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 |
|---|---|---|---|---|---|---|
How many ordered triples $(a, b, c)$ of non-zero real numbers have the property that each number is the product of the other two? | 4 | 1. **Setting up the equations:** Given the conditions $ab = c$, $bc = a$, and $ca = b$, we need to find the ordered triples $(a, b, c)$ of non-zero real numbers that satisfy these equations.
2. **Multiplying the equations:** Multiply the three given equations:
\[
ab \cdot bc \cdot ca = c \cdot a \cdot b
\]
... | 1 | 4,570.8125 | 4,570.8125 | -1 |
A fair die is rolled twice in succession, and the numbers facing up are observed and recorded as $x$ and $y$ respectively.
$(1)$ If the event "$x+y=8$" is denoted as event $A$, find the probability of event $A$ occurring;
$(2)$ If the event "$x^{2}+y^{2} \leqslant 12$" is denoted as event $B$, find the probability ... | \dfrac{1}{6} | 0.9375 | 3,063.75 | 3,121.6 | 2,196 | |
In how many ways can four people line up in a straight line if the youngest person cannot be first in line? | 18 | 1 | 1,937.1875 | 1,937.1875 | -1 | |
Grandma told her grandchildren: "Today I am 60 years and 50 months and 40 weeks and 30 days old."
How old was Grandma on her last birthday? | 65 | 0.125 | 835.1875 | 659 | 860.357143 | |
Determine the distance that the origin $O(0,0)$ moves under the dilation transformation that sends the circle of radius $4$ centered at $B(3,1)$ to the circle of radius $6$ centered at $B'(7,9)$. | 0.5\sqrt{10} | 0 | 4,492.4375 | -1 | 4,492.4375 | |
In $\triangle ABC$, $\angle ACB=60^{\circ}$, $BC > 1$, and $AC=AB+\frac{1}{2}$. When the perimeter of $\triangle ABC$ is at its minimum, the length of $BC$ is $\_\_\_\_\_\_\_\_\_\_$. | 1 + \frac{\sqrt{2}}{2} | 0 | 7,749.375 | -1 | 7,749.375 | |
Convert the complex number \(1 + i \sqrt{3}\) into its exponential form \(re^{i \theta}\) and find \(\theta\). | \frac{\pi}{3} | 0.5 | 1,572 | 1,728.5 | 1,415.5 | |
What is the average of all the integer values of $M$ such that $\frac{M}{70}$ is strictly between $\frac{2}{5}$ and $\frac{3}{10}$? | 24.5 | 0.6875 | 3,571.375 | 3,387.909091 | 3,975 | |
What is the value of the expression $x^2+ 5x-6$, when $x =-1$? | -10 | 0.9375 | 812.5625 | 852.466667 | 214 | |
The remainder can be defined for all real numbers $x$ and $y$ with $y \neq 0$ by $\text{rem} (x ,y)=x-y\left \lfloor \frac{x}{y} \right \rfloor$ where $\left \lfloor \tfrac{x}{y} \right \rfloor$ denotes the greatest integer less than or equal to $\tfrac{x}{y}$. What is the value of $\text{rem} (\tfrac{3}{8}, -\tfrac{2}... | -\frac{1}{40} | 1. **Definition of Remainder**: The remainder function $\text{rem}(x, y)$ for real numbers $x$ and $y$ (with $y \neq 0$) is defined as:
\[
\text{rem}(x, y) = x - y \left\lfloor \frac{x}{y} \right\rfloor
\]
where $\left\lfloor \cdot \right\rfloor$ denotes the greatest integer less than or equal to the enclos... | 0.9375 | 4,295.6875 | 4,035.933333 | 8,192 |
A 9x9 chessboard has its squares labeled such that the label of the square in the ith row and jth column is given by $\frac{1}{2 \times (i + j - 1)}$. We need to select one square from each row and each column. Find the minimum sum of the labels of the nine chosen squares. | \frac{1}{2} | 0.125 | 7,981.4375 | 6,507.5 | 8,192 | |
What is the $x$-intercept of the line perpendicular to the line defined by $3x-2y = 6$ and whose $y$-intercept is 2? | 3 | 0.9375 | 1,681.875 | 1,591.4 | 3,039 | |
Yu Semo and Yu Sejmo have created sequences of symbols $\mathcal{U} = (\text{U}_1, \ldots, \text{U}_6)$ and $\mathcal{J} = (\text{J}_1, \ldots, \text{J}_6)$ . These sequences satisfy the following properties.
- Each of the twelve symbols must be $\Sigma$ , $\#$ , $\triangle$ , or $\mathbb{Z}$ .
