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 |
|---|---|---|---|---|---|---|
Alice is bored in class, so she thinks of a positive integer. Every second after that, she subtracts from her current number its smallest prime divisor, possibly itself. After 2022 seconds, she realizes that her number is prime. Find the sum of all possible values of her initial number. | 8093 | Let $a_{k}$ denote Alice's number after $k$ seconds, and let $p_{k}$ be the smallest prime divisor of $a_{k}$. We are given that $a_{2022}$ is prime, and want to find $a_{0}$. If $a_{0}$ is even, then $a_{n+1}=a_{n}-2$, since every $a_{n}$ is even. Then we need $a_{2022}=2$, so $a_{0}=4046$. If $a_{0}$ is odd, then $a_... | 0.0625 | 7,815.8125 | 5,572 | 7,965.4 |
In the arithmetic sequence $\{a_n\}$, the common difference is $\frac{1}{2}$, and $a_1+a_3+a_5+\ldots+a_{99}=60$. Find the value of $a_2+a_4+a_6+\ldots+a_{100}$. | 85 | 0.4375 | 6,361.1875 | 4,007.285714 | 8,192 | |
Given $$(1+x)(2-x)^{6}=a_{0}+a_{1}(x-1)+a_{2}(x-1)^{2}+\ldots+a_{7}(x-1)^{7}$$, find the value of $a_{3}$. | -25 | 0.375 | 7,324.6875 | 5,879.166667 | 8,192 | |
A plane contains $40$ lines, no $2$ of which are parallel. Suppose that there are $3$ points where exactly $3$ lines intersect, $4$ points where exactly $4$ lines intersect, $5$ points where exactly $5$ lines intersect, $6$ points where exactly $6$ lines intersect, and no points where more than $6$ lines intersect. Fin... | 607 | In this solution, let $\boldsymbol{n}$-line points be the points where exactly $n$ lines intersect. We wish to find the number of $2$-line points.
There are $\binom{40}{2}=780$ pairs of lines. Among them:
The $3$-line points account for $3\cdot\binom32=9$ pairs of lines.
The $4$-line points account for $4\cdot\binom42... | 0.5625 | 5,856.25 | 4,039.555556 | 8,192 |
Given the hyperbola $C: \frac{x^{2}}{4} - \frac{y^{2}}{3} = 1$, with its right vertex at $P$.
(1) Find the standard equation of the circle centered at $P$ and tangent to both asymptotes of the hyperbola $C$;
(2) Let line $l$ pass through point $P$ with normal vector $\overrightarrow{n}=(1,-1)$. If there are exactly thr... | \frac{3\sqrt{2}}{2} | 0 | 8,192 | -1 | 8,192 | |
Find, with proof, the smallest real number $C$ with the following property:
For every infinite sequence $\{x_i\}$ of positive real numbers such that $x_1 + x_2 +\cdots + x_n \leq x_{n+1}$ for $n = 1, 2, 3, \cdots$, we have
\[\sqrt{x_1}+\sqrt{x_2}+\cdots+\sqrt{x_n} \leq C \sqrt{x_1+x_2+\cdots+x_n} \qquad \forall n \in ... | $C=1+\sqrt{2}$ |
To find the smallest real number \( C \) such that the inequality
\[
\sqrt{x_1} + \sqrt{x_2} + \cdots + \sqrt{x_n} \leq C \sqrt{x_1 + x_2 + \cdots + x_n}
\]
holds for every infinite sequence \(\{x_i\}\) of positive real numbers satisfying
\[
x_1 + x_2 + \cdots + x_n \leq x_{n+1}
\]
for all \( n \in \mathbb{N} \... | 0 | 8,152.75 | -1 | 8,152.75 |
A general gathers his troops. When he arranges them in groups of 2, one soldier is left over. When he arranges them in groups of 3, two soldiers are left over. When he arranges them in groups of 5, three soldiers are left over. If the general arranges his soldiers in groups of 30, how many soldiers will be left over? | 23 | 1 | 2,537.6875 | 2,537.6875 | -1 | |
Given two arithmetic sequences $\{a_n\}$ and $\{b_n\}$, their sums of the first $n$ terms are denoted as $S_n$ and $T_n$, respectively. If $\frac {S_{n}}{T_{n}}= \frac {n}{n+7}$, then calculate $\frac {a_{7}}{b_{7}}$. | \frac{13}{20} | 0.9375 | 4,965.375 | 4,750.266667 | 8,192 | |
In a school fundraising campaign, $25\%$ of the money donated came from parents. The rest of the money was donated by teachers and students. The ratio of the amount of money donated by teachers to the amount donated by students was $2:3$. What is the ratio of the amount of money donated by parents to the amount donated... | 5:9 | Since $25\%$ of the money donated came from parents, then the remaining $100\%-25\%=75\%$ came from the teachers and students. Since the ratio of the amount donated by teachers to the amount donated by students is $2:3$, then the students donated $\frac{3}{2+3}=\frac{3}{5}$ of this remaining $75\%$. This means that the... | 0.625 | 4,575.875 | 3,658.7 | 6,104.5 |
On the coordinate plane (\( x; y \)), a circle with radius 4 and center at the origin is drawn. A line given by the equation \( y = 4 - (2 - \sqrt{3}) x \) intersects the circle at points \( A \) and \( B \). Find the sum of the length of segment \( A B \) and the length of the shorter arc \( A B \). | 4\sqrt{2 - \sqrt{3}} + \frac{2\pi}{3} | 0 | 5,971.3125 | -1 | 5,971.3125 | |
Suppose that $f(x)$ and $g(x)$ are functions which satisfy the equations $f(g(x)) = 2x^2$ and $g(f(x)) = x^4$ for all $x \ge 1$. If $g(4) = 16$, compute $[g(2)]^4$. | 16 | 0 | 7,809.75 | -1 | 7,809.75 | |
Find the distance from point \( M_{0} \) to the plane passing through three points \( M_{1}, M_{2}, M_{3} \).
