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 |
|---|---|---|---|---|---|---|
Determine all pairs $(h, s)$ of positive integers with the following property: If one draws $h$ horizontal lines and another $s$ lines which satisfy (i) they are not horizontal, (ii) no two of them are parallel, (iii) no three of the $h+s$ lines are concurrent, then the number of regions formed by these $h+s$ lines is ... | (995,1),(176,10),(80,21) | Let $a_{h, s}$ the number of regions formed by $h$ horizontal lines and $s$ another lines as described in the problem. Let $\mathcal{F}_{h, s}$ be the union of the $h+s$ lines and pick any line $\ell$. If it intersects the other lines in $n$ (distinct!) points then $\ell$ is partitioned into $n-1$ line segments and 2 r... | 0 | 7,596.3125 | -1 | 7,596.3125 |
If $x \sim N(4, 1)$ and $f(x < 3) = 0.0187$, then $f(x < 5) = \_\_\_\_\_\_$. | 0.9813 | 0.0625 | 7,636.4375 | 6,977 | 7,680.4 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. The function $f(x)=2\cos x\sin (x-A)+\sin A (x\in R)$ reaches its maximum value at $x=\frac{5\pi}{12}$.
(1) Find the range of the function $f(x)$ when $x\in(0,\frac{\pi}{2})$;
(2) If $a=7$ and $\sin B+\sin C=\frac{13\... | 10\sqrt{3} | 0.75 | 6,781.5625 | 6,311.416667 | 8,192 | |
Given that the first term of a geometric sequence $\{a\_n\}$ is $\frac{3}{2}$, and the sum of the first $n$ terms is $S\_n$, where $n \in \mathbb{N}^*$. Also, $-2S\_2$, $S\_3$, and $4S\_4$ form an arithmetic sequence.
1. Find the general term formula for the sequence $\{a\_n\}$.
2. For a sequence $\{A\_n\}$, if there ... | \frac{1}{6} | 0 | 8,142.25 | -1 | 8,142.25 | |
Given that the function $y=f(x)$ is an odd function defined on $\mathbb{R}$ and $f(-1)=2$, and the period of the function is $4$, calculate the values of $f(2012)$ and $f(2013)$. | -2 | 1 | 2,273.6875 | 2,273.6875 | -1 | |
During the Double 11 shopping festival, a certain online store purchased two types of toys, $A$ and $B$, directly from the factory. The purchase price and selling price are as shown in the table: (Note: Profit = Selling Price - Purchase Price)
| | $A$ toy | $B$ toy |
|----------|---------|---------|
| Purchas... | 36 | 1 | 2,979 | 2,979 | -1 | |
There are $522$ people at a beach, each of whom owns a cat, a dog, both, or neither. If $20$ percent of cat-owners also own a dog, $70$ percent of dog-owners do not own a cat, and $50$ percent of people who don’t own a cat also don’t own a dog, how many people own neither type of pet? | 126 | 0.6875 | 5,504.4375 | 4,282.818182 | 8,192 | |
Define a $\textit{better word}$ as a sequence of letters that consists only of the letters $A$, $B$, $C$, and $D$ — some of these letters may not appear in the sequence — where $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, $C$ is never immediately followed by $D$, and $D$ is never ... | 2916 | 0.75 | 5,369.4375 | 5,181.416667 | 5,933.5 | |
From 5 cards labeled with the numbers $1, 2, 3, 4, 5$, 3 cards are randomly drawn to form a three-digit number. How many such three-digit numbers cannot be divided by 3? | 36 | 0.75 | 6,153.1875 | 5,473.583333 | 8,192 | |
Five persons wearing badges with numbers $1, 2, 3, 4, 5$ are seated on $5$ chairs around a circular table. In how many ways can they be seated so that no two persons whose badges have consecutive numbers are seated next to each other? (Two arrangements obtained by rotation around the table are considered different) | 10 | 0 | 8,192 | -1 | 8,192 | |