- In each of th... | 24 | 0 | 8,192 | -1 | 8,192 | |
Solve the equation \(\sqrt{8x+5} + 2 \{x\} = 2x + 2\). Here, \(\{x\}\) denotes the fractional part of \(x\), i.e., \(\{x\} = x - \lfloor x \rfloor\). Write down the sum of all solutions. | 0.75 | 0 | 5,016.5 | -1 | 5,016.5 | |
Which of the following is equal to $9^{4}$? | 3^{8} | Since $9=3 \times 3$, then $9^{4}=(3 \times 3)^{4}=3^{4} \times 3^{4}=3^{8}$. Alternatively, we can note that $9^{4}=9 \times 9 \times 9 \times 9=(3 \times 3) \times(3 \times 3) \times(3 \times 3) \times(3 \times 3)=3^{8}$. | 0 | 2,801.9375 | -1 | 2,801.9375 |
Given $\{1,a, \frac{b}{a}\}=\{0,a^2,a+b\}$, calculate the value of $a^{2005}+b^{2005}$. | -1 | 0.125 | 7,963 | 8,032 | 7,953.142857 | |
A school has 1200 students, and each student participates in exactly \( k \) clubs. It is known that any group of 23 students all participate in at least one club in common, but no club includes all 1200 students. Find the minimum possible value of \( k \). | 23 | 0 | 8,192 | -1 | 8,192 | |
Given a box contains $4$ shiny pennies and $5$ dull pennies, determine the probability that the third shiny penny appears on the sixth draw. | \frac{5}{21} | 0.25 | 7,435.0625 | 6,054.25 | 7,895.333333 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively, and $a^{2}$, $b^{2}$, $c^{2}$ form an arithmetic sequence. Calculate the maximum value of $\sin B$. | \dfrac{ \sqrt {3}}{2} | 0 | 6,829.3125 | -1 | 6,829.3125 | |
Every high school in the city of Euclid sent a team of $3$ students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed $37$th and $64$th, respectively. How m... | 23 | 1. **Understanding the problem setup**: Each high school sends 3 students, and each student has a unique score. Andrea's score is the median of all scores, and she has the highest score on her team. Her teammates Beth and Carla placed 37th and 64th, respectively.
2. **Determining Andrea's rank**: Since Andrea's score ... | 0.5625 | 6,452.875 | 5,443.222222 | 7,751 |
Five people are sitting at a round table. Let $f\geq 0$ be the number of people sitting next to at least 1 female and $m\geq0$ be the number of people sitting next to at least one male. The number of possible ordered pairs $(f,m)$ is
$\mathrm{(A) \ 7 } \qquad \mathrm{(B) \ 8 } \qquad \mathrm{(C) \ 9 } \qquad \mathrm{(D... | 8 | 0 | 8,192 | -1 | 8,192 | |
A metallic weight has a mass of 20 kg and is an alloy of four metals. The first metal in this alloy is one and a half times the amount of the second metal. The mass of the second metal relates to the mass of the third metal as $3:4$, and the mass of the third metal to the mass of the fourth metal as $5:6$. Determine th... | 5.89 | 0.625 | 3,671.75 | 3,575.9 | 3,831.5 | |
Suppose two arithmetic sequences $\{a_n\}$ and $\{b_n\}$ have the sum of their first $n$ terms as $S_n$ and $T_n$, respectively. Given that $\frac{S_n}{T_n} = \frac{7n}{n+3}$, find the value of $\frac{a_5}{b_5}$. | \frac{21}{4} | 0.5 | 5,514.1875 | 4,181.875 | 6,846.5 | |
Let $d$ and $e$ denote the solutions of $3x^2+10x-25=0$. Find $(d-e)^2$. | \frac{400}{9} | 1 | 2,196.1875 | 2,196.1875 | -1 | |
In parallelogram ABCD, $\angle BAD=60^\circ$, $AB=1$, $AD=\sqrt{2}$, and P is a point inside the parallelogram such that $AP=\frac{\sqrt{2}}{2}$. If $\overrightarrow{AP}=\lambda\overrightarrow{AB}+\mu\overrightarrow{AD}$ ($\lambda,\mu\in\mathbb{R}$), then the maximum value of $\lambda+\sqrt{2}\mu$ is \_\_\_\_\_\_. | \frac{\sqrt{6}}{3} | 0 | 7,912.9375 | -1 | 7,912.9375 | |
Each of the points $A,B,C,D,E,$ and $F$ in the figure below represents a different digit from $1$ to $6.$ Each of the five lines shown passes through some of these points. The digits along each line are added to produce five sums, one for each line. The total of the five sums is $47.$ What is the digit represented by $... | 5 | 1. **Identify the sums along each line**: Given the points $A, B, C, D, E, F$, each representing a unique digit from 1 to 6, and the lines connecting them, we can write the sums for each line as:
- Line through $A, B, C$: $A + B + C$
- Line through $A, E, F$: $A + E + F$
- Line through $C, D, E$: $C + D + E$
... | 0.1875 | 7,852.6875 | 6,382.333333 | 8,192 |
$B$ and $C$ trisect $\overline{AD}$ and $M$ is the midpoint of $\overline{AD}$. $MC = 8$. How many units are in the length of $\overline{AD}$? | 48 | 1 | 1,740.5625 | 1,740.5625 | -1 | |
Two identical CDs regularly cost a total of $\$28.$ What is the cost in dollars of five of these CDs? | 70 | 1 | 1,267.9375 | 1,267.9375 | -1 | |
What is the smallest integer that can be placed in the box so that $\frac{1}{2} < \frac{\square}{9}$? | 5 | We know that $\frac{1}{2} = 0.5$. Since $\frac{4}{9} \approx 0.44$ is less than $\frac{1}{2} = 0.5$, then 4 cannot be placed in the box. (No integer smaller than 4 can be placed in the box either.) Since $\frac{5}{9} \approx 0.56$ is greater than $\frac{1}{2} = 0.5$, then the smallest integer that can be placed in the ... | 1 | 1,054.875 | 1,054.875 | -1 |
The $\$4.55$ in Carol's piggy bank consists of quarters and nickels. There are seven more nickels than quarters. How many nickels does Carol have in her bank? | 21 | 1 | 1,454.4375 | 1,454.4375 | -1 | |
Let $A=(0,9)$ and $B=(0,12)$. Points $A'$ and $B'$ are on the line $y=x$, and $\overline{AA'}$ and $\overline{BB'}$ intersect at $C=(2,8)$. What is the length of $\overline{A'B'}$? | 2\sqrt{2} | 1 | 3,094.25 | 3,094.25 | -1 | |
Let $\triangle ABC$ have $\angle ABC=67^{\circ}$ . Point $X$ is chosen such that $AB = XC$ , $\angle{XAC}=32^\circ$ , and $\angle{XCA}=35^\circ$ . Compute $\angle{BAC}$ in degrees.
*Proposed by Raina Yang* | 81 | 0.4375 | 6,821.9375 | 5,315.428571 | 7,993.666667 | |
In each cell of a $4 \times 4$ grid, one of the two diagonals is drawn uniformly at random. Compute the probability that the resulting 32 triangular regions can be colored red and blue so that any two regions sharing an edge have different colors. | \frac{1}{512} | Give each cell coordinates from $(1,1)$ to $(4,4)$. Claim. The grid has a desired coloring if and only if every vertex not on the boundary meets an even number of edges and diagonals. Proof. If this were not the case, the odd number of regions around the vertex would have to alternate between the two colors, which is c... | 0 | 8,192 | -1 | 8,192 |
Consider the sequence
$$
a_{n}=\cos (\underbrace{100 \ldots 0^{\circ}}_{n-1})
$$
For example, $a_{1}=\cos 1^{\circ}, a_{6}=\cos 100000^{\circ}$.
How many of the numbers $a_{1}, a_{2}, \ldots, a_{100}$ are positive? | 99 | 0.5 | 6,566.1875 | 5,761.25 | 7,371.125 | |
Alice's favorite number has the following properties:
- It has 8 distinct digits.
- The digits are decreasing when read from left to right.
- It is divisible by 180.
What is Alice's favorite number?
*Author: Anderson Wang* | 97654320 | 0.0625 | 8,169.625 | 7,834 | 8,192 | |
If $13^{3n}=\left(\frac{1}{13}\right)^{n-24}$, find $n$. | 6 | 1 | 1,712.3125 | 1,712.3125 | -1 | |
The points $(1, 7), (13, 16)$ and $(5, k)$, where $k$ is an integer, are vertices of a triangle. What is the sum of the values of $k$ for which the area of the triangle is a minimum? | 20 | 0.625 | 3,995.8125 | 3,294 | 5,165.5 | |
If $A*B$ means $\frac{A+B}{2}$, then $(3*5)*8$ is | 6 | 1. **Interpret the operation $*$**: Given that $A*B = \frac{A+B}{2}$, we need to apply this operation to the numbers in the expression $(3*5)*8$.
2. **Evaluate $3*5$**:
\[
3*5 = \frac{3+5}{2} = \frac{8}{2} = 4
\]
Here, we added 3 and 5, then divided by 2 as per the definition of the operation $*$.