\( M_{1}(0, -1, -1) \)
\( M_{2}(-2, 3, 5) \)
\( M_{3}(1, -5, -9) \)
\( M_{0}(-4, -13, 6) \) | 2 \sqrt{45} | 0 | 4,838.5625 | -1 | 4,838.5625 | |
The intersection of two squares with perimeter $8$ is a rectangle with diagonal length $1$ . Given that the distance between the centers of the two squares is $2$ , the perimeter of the rectangle can be expressed as $P$ . Find $10P$ . | 25 | 0.125 | 7,878.375 | 5,683 | 8,192 | |
Find the smallest natural number whose digits sum up to 47. | 299999 | 0.1875 | 7,687.5625 | 5,501.666667 | 8,192 | |
Suppose that $n$ persons meet in a meeting, and that each of the persons is acquainted to exactly $8$ others. Any two acquainted persons have exactly $4$ common acquaintances, and any two non-acquainted persons have exactly $2$ common acquaintances. Find all possible values of $n$ . | 21 | 0.375 | 7,213.625 | 5,583 | 8,192 | |
Which one satisfies $n^{29} \equiv 7 \pmod {65}$? | 37 | 0.875 | 4,469.125 | 3,937.285714 | 8,192 | |
In triangle $\triangle ABC$, $A=60^{\circ}$, $a=\sqrt{6}$, $b=2$.
$(1)$ Find $\angle B$;
$(2)$ Find the area of $\triangle ABC$. | \frac{3 + \sqrt{3}}{2} | 0 | 5,793.125 | -1 | 5,793.125 | |
Ana and Banana are rolling a standard six-sided die. Ana rolls the die twice, obtaining $a_{1}$ and $a_{2}$, then Banana rolls the die twice, obtaining $b_{1}$ and $b_{2}$. After Ana's two rolls but before Banana's two rolls, they compute the probability $p$ that $a_{1} b_{1}+a_{2} b_{2}$ will be a multiple of 6. What ... | \frac{2}{3} | If either $a_{1}$ or $a_{2}$ is relatively prime to 6, then $p=\frac{1}{6}$. If one of them is a multiple of 2 but not 6, while the other is a multiple of 3 but not 6, we also have $p=\frac{1}{6}$. In other words, $p=\frac{1}{6}$ if $\operatorname{gcd}(a_{1}, a_{2})$ is coprime to 6, and otherwise $p \neq \frac{1}{6}$.... | 0 | 8,192 | -1 | 8,192 |
Given that $A, B, C, D, E, F$ are the vertices of a regular hexagon with a side length of 2, and a parabola passes through the points $A, B, C, D$, find the distance from the focus of the parabola to its directrix. | \frac{\sqrt{3}}{2} | 0 | 6,305.1875 | -1 | 6,305.1875 | |
Trapezoid $EFGH$ has sides $EF=105$, $FG=45$, $GH=21$, and $HE=80$, with $EF$ parallel to $GH$. A circle with center $Q$ on $EF$ is drawn tangent to $FG$ and $HE$. Find the exact length of $EQ$ using fractions. | \frac{336}{5} | 0.125 | 7,999.9375 | 6,655.5 | 8,192 | |
A right trapezoid has an upper base that is 60% of the lower base. If the upper base is increased by 24 meters, it becomes a square. What was the original area of the right trapezoid in square meters? | 2880 | 0.3125 | 4,907.5 | 3,588.4 | 5,507.090909 | |
The probability that three friends, Al, Bob, and Carol, will be assigned to the same lunch group is approximately what fraction. | \frac{1}{9} | 0.125 | 5,914.1875 | 6,005.5 | 5,901.142857 | |
Given the function $f(x)=x^{2}-6x+4\ln x$, find the x-coordinate of the quasi-symmetric point of the function. | \sqrt{2} | 0 | 7,999.0625 | -1 | 7,999.0625 | |
Jo is thinking of a positive integer less than 100. It is one less than a multiple of 8, and it is three less than a multiple of 7. What is the greatest possible integer Jo could be thinking of? | 95 | 1 | 2,703.5625 | 2,703.5625 | -1 | |
Let $ a_1, a_2,\ldots ,a_8$ be $8$ distinct points on the circumference of a circle such that no three chords, each joining a pair of the points, are concurrent. Every $4$ of the $8$ points form a quadrilateral which is called a *quad*. If two chords, each joining a pair of the $8$ points, intersect, the poi... | 14 | 0.25 | 7,833.9375 | 6,759.75 | 8,192 | |
Let $[x]$ denote the greatest integer less than or equal to $x$. When $0 \leqslant x \leqslant 10$, find the number of distinct integers represented by the function $f(x) = [x] + [2x] + [3x] + [4x]$. | 61 | 0 | 8,192 | -1 | 8,192 | |
What is $\frac{1}{(-5^2)^3} \cdot (-5)^8 \cdot \sqrt{5}$? | 5^{5/2} | 0 | 4,992.6875 | -1 | 4,992.6875 | |