Let $A B C$ be an acute scalene triangle with circumcenter $O$ and centroid $G$. Given that $A G O$ is a right triangle, $A O=9$, and $B C=15$, let $S$ be the sum of all possible values for the area of triangle $A G O$. Compute $S^{2}$. | 288 | Note that we know that $O, H$, and $G$ are collinear and that $H G=2 O G$. Thus, let $O G=x$ and $H G=2 x$. We also have $\sin A=\frac{B C}{2 R}=\frac{5}{6}$, so $\cos A=\frac{\sqrt{11}}{6}$. Then, if $A G \perp O G$, then we have $x^{2}+A G^{2}=O G^{2}+A G^{2}=A O^{2}=81$ and $H G^{2}+A G^{2}=4 x^{2}+A G^{2}=A H^{2}=(... | 0 | 8,192 | -1 | 8,192 |
Given that $a$, $b$, and $c$ are integers, and $a-2b=4$, $ab+c^2-1=0$, find the value of $a+b+c$. | -3 | 0 | 8,192 | -1 | 8,192 | |
If $n$ is a positive integer, the notation $n$! (read " $n$ factorial") is used to represent the product of the integers from 1 to $n$. That is, $n!=n(n-1)(n-2) \cdots(3)(2)(1)$. For example, $4!=4(3)(2)(1)=24$ and $1!=1$. If $a$ and $b$ are positive integers with $b>a$, what is the ones (units) digit of $b!-a$! that c... | 7 | The first few values of $n$! are
\begin{aligned}
& 1!=1 \\
& 2!=2(1)=2 \\
& 3!=3(2)(1)=6 \\
& 4!=4(3)(2)(1)=24 \\
& 5!=5(4)(3)(2)(1)=120
\end{aligned}
We note that
\begin{aligned}
& 2!-1!=1 \\
& 4!-1!=23 \\
& 3!-1!=5 \\
& 5!-1!=119
\end{aligned}
This means that if $a$ and $b$ are positive integers with $b>a$, then ... | 0.1875 | 7,622.875 | 5,535.333333 | 8,104.615385 |
Quadrilateral $EFGH$ is a parallelogram. A line through point $G$ makes a $30^\circ$ angle with side $GH$. Determine the degree measure of angle $E$.
[asy]
size(100);
draw((0,0)--(5,2)--(6,7)--(1,5)--cycle);
draw((5,2)--(7.5,3)); // transversal line
draw(Arc((5,2),1,-60,-20)); // transversal angle
label("$H$",(0,0),SW... | 150 | 0.125 | 7,386.0625 | 4,665.5 | 7,774.714286 | |
The Minions need to make jam within the specified time. Kevin can finish the job 4 days earlier if he works alone, while Dave would finish 6 days late if he works alone. If Kevin and Dave work together for 4 days and then Dave completes the remaining work alone, the job is completed exactly on time. How many days would... | 12 | 0.25 | 6,701 | 4,632.5 | 7,390.5 | |
A student is given a budget of $10,000 to produce a rectangular banner for a school function. The length and width (in meters) of the banner must be integers. If each meter in length costs $330 while each meter in width costs $450, what is the maximum area (in square meters) of the banner that can be produced? | 165 | 0 | 8,192 | -1 | 8,192 | |
Amy rolls six fair 8-sided dice, each numbered from 1 to 8. What is the probability that exactly three of the dice show a prime number and at least one die shows an 8? | \frac{2899900}{16777216} | 0 | 6,060.125 | -1 | 6,060.125 | |
In the table, 8 is located in the 3rd row and 2nd column, 2017 is located in the $a$th row and $b$th column. Find $a - b = \quad$
\begin{tabular}{|c|c|c|}
\hline 1 & 4 & 5 \\
\hline 2 & 3 & 6 \\
\hline 9 & 8 & 7 \\
\hline 10 & 13 & 14 \\
\hline 11 & 12 & 15 \\
\hline 18 & 17 & 16 \\
\hline 19 & 22 & 23 \\
\hline 20 & 2... | 672 | 0 | 8,192 | -1 | 8,192 | |
There are 9 digits: 0, 1, 2, …, 8. Using five cards, with the two sides respectively marked as 0/8, 1/7, 2/5, 3/4, 6/9; and 6 can be used as 9. How many different four-digit numbers can be formed with these five cards? | 1728 | 0.0625 | 8,177.625 | 7,962 | 8,192 | |
How many four-digit positive integers are there with thousands digit $2?$ | 1000 | 1 | 1,829.6875 | 1,829.6875 | -1 | |
Given the function $f(x)=\cos x\cos \left( x+\dfrac{\pi}{3} \right)$.