3. **Us... | 1 | 2,553.75 | 2,553.75 | -1 |
Let $\triangle ABC$ be a right triangle with $B$ as the right angle. A circle with diameter $AC$ intersects side $BC$ at point $D$. If $AB = 18$ and $AC = 30$, find the length of $BD$. | 14.4 | 0 | 8,051.9375 | -1 | 8,051.9375 | |
In the Cartesian coordinate system \(xOy\), there is a point \(P(0, \sqrt{3})\) and a line \(l\) with the parametric equations \(\begin{cases} x = \dfrac{1}{2}t \\ y = \sqrt{3} + \dfrac{\sqrt{3}}{2}t \end{cases}\) (where \(t\) is the parameter). Using the origin as the pole and the non-negative half-axis of \(x\) to es... | \sqrt{14} | 0.625 | 6,802.25 | 6,442.5 | 7,401.833333 | |
Use the Horner's method to write out the process of calculating the value of $f(x) = 1 + x + 0.5x^2 + 0.16667x^3 + 0.04167x^4 + 0.00833x^5$ at $x = -0.2$. | 0.81873 | 0.125 | 7,740.375 | 7,843 | 7,725.714286 | |
Jamal wants to save 30 files onto disks, each with 1.44 MB space. 3 of the files take up 0.8 MB, 12 of the files take up 0.7 MB, and the rest take up 0.4 MB. It is not possible to split a file onto 2 different disks. What is the smallest number of disks needed to store all 30 files? | 13 | 1. **Analyze the distribution of file sizes:**
- There are 3 files of 0.8 MB each.
- There are 12 files of 0.7 MB each.
- The remaining files are 15 files of 0.4 MB each (since 3 + 12 = 15 files are accounted for, and there are 30 files in total).
2. **Optimize the storage of 0.8 MB files:**
- Each 0.8 MB ... | 0 | 8,184.0625 | -1 | 8,184.0625 |
Let $p$, $q$, and $r$ be the distinct roots of the polynomial $x^3 - 22x^2 + 80x - 67$. It is given that there exist real numbers $A$, $B$, and $C$ such that \begin{equation*} \frac{1}{s^3 - 22s^2 + 80s - 67} = \frac{A}{s-p} + \frac{B}{s-q} + \frac{C}{s-r} \end{equation*} for all $s\not\in\{p,q,r\}$. What is $\tfrac1A+... | 244 | We start by expressing the given partial fraction decomposition:
\[
\frac{1}{s^3 - 22s^2 + 80s - 67} = \frac{A}{s-p} + \frac{B}{s-q} + \frac{C}{s-r}
\]
for all \( s \not\in \{p, q, r\} \), where \( p, q, r \) are the roots of the polynomial \( x^3 - 22x^2 + 80x - 67 \).
Multiplying through by the denominator on the le... | 0.8125 | 4,768.375 | 4,427.692308 | 6,244.666667 |
A parallelogram is defined by the vectors $\begin{pmatrix} 3 \\ 0 \\ 4 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$. Determine the cosine of the angle $\theta$ between the diagonals of this parallelogram. | \frac{-11}{3\sqrt{69}} | 0 | 5,345.125 | -1 | 5,345.125 | |
Given real numbers \(a\) and \(b\) that satisfy \(0 \leqslant a, b \leqslant 8\) and \(b^2 = 16 + a^2\), find the sum of the maximum and minimum values of \(b - a\). | 12 - 4\sqrt{3} | 0.625 | 5,800.5625 | 5,400 | 6,468.166667 | |
Given the function $f(x) = (2 - a)(x - 1) - 2 \ln x (a \in \mathbb{R})$.