On graph paper, large and small triangles are drawn (all cells are square and of the same size). How many small triangles can be cut out from the large triangle? Triangles cannot be rotated or flipped (the large triangle has a right angle in the bottom left corner, the small triangle has a right angle in the top right ... | 12 | 0 | 7,296.9375 | -1 | 7,296.9375 | |
Find the smallest natural number $m$ for which there exists a natural number $n$ such that the sets of the last 2014 digits in the decimal representation of the numbers $a=2015^{3 m+1}$ and $b=2015^{6 n+2}$ are identical, with the condition that $a<b$. | 671 | 0 | 8,192 | -1 | 8,192 | |
How many perfect squares divide $2^{3} \cdot 3^{5} \cdot 5^{7} \cdot 7^{9}$? | 120 | The number of such perfect squares is $2 \cdot 3 \cdot 4 \cdot 5$, since the exponent of each prime can be any nonnegative even number less than the given exponent. | 0.9375 | 2,405.125 | 2,019.333333 | 8,192 |
Let $S$ be the set of positive real numbers. Let $f : S \to \mathbb{R}$ be a function such that
\[f(x) f(y) = f(xy) + 2005 \left( \frac{1}{x} + \frac{1}{y} + 2004 \right)\]for all $x,$ $y > 0.$
Let $n$ be the number of possible values of $f(2),$ and let $s$ be the sum of all possible values of $f(2).$ Find $n \times... | \frac{4011}{2} | 0.1875 | 8,096.875 | 8,192 | 8,074.923077 | |
There is a rectangular field that measures $20\text{m}$ by $15\text{m}$ . Xiaoyu the butterfly is sitting at the perimeter of the field on one of the $20\text{m}$ sides such that he is $6\text{m}$ from a corner. He flies in a straight line to another point on the perimeter. His flying path splits the field into ... | 17 | 0.3125 | 7,695.875 | 6,604.4 | 8,192 | |
The polar equation of circle C is $\rho = 2\cos(θ + \frac{π}{4})$. The parametric equation of line l is
$$
\begin{cases}
x= \sqrt{2}t \\
y= \sqrt{2}t+4\sqrt{2}
\end{cases}
$$
(where t is the parameter). A tangent is drawn from a point P on line l to circle C at the point A, find the minimum value of the length of ... | 2\sqrt{6} | 0.4375 | 5,379.4375 | 4,303 | 6,216.666667 | |
In the rectangular coordinate system $xOy$, the parametric equation of line $l$ is $\begin{cases} x=t \\ y=t+1 \end{cases}$ (where $t$ is the parameter), and the parametric equation of curve $C$ is $\begin{cases} x=2+2\cos \phi \\ y=2\sin \phi \end{cases}$ (where $\phi$ is the parameter). Establish a polar coordinate s... | 2\sqrt{2} | 1 | 4,243.875 | 4,243.875 | -1 | |
Given that $\binom{17}{7}=19448$, $\binom{17}{8}=24310$ and $\binom{17}{9}=24310$, calculate $\binom{19}{9}$. | 92378 | 0.8125 | 5,652.625 | 5,066.615385 | 8,192 | |
Mickey's age is 4 years less than $300\%$ of Jerry's age. If Mickey is 14 years old, how old is Jerry? | 6 | 1 | 1,371.6875 | 1,371.6875 | -1 | |
For which smallest \( n \) do there exist \( n \) numbers in the interval \( (-1, 1) \) such that their sum is 0, and the sum of their squares is 36? | 38 | 0.0625 | 7,760.0625 | 8,192 | 7,731.266667 | |
If \(\sin A = \frac{p}{5}\) and \(\frac{\cos A}{\tan A} = \frac{q}{15}\), find \(q\). | 16 | 0 | 7,666.375 | -1 | 7,666.375 | |
$(1)\sqrt{5}-27+|2-\sqrt{5}|-\sqrt{9}+(\frac{1}{2})^{2}$；<br/>$(2)2\sqrt{40}-5\sqrt{\frac{1}{10}}-\sqrt{10}$；<br/>$(3)(3\sqrt{12}-2\sqrt{\frac{1}{3}}-\sqrt{48})÷4\sqrt{3}-{(\sqrt{2}-1)^0}$；<br/>$(4)(-\sqrt{5}+\sqrt{3})(\sqrt{5}+\sqrt{3})+(-\sqrt{3}-1)^{2}$. | 2+2\sqrt{3} | 0.875 | 4,228.8125 | 4,328.785714 | 3,529 | |
From the numbers \(1,2, \cdots, 14\), select \(a_{1}, a_{2}, a_{3}\) in ascending order such that \(a_{2} - a_{1} \geq 3\) and \(a_{3} - a_{2} \geq 3\). How many different ways are there to select the numbers satisfying these conditions? | 120 | 0.25 | 7,704.3125 | 6,241.25 | 8,192 | |
In the adjoining figure, two circles with radii $8$ and $6$ are drawn with their centers $12$ units apart. At $P$, one of the points of intersection, a line is drawn in such a way that the chords $QP$ and $PR$ have equal length. Find the square of the length of $QP$.