(1) Find the smallest positive period of $f(x)$;
(2) In $\triangle ABC$, angles $A$, $B$, $C$ correspond to sides $a$, $b$, $c$, respectively. If $f(C)=-\dfrac{1}{4}$, $a=2$, and the area of $\triangle ABC$ is $2\sqrt{3}$, find the value of side le... | 2 \sqrt {3} | 0 | 4,387.1875 | -1 | 4,387.1875 | |
Let \( a, b, c \) be real numbers satisfying \( 9a^2 + 4b^2 + 36c^2 = 4 \). Find the minimum value of \( 3a + 6b + 12c \). | -2\sqrt{14} | 0.75 | 6,233 | 5,580 | 8,192 | |
The road from Petya's house to the school takes 20 minutes. One day, on his way to school, he remembered that he forgot his pen at home. If he continues his journey at the same speed, he will arrive at school 3 minutes before the bell rings. However, if he returns home to get the pen and then goes to school at the sam... | \frac{7}{20} | 0.0625 | 8,115 | 6,960 | 8,192 | |
Given a right triangle \( ABC \) with \(\angle A = 60^\circ\) and hypotenuse \( AB = 2 + 2\sqrt{3} \), a line \( p \) is drawn through vertex \( B \) parallel to \( AC \). Points \( D \) and \( E \) are placed on line \( p \) such that \( AB = BD \) and \( BC = BE \). Let \( F \) be the intersection point of lines \( A... | 1 + \sqrt{3} + \sqrt{6} | 0 | 8,192 | -1 | 8,192 | |
A cone is perfectly fitted inside a cube such that the cone's base is one face of the cube and its vertex touches the opposite face. A sphere is inscribed in the same cube. Given that one edge of the cube is 8 inches, calculate:
1. The volume of the inscribed sphere.
2. The volume of the inscribed cone.
Express your a... | \frac{128}{3}\pi | 0.9375 | 1,960.0625 | 1,970.8 | 1,799 | |
Suppose that $|x_i| < 1$ for $i = 1, 2, \dots, n$. Suppose further that $|x_1| + |x_2| + \dots + |x_n| = 19 + |x_1 + x_2 + \dots + x_n|.$ What is the smallest possible value of $n$? | 20 | Since $|x_i| < 1$ then
\[|x_1| + |x_2| + \dots + |x_n| = 19 + |x_1 + x_2 + \dots + x_n| < n.\]
So $n \ge 20$. We now just need to find an example where $n = 20$: suppose $x_{2k-1} = \frac{19}{20}$ and $x_{2k} = -\frac{19}{20}$; then on the left hand side we have $\left|\frac{19}{20}\right| + \left|-\frac{19}{20}\right... | 0.5 | 7,066.1875 | 5,976.75 | 8,155.625 |
Given that $\tan \alpha =2$, find the value of $\frac{4\sin^{3} \alpha -2\cos \alpha }{5\cos \alpha +3\sin \alpha }$. | \frac{2}{5} | 1 | 4,240.25 | 4,240.25 | -1 | |
For how many non-negative real values of $x$ is $\sqrt{144-\sqrt[3]{x}}$ an integer? | 13 | 1 | 2,185 | 2,185 | -1 | |
Positive real numbers $r,s$ satisfy the equations $r^2 + s^2 = 1$ and $r^4 + s^4= \frac{7}{8}$. Find $rs$. | \frac{1}{4} | 1 | 2,079.1875 | 2,079.1875 | -1 | |
A container holds one liter of wine, and another holds one liter of water. From the first container, we pour one deciliter into the second container and mix thoroughly. Then, we pour one deciliter of the mixture back into the first container. Calculate the limit of the amount of wine in the first container if this proc... | \frac{1}{2} | 0.1875 | 7,583.25 | 5,789 | 7,997.307692 | |
Every week, Judy goes to the supermarket and buys the following: $5$ carrots at $\$1$ each, $3$ bottles of milk at $\$3$ each, $2$ pineapples at $\$4$ each, $2$ bags of flour at $\$5$ each, and a giant $\$7$ container of ice cream. This week the store has a sale and pineapples are half price. Judy also has a coupon for... | \$30 | 1 | 1,699.3125 | 1,699.3125 | -1 | |
For some complex number $\omega$ with $|\omega| = 2016$ , there is some real $\lambda>1$ such that $\omega, \omega^{2},$ and $\lambda \omega$ form an equilateral triangle in the complex plane. Then, $\lambda$ can be written in the form $\tfrac{a + \sqrt{b}}{c}$ , where $a,b,$ and $c$ are positive intege... | 4032 | 0 | 8,192 | -1 | 8,192 | |
Given the quadratic function $f(x)=x^{2}-x+k$, where $k\in\mathbb{Z}$, if the function $g(x)=f(x)-2$ has two distinct zeros in the interval $(-1, \frac{3}{2})$, find the minimum value of $\frac{[f(x)]^{2}+2}{f(x)}$. | \frac{81}{28} | 0.8125 | 5,721.3125 | 5,151.153846 | 8,192 | |
Right triangle $ABC$ has one leg of length 9 cm, another leg of length 12 cm, and a right angle at $A$. A square has one side on the hypotenuse of triangle $ABC$ and a vertex on each of the two legs of triangle $ABC$. What is the length of one side of the square, in cm? Express your answer as a common fraction. | \frac{45}{8} | 0 | 7,993.1875 | -1 | 7,993.1875 | |
Numbers $m$ and $n$ are on the number line. What is the value of $n-m$? | 55 | On a number line, the markings are evenly spaced. Since there are 6 spaces between 0 and 30, each space represents a change of $\frac{30}{6}=5$. Since $n$ is 2 spaces to the right of 60, then $n=60+2 \times 5=70$. Since $m$ is 3 spaces to the left of 30, then $m=30-3 \times 5=15$. Therefore, $n-m=70-15=55$. | 0 | 7,025 | -1 | 7,025 |
Given the function $f(x)=\sin \left(2x+\frac{\pi }{3}\right)+\sin \left(2x-\frac{\pi }{3}\right)+2\cos ^{2}(x)-1$, where $x\in R$.