(1) If the tangent line of the curve $g(x) = f(x) + x$ at the point $(1, g(1))$ passes through the point $(0, 2)$, find the monotonically decreasing interval of the function $g(x)$;
(2) If the function $y = f(x)$ has no zeros in the interval $(0... | 2 - 4 \ln 2 | 0.375 | 7,746.375 | 7,192 | 8,079 | |
Two square napkins with dimensions \(1 \times 1\) and \(2 \times 2\) are placed on a table so that the corner of the larger napkin falls into the center of the smaller napkin. What is the maximum area of the table that the napkins can cover? | 4.75 | 0 | 7,854.875 | -1 | 7,854.875 | |
A square sheet contains 1000 points, with any three points, including the vertices of the square, not being collinear. Connect some of these points and the vertices of the square with line segments to divide the entire square into smaller triangles (using the connected line segments and square edges as sides, and ensur... | 2002 | 0.5625 | 5,842.3125 | 4,975.666667 | 6,956.571429 | |
Simplify $5(3-i)+3i(5-i)$. | 18+10i | 0.9375 | 2,889 | 2,535.466667 | 8,192 | |
A burger at Ricky C's now weighs 180 grams, of which 45 grams are filler. What percent of the burger is not filler? Additionally, what percent of the burger is filler? | 25\% | 0.5625 | 861.0625 | 1,136.333333 | 507.142857 | |
Let $p$, $q$, $r$, $s$, and $t$ be distinct integers such that
$(8-p)(8-q)(8-r)(8-s)(8-t) = 120$, find the sum of $p$, $q$, $r$, $s$, and $t$. | 35 | 0 | 7,954.375 | -1 | 7,954.375 | |
Given a tetrahedron \(ABCD\), in what ratio does the plane passing through the intersection points of the medians of the faces \(ABC\), \(ABD\), and \(BCD\) divide the edge \(BD\)? | 1:2 | 0 | 6,338.3125 | -1 | 6,338.3125 | |
Given that the terminal side of angle $\alpha$ passes through the point $P(\sqrt{3}, m)$ ($m \neq 0$), and $\cos\alpha = \frac{m}{6}$, then $\sin\alpha = \_\_\_\_\_\_$. | \frac{\sqrt{3}}{2} | 0 | 6,714.3125 | -1 | 6,714.3125 | |
Let \( a \in \mathbf{R} \). A complex number is given by \(\omega = 1 + a\mathrm{i}\). A complex number \( z \) satisfies \( \overline{\omega} z - \omega = 0 \). Determine the value of \( a \) such that \(|z^2 - z + 2|\) is minimized, and find this minimum value. | \frac{\sqrt{14}}{4} | 0 | 8,192 | -1 | 8,192 | |
Find the number of ordered triples of positive integers $(a, b, c)$ such that $6a+10b+15c=3000$. | 4851 | Note that $6a$ must be a multiple of 5, so $a$ must be a multiple of 5. Similarly, $b$ must be a multiple of 3, and $c$ must be a multiple of 2. Set $a=5A, b=3B, c=2C$. Then the equation reduces to $A+B+C=100$. This has $\binom{99}{2}=4851$ solutions. | 0.375 | 7,183.4375 | 5,502.5 | 8,192 |
For a set of five distinct lines in a plane, there are exactly $M$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $M$? | 37 | 0 | 8,188.5 | -1 | 8,188.5 | |
The altitude to the hypotenuse of a triangle with angles of 30 and 60 degrees is 3 units. What is the area of the triangle, in square units? Express your answer in simplest radical form. [asy]
unitsize(6mm);
defaultpen(linewidth(.7pt)+fontsize(8pt));
real r=2*sqrt(3);
pair A=r*dir(0), B=r*dir(60), C=r*dir(180);
pair ... | 6\sqrt{3} | 0.875 | 3,392.125 | 2,706.428571 | 8,192 | |
In the coordinate plane \(xOy\), given points \(A(1,3)\), \(B\left(8 \frac{1}{3}, 1 \frac{2}{3}\right)\), and \(C\left(7 \frac{1}{3}, 4 \frac{2}{3}\right)\), the extended lines \(OA\) and \(BC\) intersect at point \(D\). Points \(M\) and \(N\) are on segments \(OD\) and \(BD\) respectively, with \(OM = MN = BN\). Find ... | \frac{5 \sqrt{10}}{3} | 0 | 7,257.125 | -1 | 7,257.125 | |
Trapezoid $PQRS$ has vertices $P(-3, 3)$, $Q(3, 3)$, $R(5, -1)$, and $S(-5, -1)$. If a point is selected at random from the region determined by the trapezoid, what is the probability that the point is above the $x$-axis? | \frac{3}{4} | 0 | 7,588.5625 | -1 | 7,588.5625 | |
Two students were asked to add two positive integers. Alice subtracted the two numbers by mistake and obtained 3. Bob mistakenly multiplied the same two integers and got 63. What was the correct sum of the two integers? | 17 | 0 | 8,192 | -1 | 8,192 | |
Let $S$ be a subset of the set $\{1,2,3, \ldots, 2015\}$ such that for any two elements $a, b \in S$, the difference $a-b$ does not divide the sum $a+b$. Find the maximum possible size of $S$. | 672 | From each of the sets $\{1,2,3\},\{4,5,6\},\{7,8,9\}, \ldots$ at most 1 element can be in $S$. This leads to an upper bound of $\left\lceil\frac{2015}{3}\right\rceil=672$ which we can obtain with the set $\{1,4,7, \ldots, 2014\}$. | 0 | 8,192 | -1 | 8,192 |
Solve the quadratic equation $x^{2}-2x+3=4x$. | 3-\sqrt{6} | 0.875 | 1,781.5625 | 1,921.785714 | 800 | |
Each of the equations \( a x^{2} - b x + c = 0 \) and \( c x^{2} - a x + b = 0 \) has two distinct real roots. The sum of the roots of the first equation is non-negative, and the product of the roots of the first equation is 9 times the sum of the roots of the second equation. Find the ratio of the sum of the roots of ... | -3 | 0.5 | 6,739.0625 | 6,463.875 | 7,014.25 | |