[asy]size(160); defaultpen(linewidth(.8pt)+fontsize(... | 130 | 0.375 | 7,720.375 | 6,934.333333 | 8,192 | |
Let $S$ be the set of all polynomials of the form $z^3 + az^2 + bz + c$, where $a$, $b$, and $c$ are integers. Find the number of polynomials in $S$ such that each of its roots $z$ satisfies either $|z| = 20$ or $|z| = 13$. | 540 | There are two cases: either all the roots are real, or one is real and two are imaginary.
Case 1: All roots are real. Then each of the roots is a member of the set $\{-20, 20, -13, 13\}$. It splits into three sub-cases: either no two are the same, exactly two are the same, or all three are the same.
Sub-case 1.1: No tw... | 0 | 8,192 | -1 | 8,192 |
Given vectors $\overrightarrow{a}=(\cos 25^{\circ},\sin 25^{\circ})$ and $\overrightarrow{b}=(\sin 20^{\circ},\cos 20^{\circ})$, let $t$ be a real number and $\overrightarrow{u}=\overrightarrow{a}+t\overrightarrow{b}$. Determine the minimum value of $|\overrightarrow{u}|$. | \frac{\sqrt{2}}{2} | 0 | 4,568.3125 | -1 | 4,568.3125 | |
A pedestrian and a cyclist set off from point $A$ to point $B$ simultaneously. At point $B$, the cyclist turns back and meets the pedestrian 20 minutes after starting. Without stopping, the cyclist continues to point $A$, turns around, and catches up with the pedestrian 10 minutes after their first meeting. How long wi... | 60 | 0 | 8,192 | -1 | 8,192 | |
Given that $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=2$, and $(\overrightarrow{a}+\overrightarrow{b})\perp\overrightarrow{a}$, determine the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{2\pi}{3} | 0.0625 | 2,584.5625 | 1,660 | 2,646.2 | |
A natural number \( A \) is written twice consecutively (for example, writing 12 as 1212) to obtain a new number \( B \). If \( B \) is a multiple of 2016, then the smallest value of \( A \) is ______. | 288 | 0.3125 | 7,763.375 | 6,820.4 | 8,192 | |
The function f is defined on the set of integers and satisfies $f(n)=\begin{cases} n-3&\mbox{if}\ n\ge 1000\\ f(f(n+5))&\mbox{if}\ n<1000\end{cases}$
Find $f(84)$. | 997 | Assume that $f(84)$ is to be performed $n+1$ times. Then we have \[f(84)=f^{n+1}(84)=f(f^n(84+5))\] In order to find $f(84)$, we want to know the smallest value of \[f^n(84+5)\ge1000\] Because then \[f(84)=f(f^n(84+5))=(f^n(84+5))-3\] From which we'll get a numerical value for $f(84)$.
Notice that the value of $n$ we ... | 0 | 8,102.9375 | -1 | 8,102.9375 |
From the $8$ vertices of a cube, select $4$ vertices. The probability that these $4$ vertices lie in the same plane is ______. | \frac{6}{35} | 0.5 | 7,323.0625 | 6,454.125 | 8,192 | |
How many positive factors of 36 are also multiples of 4? | 3 | To solve this problem, we first need to find the prime factorization of 36. Then, we will determine which factors of 36 are also multiples of 4.
1. **Prime Factorization of 36**:
\[
36 = 2^2 \cdot 3^2
\]
This factorization shows that 36 is composed of the square of 2 and the square of 3.
