$. Consider the distance from which the larger tube first contacts the smaller tube, until when it completely loses contact with the smaller tube.
Drawing the radii as shown in the diagram, notice that the hypotenuse of the right triangle in the dia... | 0 | 7,904.5625 | -1 | 7,904.5625 |
In the diagram below, lines $k$ and $\ell$ are parallel. Find the measure of angle $x$ in degrees.
[asy]
size(200);
import markers;
pair A = dir(-22)*(0,0);
pair B = dir(-22)*(4,0);
pair C = dir(-22)*(4,2);
pair D = dir(-22)*(0,2);
pair F = dir(-22)*(0,1.3);
pair G = dir(-22)*(4,1.3);
pair H = dir(-22)*(2,1);
//mark... | 60^\circ | 0.25 | 7,518.75 | 5,499 | 8,192 | |
If $\|\mathbf{a}\| = 3$ and $\|\mathbf{b}\| = 6,$ then find $(\mathbf{a} + \mathbf{b}) \cdot (\mathbf{a} - \mathbf{b}).$ | -27 | 1 | 1,715.4375 | 1,715.4375 | -1 | |
Given that $x$ is a multiple of $18720$, what is the greatest common divisor of $f(x)=(5x+3)(8x+2)(12x+7)(3x+11)$ and $x$? | 462 | 0 | 5,876.0625 | -1 | 5,876.0625 | |
Out of 10 distinct positive integers, the product of any 5 of them is even, and the sum of all 10 numbers is odd. What is the minimum sum of these 10 positive integers? | 65 | 0.375 | 7,438.4375 | 6,182.5 | 8,192 |
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