Given four distinct real numbers \( a, b, c, d \) such that \(\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} = 4\) and \( ac = bd \), find the maximum value of \(\frac{a}{c} + \frac{b}{d} + \frac{c}{a} + \frac{d}{b} \). | -12 | 0.0625 | 8,192 | 8,192 | 8,192 | |
The four zeros of the polynomial $x^4 + jx^2 + kx + 256$ are distinct real numbers in arithmetic progression. Compute the value of $j$. | -40 | 0 | 4,715.3125 | -1 | 4,715.3125 | |
On a sheet of paper, points \( A, B, C, D \) are marked. A recognition device can perform two types of operations with absolute precision: a) measuring the distance in centimeters between two given points; b) comparing two given numbers. What is the minimum number of operations needed for this device to definitively de... | 10 | 0 | 8,192 | -1 | 8,192 | |
If \(\frac{\left(\frac{a}{c}+\frac{a}{b}+1\right)}{\left(\frac{b}{a}+\frac{b}{c}+1\right)}=11\), where \(a, b\), and \(c\) are positive integers, find the number of different ordered triples \((a, b, c)\) such that \(a+2b+c \leq 40\). | 42 | 0.0625 | 8,120.25 | 7,044 | 8,192 | |
If a regular polygon has a total of nine diagonals, how many sides does it have? | 6 | 1 | 1,867.25 | 1,867.25 | -1 | |
A company organizes its employees into 7 distinct teams for a cycling event. The employee count is between 200 and 300. If one employee takes a day off, the teams are still equally divided among all present employees. Determine the total number of possible employees the company could have. | 3493 | 0 | 7,367.4375 | -1 | 7,367.4375 | |
How many digits does the number \(2^{100}\) have? What are its last three digits? (Give the answers without calculating the power directly or using logarithms!) If necessary, how could the power be quickly calculated? | 376 | 0.875 | 4,986.625 | 4,717.571429 | 6,870 | |
Al gets the disease algebritis and must take one green pill and one pink pill each day for two weeks. A green pill costs $1 more than a pink pill, and Al's pills cost a total of $546 for the two weeks. How much does one green pill cost? | $20 | 1. **Identify the total number of days and daily cost:**
Al needs to take one green pill and one pink pill each day for two weeks. Since there are 14 days in two weeks, we first calculate the total daily cost of the pills:
\[
\text{Daily cost} = \frac{\text{Total cost for two weeks}}{\text{Number of days}} = \... | 0 | 1,459.3125 | -1 | 1,459.3125 |
In this problem assume $s_{1}=3$ and $s_{2}=2$. Determine, with proof, the nonnegative integer $k$ with the following property: 1. For every board configuration with strictly fewer than $k$ blank squares, the first player wins with probability strictly greater than $\frac{1}{2}$; but 2. there exists a board configurati... | \[ k = 3 \] | The answer is $k=\mathbf{3}$. Consider the configuration whose blank squares are 2,6, and 10. Because these numbers represent all congruence classes modulo 3, player 1 cannot win on his first turn: he will come to rest on one of the blank squares. But player 2 will win on her first turn if she rolls a 1, for 2,6, and 1... | 0 | 7,916.4375 | -1 | 7,916.4375 |
Square $IJKL$ is contained within square $WXYZ$ such that each side of $IJKL$ can be extended to pass through a vertex of $WXYZ$. The side length of square $WXYZ$ is $\sqrt{98}$, and $WI = 2$. What is the area of the inner square $IJKL$?
A) $62$
B) $98 - 4\sqrt{94}$
C) $94 - 4\sqrt{94}$
D) $98$
E) $100$ | 98 - 4\sqrt{94} | 0 | 8,192 | -1 | 8,192 | |
Let the sequence $\{a_n\}$ satisfy that the sum of the first $n$ terms $S_n$ fulfills $S_n + a_1 = 2a_n$, and $a_1$, $a_2 + 1$, $a_3$ form an arithmetic sequence. Find the value of $a_1 + a_5$. | 34 | 1 | 3,798.8125 | 3,798.8125 | -1 | |
Find all four-digit numbers that are 9 times larger than their reversed counterparts. | 9801 | 0.1875 | 7,674.0625 | 5,429.666667 | 8,192 | |
Determine the number of ways to select a sequence of 8 sets $A_{1}, A_{2}, \ldots, A_{8}$, such that each is a subset (possibly empty) of \{1,2\}, and $A_{m}$ contains $A_{n}$ if $m$ divides $n$. | 2025 | Consider an arbitrary $x \in\{1,2\}$, and let us consider the number of ways for $x$ to be in some of the sets so that the constraints are satisfied. We divide into a few cases: - Case: $x \notin A_{1}$. Then $x$ cannot be in any of the sets. So there is one possibility. - Case: $x \in A_{1}$ but $x \notin A_{2}$. Then... | 0 | 8,192 | -1 | 8,192 |
Given that points $A$ and $B$ lie on the curves $C_{1}: x^{2}-y+1=0$ and $C_{2}: y^{2}-x+1=0$ respectively, what is the minimum value of the distance $|AB|$? | \frac{3\sqrt{2}}{4} | 0 | 7,486.625 | -1 | 7,486.625 | |
Monic quadratic polynomial $P(x)$ and $Q(x)$ have the property that $P(Q(x))$ has zeros at $x=-23, -21, -17,$ and $-15$, and $Q(P(x))$ has zeros at $x=-59,-57,-51$ and $-49$. What is the sum of the minimum values of $P(x)$ and $Q(x)$? | -100 | 1. **Define the polynomials**: Let $P(x) = x^2 + Bx + C$ and $Q(x) = x^2 + Ex + F$, where both are monic quadratic polynomials.