2. **Factors of 3... | 1 | 2,459.25 | 2,459.25 | -1 |
From milk with a fat content of $5\%$, cottage cheese with a fat content of $15.5\%$ is produced, while there remains whey with a fat content of $0.5\%$. How much cottage cheese is obtained from 1 ton of milk? | 0.3 | 0.1875 | 2,719.3125 | 1,323.666667 | 3,041.384615 | |
The slope angle of the line passing through points M(-3, 2) and N(-2, 3) is equal to what angle, measured in radians. | \frac{\pi}{4} | 0.9375 | 1,744.9375 | 1,315.133333 | 8,192 | |
Calculate \(\frac{2}{3} \cdot \frac{4}{7} \cdot \frac{5}{9} \cdot \frac{11}{13}\). | \frac{440}{2457} | 0.875 | 4,213.75 | 3,645.428571 | 8,192 | |
In triangle $ABC,$ it is given that angles $B$ and $C$ are congruent. Points $P$ and $Q$ lie on $\overline{AC}$ and $\overline{AB},$ respectively, so that $AP = PQ = QB = BC.$ Angle $ACB$ is $r$ times as large as angle $APQ,$ where $r$ is a positive real number. Find $\lfloor 1000r \rfloor$. | 571 | Let $\angle BAC= 2\theta$ and $AP=PQ=QB=BC=x$. $\triangle APQ$ is isosceles, so $AQ=2x\cos 2\theta =2x(1-2\sin^2\theta)$ and $AB= AQ+x=x\left(3-4\sin^2\theta\right)$. $\triangle{ABC}$ is isosceles too, so $x=BC=2AB\sin\theta$. Using the expression for $AB$, we get \[1=2\left(3\sin\theta-4\sin^3\theta\right)=2\sin3\thet... | 0 | 8,192 | -1 | 8,192 |
The graphs of $y = -|x-a| + b$ and $y = |x-c| + d$ intersect at points $(2,5)$ and $(8,3)$. Find $a+c$. | 10 | 1. **Identify the vertex of each graph:**
- The graph of $y = -|x-a| + b$ has a vertex at $(a, b)$, where the graph reaches its maximum value $b$.
- The graph of $y = |x-c| + d$ has a vertex at $(c, d)$, where the graph reaches its minimum value $d$.
2. **Understand the intersection points:**
- The given inte... | 1 | 2,974 | 2,974 | -1 |
A shirt regularly priced at $\$30$ is on sale for $20\%$ off. Mary has a coupon that will take another $25\%$ off of the sale price. What single percent discount would give the same final price as the two successive discounts? | 40\% | 1 | 1,788.5 | 1,788.5 | -1 | |
Given a parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$, the point $X$ is chosen on the edge $A_{1} D_{1}$ and the point $Y$ is chosen on the edge $B C$. It is known that $A_{1} X=5$, $B Y=3$, and $B_{1} C_{1}=14$. The plane $C_{1} X Y$ intersects the ray $D A$ at point $Z$. Find $D Z$. | 20 | 0.25 | 7,535.3125 | 5,977 | 8,054.75 | |
In triangle $ABC$, $AB=10$, $BC=12$ and $CA=14$. Point $G$ is on $\overline{AB}$, $H$ is on $\overline{BC}$, and $I$ is on $\overline{CA}$. Let $AG=s\cdot AB$, $BH=t\cdot BC$, and $CI=u\cdot CA$, where $s$, $t$, and $u$ are positive and satisfy $s+t+u=3/4$ and $s^2+t^2+u^2=3/7$. The ratio of the area of triangle $GHI$ ... | 295 | 0.375 | 7,737.625 | 6,980.333333 | 8,192 | |
1. \(\lim _{x \rightarrow 1}(1-x) \operatorname{tg} \frac{\pi x}{2}\)
2. \(\lim _{x \rightarrow \frac{\pi}{4}}\left(\frac{\pi}{4}-x\right) \operatorname{cosec}\left(\frac{3}{4} \pi+x\right)\)
3. \(\lim _{x \rightarrow+\infty} x \operatorname{arcctg} x\)
4. \(\lim _{x \rightarrow-\infty} x\left(\frac{\pi}{2}+\operatorna... | -1 | 0.8125 | 5,969.0625 | 5,456.076923 | 8,192 | |
A circle with radius 6 cm is tangent to three sides of a rectangle. The area of the rectangle is three times the area of the circle. Determine the length of the longer side of the rectangle, expressed in centimeters and in terms of $\pi$. | 9\pi | 0.4375 | 6,643.5 | 4,652.571429 | 8,192 | |
There are 4 spheres in space with radii 2, 2, 3, and 3, respectively. Each sphere is externally tangent to the other 3 spheres. Additionally, there is a small sphere that is externally tangent to all 4 of these spheres. Find the radius of the small sphere. | 6/11 | 0.0625 | 8,192 | 8,192 | 8,192 | |
Circle $C$ with radius 6 has diameter $\overline{AB}$. Circle $D$ is internally tangent to circle $C$ at $A$. Circle $E$ is internally tangent to circle $C$, externally tangent to circle $D$, and tangent to $\overline{AB}$. The radius of circle $D$ is twice the radius of circle $E$, and can be written in the form $\sqr... | 186 | 0 | 7,981.9375 | -1 | 7,981.9375 | |
Let
\[f(x) = \sqrt{x(50 - x)} + \sqrt{x(2 - x)}\]for $0 \le x \le 2.$ Let $M$ be the maximum value of $f(x),$ and let $x = x_0$ be the point where the maximum value is attained. Enter the ordered pair $(x_0,M).$ | \left( \frac{25}{13}, 10 \right) | 0.625 | 6,177.1875 | 5,367.2 | 7,527.166667 | |
Given the function f(x) = |lnx|, and real numbers m and n that satisfy 0 < m < n and f(m) = f(n). If the maximum value of f(x) in the interval [m^2, n] is 2, find the value of $\frac{n}{m}$. | e^2 | 0.75 | 3,861.5625 | 2,981.083333 | 6,503 | |
Find the least positive four-digit solution to the following system of congruences. \begin{align*}
7x &\equiv 21 \pmod{14} \\
2x+13 &\equiv 16 \pmod{9} \\
-2x+1 &\equiv x \pmod{25} \\
\end{align*} | 1167 | 0.625 | 5,009.875 | 3,525.9 | 7,483.166667 | |
For how many integers $n$ between 1 and 20 (inclusive) is $\frac{n}{42}$ a repeating decimal? | 20 | 0.625 | 5,330.8125 | 4,591.9 | 6,562.333333 | |
The sum of two angles of a triangle is $\frac{6}{5}$ of a right angle, and one of these two angles is $30^{\circ}$ larger than the other. What is the degree measure of the largest angle in the triangle? | 72 | 1. **Identify the given information and the problem statement**: We are given that the sum of two angles of a triangle is $\frac{6}{5}$ of a right angle, and one of these angles is $30^\circ$ larger than the other. We need to find the degree measure of the largest angle in the triangle.
2. **Convert the fraction of th... | 1 | 2,422.75 | 2,422.75 | -1 |
My friend Julia likes numbers that are divisible by 4. How many different last two digits are possible in numbers that Julia likes? | 25 | 0.4375 | 4,235.5 | 2,503.571429 | 5,582.555556 | |
Given an ellipse $C$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ passing through the point $(0,4)$, with an eccentricity of $\frac{3}{5}$
1. Find the equation of $C$;
2. Find the length of the line segment intercepted by $C$ on the line passing through point $(3,0)$ with a slope of $\frac{4}{5}$. | \frac{41}{5} | 1 | 4,274 | 4,274 | -1 | |
Calculate $\sqrt{10p} \cdot \sqrt{5p^2} \cdot \sqrt{6p^4}$ . Express your answer in simplest radical form in terms of $p$.
Note: When entering a square root with more than one character, you must use parentheses or brackets. For example, you should enter $\sqrt{14}$ as "sqrt(14)" or "sqrt{14}". | 10p^3 \sqrt{3p} | 1 | 723.25 | 723.25 | -1 | |
Define a natural number to be "super-nice" if it is either the product of exactly three distinct prime numbers or the fourth power of a prime number. What is the sum of the first eight super-nice numbers? | 520 | 0 | 7,133.6875 | -1 | 7,133.6875 | |
Let $f(x)=2\sqrt{x} + \frac{12}{\sqrt{x}}$ and let $g(x)=2x^2-2x-3$. What is $f(g(3))$? | 10 | 1 | 2,233.8125 | 2,233.8125 | -1 | |
On the sides \( AB \) and \( AD \) of the square \( ABCD \), points \( E \) and \( F \) are marked such that \( BE : EA = AF : FD = 2022 : 2023 \). The segments \( EC \) and \( FC \) intersect the diagonal of the square \( BD \) at points \( G \) and \( H \) respectively. Find \( \frac{GH}{BD} \). | \frac{12271519}{36814556} | 0 | 8,192 | -1 | 8,192 | |
The minimum distance from any integer-coordinate point on the plane to the line \( y = \frac{5}{3} x + \frac{4}{5} \) is to be determined. | \frac{\sqrt{34}}{85} | 0 | 7,007.1875 | -1 | 7,007.1875 | |
How many $3$-digit positive integers have digits whose product equals $36$? | 21 | 0.3125 | 7,468.5625 | 6,525.2 | 7,897.363636 | |
Let $S$ be the set of all positive integer divisors of $100,000.$ How many numbers are the product of two distinct elements of $S?$ | 117 | 1. **Identify the prime factorization of 100,000**:
The prime factorization of $100,000$ is $100,000 = 10^5 = (2 \cdot 5)^5 = 2^5 \cdot 5^5$.
2. **Determine the set $S$ of divisors**:
The set $S$ consists of all divisors of $100,000$, which can be expressed in the form $2^a5^b$ where $0 \leq a, b \leq 5$.