2. **Expression for $P(Q(x))$**:
\[
P(Q(x)) = (x^2 + Ex + F)^2 + B(x^2 + Ex + F) + C
\]
Expanding and simplifying, we get:
\[
P(Q(x)) = x^4 + 2Ex^3 + (E^2 + 2F + B)x^2 + (2E... | 0.0625 | 7,804.0625 | 4,043 | 8,054.8 |
Consider an arithmetic sequence $\{a\_n\}$ with the sum of its first $n$ terms denoted as $S\_n$. When $k \geqslant 2$, if $S\_{k-1}=8$, $S\_k=0$, and $S\_{k+1}=-10$, what is the maximum value of $S\_n$? | 20 | 1 | 3,759 | 3,759 | -1 | |
Consider the numbers $\{24,27,55,64,x\}$ . Given that the mean of these five numbers is prime and the median is a multiple of $3$ , compute the sum of all possible positive integral values of $x$ . | 60 | 0.375 | 7,178.375 | 6,109 | 7,820 | |
If three different natural numbers $a$, $b$ and $c$ each have exactly four natural-number factors, how many factors does $a^3b^4c^5$ have? | 2080 | 0 | 7,980.8125 | -1 | 7,980.8125 | |
Find the equation of the directrix of the parabola \( y = \frac{x^2 - 8x + 12}{16} \). | y = -\frac{1}{2} | 0 | 3,669.5 | -1 | 3,669.5 | |
Convert $10101_3$ to a base 10 integer. | 91 | 1 | 3,176.75 | 3,176.75 | -1 | |
Let $S_n$ denote the sum of the first $n$ terms of an arithmetic sequence with a common difference of 5. If $\frac{S_{2n}}{S_n}$ is a constant that does not depend on $n$ for all positive integers $n$, find the first term. | 2.5 | 0 | 4,501.75 | -1 | 4,501.75 | |
What is the value of $x$ if the three numbers $2, x$, and 10 have an average of $x$? | 6 | Since the average of $2, x$ and 10 is $x$, then $\frac{2 + x + 10}{3} = x$. Multiplying by 3, we obtain $2 + x + 10 = 3x$. Re-arranging, we obtain $x + 12 = 3x$ and then $2x = 12$ which gives $x = 6$. | 1 | 1,382.875 | 1,382.875 | -1 |
To arrange a class schedule for one day with the subjects Chinese, Mathematics, Politics, English, Physical Education, and Art, where Mathematics must be in the morning and Physical Education in the afternoon, determine the total number of different arrangements. | 192 | 0 | 6,593.1875 | -1 | 6,593.1875 | |
In the diagram, $CE$ and $DE$ are two equal chords of circle $O$. The arc $\widehat{AB}$ is $\frac{1}{4}$ of the circumference. Find the ratio of the area of $\triangle CED$ to the area of $\triangle AOB$. | 2: 1 | 0 | 8,192 | -1 | 8,192 | |
The domain of the function $f(x) = \arcsin(\log_{m}(nx))$ is a closed interval of length $\frac{1}{2013}$ , where $m$ and $n$ are positive integers and $m>1$. Find the remainder when the smallest possible sum $m+n$ is divided by 1000. | 371 | We start with the same method as above. The domain of the arcsin function is $[-1, 1]$, so $-1 \le \log_{m}(nx) \le 1$.