3. *... | 0 | 8,192 | -1 | 8,192 |
Jim wrote a sequence of symbols a total of 50 times. How many more of one symbol than another did he write? | 150 | The sequence of symbols includes 5 of one symbol and 2 of another. This means that, each time the sequence is written, there are 3 more of one symbol written than the other. When the sequence is written 50 times, in total there are \( 50 \times 3 = 150 \) more of one symbol written than the other. | 0 | 370.1875 | -1 | 370.1875 |
How many three-digit numbers satisfy the property that the middle digit is equal to the sum of the first and last digits minus one? | 54 | 0.5625 | 6,877.4375 | 5,855 | 8,192 | |
If $a$ and $b$ are positive integers, the operation $
abla$ is defined by $a
abla b=a^{b} imes b^{a}$. What is the value of $2
abla 3$? | 72 | Since $a
abla b=a^{b} imes b^{a}$, then $2
abla 3=2^{3} imes 3^{2}=8 imes 9=72$. | 1 | 305.4375 | 305.4375 | -1 |
Given the function f(x) = 3^x with a range of M, where x < -1, find the probability that a number x randomly chosen from the interval (-1, 1) belongs to M. | \frac{1}{6} | 0.625 | 3,055.875 | 2,769.8 | 3,532.666667 | |
Eight semicircles line the inside of a square with side length 2 as shown. What is the radius of the circle tangent to all of these semicircles? | \frac{\sqrt{5}-1}{2} | 1. **Understanding the Configuration**:
- The square has a side length of 2.
- Each semicircle inside the square has a diameter of 1 (since 4 semicircles fit along one side of the square, and the side length is 2).
- Therefore, the radius of each semicircle is $\frac{1}{2}$.
2. **Positioning the Inner Circle... | 0 | 7,600.125 | -1 | 7,600.125 |
Given that the four real roots of the quartic polynomial $f(x)$ form an arithmetic sequence with a common difference of $2$, calculate the difference between the maximum root and the minimum root of $f'(x)$. | 2\sqrt{5} | 0.625 | 6,006.0625 | 4,694.5 | 8,192 | |
(Full score: 8 points)
During the 2010 Shanghai World Expo, there were as many as 11 types of admission tickets. Among them, the price for a "specified day regular ticket" was 200 yuan per ticket, and the price for a "specified day concession ticket" was 120 yuan per ticket. A ticket sales point sold a total of 1200 ... | 300 | 1 | 1,906.75 | 1,906.75 | -1 | |
\begin{align*}
2a + 3b + 5c + 7d &= 34 \\
3(d+c) &= b \\
3b + c &= a \\
c - 1 &= d \\
\end{align*} Given the above system of equations, find $a \cdot b \cdot c \cdot d$. | 0 | 1 | 2,313.25 | 2,313.25 | -1 | |
Seven sticks with lengths 2, 3, 5, 7, 11, 13 and 17 inches are placed in a box. Three of the sticks are randomly selected. What is the probability that a triangle can be formed by joining the endpoints of the sticks? Express your answer as a common fraction. | \frac{9}{35} | 0.1875 | 7,647.4375 | 6,511.333333 | 7,909.615385 | |
Two boards, one four inches wide and the other six inches wide, are nailed together to form an X. The angle at which they cross is 60 degrees. If this structure is painted and the boards are separated what is the area of the unpainted region on the four-inch board? (The holes caused by the nails are negligible.) Expres... | 16\sqrt{3} | 0.0625 | 6,552.4375 | 2,647 | 6,812.8 | |
Given the function $f(x)= \begin{cases}x-1,0 < x\leqslant 2 \\ -1,-2\leqslant x\leqslant 0 \end{cases}$, and $g(x)=f(x)+ax$, where $x\in[-2,2]$, if $g(x)$ is an even function, find the value of the real number $a$. | -\dfrac{1}{2} | 1 | 3,214.6875 | 3,214.6875 | -1 | |
Given the function $f(x) = x^3 - 3x$,
(Ⅰ) Find the intervals of monotonicity for $f(x)$;
(Ⅱ) Find the maximum and minimum values of $f(x)$ in the interval $[-3,2]$. | -18 | 1 | 2,523.75 | 2,523.75 | -1 | |
Find the distance between the points (0,15) and (8,0). | 17 | 1 | 1,400.625 | 1,400.625 | -1 | |
Let $a,$ $b,$ $c$ be the roots of $x^3 + px + q = 0.$ Compute the determinant
\[\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}.\] | 0 | 0.9375 | 4,510.1875 | 4,264.733333 | 8,192 | |
A dress originally priced at $80$ dollars was put on sale for $25\%$ off. If $10\%$ tax was added to the sale price, then the total selling price (in dollars) of the dress was | 54 | 1. **Calculate the Sale Price After Discount:**
The original price of the dress is $80$ dollars. The discount offered is $25\%$. The amount of discount in dollars can be calculated as:
\[
\text{Discount} = 25\% \times 80 = 0.25 \times 80 = 20 \text{ dollars}
\]
Therefore, the sale price after applying th... | 0 | 2,321.6875 | -1 | 2,321.6875 |
Let $\triangle A_0B_0C_0$ be a triangle whose angle measures are exactly $59.999^\circ$, $60^\circ$, and $60.001^\circ$. For each positive integer $n$, define $A_n$ to be the foot of the altitude from $A_{n-1}$ to line $B_{n-1}C_{n-1}$. Likewise, define $B_n$ to be the foot of the altitude from $B_{n-1}$ to line $A_{n-... | 15 | 1. **Define the angles and setup the problem:** Let $\triangle A_0B_0C_0$ be a triangle with angles $\angle C_0A_0B_0 = x_0 = 59.999^\circ$, $\angle A_0B_0C_0 = y_0 = 60^\circ$, and $\angle B_0C_0A_0 = z_0 = 60.001^\circ$. For each positive integer $n$, define $A_n$, $B_n$, and $C_n$ as the feet of the altitudes from t... | 0.0625 | 7,767.4375 | 4,994 | 7,952.333333 |
Mary and James each sit in a row of 7 chairs. They choose their seats at random. What is the probability that they don't sit next to each other? | \frac{5}{7} | 0.8125 | 4,422.1875 | 3,552.230769 | 8,192 | |
How many positive integers, including $1,$ are divisors of both $40$ and $72?$ | \mbox{four} | 0 | 1,353.625 | -1 | 1,353.625 | |
A circle with radius $6$ is externally tangent to a circle with radius $24$. Find the area of the triangular region bounded by the three common tangent lines of these two circles. | 192 | [asy] //Created by isabelchen size(12cm, 12cm); draw(circle((0,0),24)); draw(circle((30,0),6)); draw((72/5, 96/5) -- (40,0)); draw((72/5, -96/5) -- (40,0)); draw((24, 12) -- (24, -12)); draw((0, 0) -- (40, 0)); draw((72/5, 96/5) -- (0,0)); draw((168/5, 24/5) -- (30,0)); draw((54/5, 72/5) -- (30,0)); dot((72/5, 96/5)); ... | 0.375 | 6,614.0625 | 5,941.333333 | 7,017.7 |
Given the parabola \( y^{2} = 2 p x \) with focus \( F \) and directrix \( l \), a line passing through \( F \) intersects the parabola at points \( A \) and \( B \) such that \( |AB| = 3p \). Let \( A' \) and \( B' \) be the projections of \( A \) and \( B \) onto \( l \), respectively. If a point \( M \) is randomly ... | 1/3 | 0 | 8,192 | -1 | 8,192 | |
Suppose $ n$ is a product of four distinct primes $ a,b,c,d$ such that:
$ (i)$ $ a\plus{}c\equal{}d;$
$ (ii)$ $ a(a\plus{}b\plus{}c\plus{}d)\equal{}c(d\minus{}b);$
$ (iii)$ $ 1\plus{}bc\plus{}d\equal{}bd$ .
Determine $ n$ . | 2002 | 0.75 | 6,187.5 | 5,519.333333 | 8,192 | |
Let $ABC$ be an equilateral triangle. Extend side $\overline{AB}$ beyond $B$ to a point $B'$ so that $BB'=3 \cdot AB$. Similarly, extend side $\overline{BC}$ beyond $C$ to a point $C'$ so that $CC'=3 \cdot BC$, and extend side $\overline{CA}$ beyond $A$ to a point $A'$ so that $AA'=3 \cdot CA$. What is the ratio of the... | 16 | 1. **Define the side length of $\triangle ABC$:** Let $AB = BC = CA = x$.
2. **Extend the sides of $\triangle ABC$:**
- Extend $AB$ to $B'$ such that $BB' = 3AB$. Thus, $AB' = AB + BB' = x + 3x = 4x$.
- Extend $BC$ to $C'$ such that $CC' = 3BC$. Thus, $BC' = BC + CC' = x + 3x = 4x$.
- Extend $CA$ to $A'$ suc... | 0 | 6,119.875 | -1 | 6,119.875 |
The numbers $3, 5, 7, a,$ and $b$ have an average (arithmetic mean) of $15$. What is the average of $a$ and $b$? | 30 | 1. **Calculate the total sum of the numbers given their average**:
The average (arithmetic mean) of the numbers $3, 5, 7, a,$ and $b$ is given as $15$. The formula for the average of a set of numbers is:
\[
\text{Average} = \frac{\text{Sum of all numbers}}{\text{Number of numbers}}
\]
Here, the sum of t... | 1 | 1,207.125 | 1,207.125 | -1 |
In triangle $ABC$, given that $a + c = 2b$, find the value of $\operatorname{tg} \frac{A}{2} \operatorname{tg} \frac{C}{2}$. | \frac{1}{3} | 0.625 | 6,260.1875 | 5,101.1 | 8,192 | |
Given a regular quadrilateral pyramid $S-ABCD$ with lateral edge length of $4$ and $\angle ASB = 30^\circ$, points $E$, $F$, and $G$ are taken on lateral edges $SB$, $SC$, and $SD$ respectively. Find the minimum value of the perimeter of the spatial quadrilateral $AEFG$. | 4\sqrt{3} | 0 | 8,192 | -1 | 8,192 |
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