\[\frac{1}{m} \le nx \le m\] \[\frac{1}{mn} \le x \le \frac{m}{n}\] \[\frac{m}{n} - \frac{1}{mn} = \frac{1}{2013}\] \[n = 2013m - \frac{2013}{m}\]
For $n$ to be an integer, $m$ must divide $2013$, an... | 0.8125 | 5,544.6875 | 4,933.769231 | 8,192 |
A circle having radius $r_{1}$ centered at point $N$ is tangent to a circle of radius $r_{2}$ centered at $M$. Let $l$ and $j$ be the two common external tangent lines to the two circles. A circle centered at $P$ with radius $r_{2}$ is externally tangent to circle $N$ at the point at which $l$ coincides with circle $N$... | 3 | Suppose the lines are parallel. Draw the other tangent line to $N$ and $P$ - since $M$ and $P$ have the same radius, it is tangent to all three circles. Let $j$ and $k$ meet circle $N$ at $A$ and $B$, respectively. Then by symmetry we see that $\angle A N M=\angle M N P=\angle P N B=60^{\circ}$ since $A, N$, and $B$ ar... | 0 | 8,192 | -1 | 8,192 |
Given the function $f(x)=\ln x-mx (m\in R)$.
(I) Discuss the monotonic intervals of the function $f(x)$;
(II) When $m\geqslant \frac{ 3 \sqrt {2}}{2}$, let $g(x)=2f(x)+x^{2}$ and its two extreme points $x\_1$, $x\_2 (x\_1 < x\_2)$ are exactly the zeros of $h(x)=\ln x-cx^{2}-bx$. Find the minimum value of $y=(x\_1-x\_2)... | -\frac{2}{3} + \ln 2 | 0 | 8,192 | -1 | 8,192 | |
Let's divide a sequence of natural numbers into groups:
\((1), (2,3), (4,5,6), (7,8,9,10), \ldots\)
Let \( S_{n} \) denote the sum of the \( n \)-th group of numbers. Find \( S_{16} - S_{4} - S_{1} \). | 2021 | 0.8125 | 4,567.75 | 3,731.384615 | 8,192 | |
Let $x_{1}$ be a positive real number and for every integer $n \geq 1$ let $x_{n+1} = 1 + x_{1}x_{2}\ldots x_{n-1}x_{n}$ .
If $x_{5} = 43$ , what is the sum of digits of the largest prime factors of $x_{6}$ ? | 13 | 0.6875 | 5,347.125 | 4,271.545455 | 7,713.4 | |
Every June 1, an ecologist takes a census of the number of wrens in a state park. She noticed that the number is decreasing by $40\%$ each year. If this trend continues, in what year will the census show that the number of wrens is less than $10\%$ of what it was on June 1, 2004? | 2009 | 1 | 2,799.5625 | 2,799.5625 | -1 | |
Julia is learning how to write the letter C. She has 6 differently-colored crayons, and wants to write Cc Cc Cc Cc Cc. In how many ways can she write the ten Cs, in such a way that each upper case C is a different color, each lower case C is a different color, and in each pair the upper case C and lower case C are diff... | 222480 | Suppose Julia writes Cc a sixth time, coloring the upper-case C with the unique color different from that of the first five upper-case Cs, and doing the same with the lower-case C (note: we allow the sixth upper-case C and lower-case c to be the same color). Note that because the colors on the last Cc are forced, and a... | 0.0625 | 7,292.625 | 6,250 | 7,362.133333 |
Find the difference between $1000_7$ and $666_7$ in base $7$. | 1_7 | 0.3125 | 7,216.9375 | 5,224.6 | 8,122.545455 | |
Find the sum of all distinct possible values of $x^2-4x+100$ , where $x$ is an integer between 1 and 100, inclusive.
*Proposed by Robin Park* | 328053 | 0.0625 | 8,057.5 | 7,194 | 8,115.066667 | |
A regular triangular pyramid \(SABC\) is given, with the edge of its base equal to 1. Medians of the lateral faces are drawn from the vertices \(A\) and \(B\) of the base \(ABC\), and these medians do not intersect. It is known that the edges of a certain cube lie on the lines containing these medians. Find the length ... | \frac{\sqrt{6}}{2} | 0 | 8,192 | -1 | 8,192 | |
Calculate the product: $100 \times 29.98 \times 2.998 \times 1000 = $ | 2998^2 | 0 | 513.75 | -1 | 513.75 | |
Given $(1-2x)^7 = a + a_1x + a_2x^2 + \ldots + a_7x^7$, calculate the value of $a_2 + a_3 + a_4 + a_5 + a_6 + a_7$. | 12 | 1 | 3,056.4375 | 3,056.4375 | -1 | |
A and B play a guessing game where A first thinks of a number denoted as $a$, and then B guesses the number A thought of, denoting B's guess as $b$. Both $a$ and $b$ belong to the set $\{0,1,2,…,9\}$. If $|a-b| \leqslant 1$, then A and B are considered to have a "telepathic connection". If two people are randomly chose... | \frac{7}{25} | 0.9375 | 4,767.3125 | 4,539 | 8,192 |
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