(1) Simplify the function $f(x)$ in the form of $A\sin (\omega x+\phi )$ $(A,\omega > 0,0 < \phi < \frac{\pi }{2})$
(2) Find the maximum and minimum values of the function $f(x)$ in the i... | -1 | 1 | 3,953.1875 | 3,953.1875 | -1 | |
We want to set up an electric bell. The location of the bell is at a distance of $30 \mathrm{~m}$ from the ringing spot. The internal resistance of the bell is 2 ohms. We plan to use 2 Leclanché cells connected in series, each with an electromotive force of 1.5 Volts and an internal resistance of 1 ohm. What diameter ... | 0.63 | 0 | 7,160.625 | -1 | 7,160.625 | |
Assume that $f$ and $g$ are functions for which $f^{-1}(g(x))=5x+3$. Find $g^{-1}(f(-7))$. | -2 | 1 | 2,346.0625 | 2,346.0625 | -1 | |
$N$ students are seated at desks in an $m \times n$ array, where $m, n \ge 3$ . Each student shakes hands with the students who are adjacent horizontally, vertically or diagonally. If there are $1020 $ handshakes, what is $N$ ? | 280 | 0.3125 | 7,446 | 6,037.4 | 8,086.272727 | |
Compute
\[\frac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)(52^4+324)}.\] | 373 | 0.0625 | 7,741.375 | 6,725 | 7,809.133333 | |
Point $P$ is on the circle $C_{1}: x^{2}+y^{2}-8x-4y+11=0$, and point $Q$ is on the circle $C_{2}: x^{2}+y^{2}+4x+2y+1=0$. What is the minimum value of $|PQ|$? | 3\sqrt{5} - 5 | 0.75 | 5,638.375 | 4,787.166667 | 8,192 | |
The ratio of the radii of two concentric circles is $1:3$. If $\overline{AC}$ is a diameter of the larger circle, $\overline{BC}$ is a chord of the larger circle that is tangent to the smaller circle, and $AB=12$, then the radius of the larger circle is | 18 | 1. **Identify the Geometry and Setup the Problem:**
- We have two concentric circles with radii ratio $1:3$. Let the radius of the smaller circle be $r$ and the radius of the larger circle be $3r$.
- $\overline{AC}$ is a diameter of the larger circle, so $AC = 2 \times 3r = 6r$.
- $\overline{BC}$ is a chord of... | 0.8125 | 4,258.1875 | 3,350.384615 | 8,192 |
What is the degree measure of an angle whose measure is double the measure of its complement? | 60 | 1 | 1,295.75 | 1,295.75 | -1 | |
The sum of two numbers is $19$ and their difference is $5$. What is their product? | 84 | 1 | 2,097.25 | 2,097.25 | -1 | |
A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola $x^2 - y^2 = 2000^2$? | 98 | \[(x-y)(x+y)=2000^2=2^8 \cdot 5^6\]
Note that $(x-y)$ and $(x+y)$ have the same parities, so both must be even. We first give a factor of $2$ to both $(x-y)$ and $(x+y)$. We have $2^6 \cdot 5^6$ left. Since there are $7 \cdot 7=49$ factors of $2^6 \cdot 5^6$, and since both $x$ and $y$ can be negative, this gives us $... | 0 | 8,192 | -1 | 8,192 |
If the sum of the lengths of the six edges of a trirectangular tetrahedron $PABC$ (i.e., $\angle APB=\angle BPC=\angle CPA=90^o$ ) is $S$ , determine its maximum volume. | \[
\frac{S^3(\sqrt{2}-1)^3}{162}
\] | Let the side lengths of $AP$ , $BP$ , and $CP$ be $a$ , $b$ , and $c$ , respectively. Therefore $S=a+b+c+\sqrt{a^2+b^2}+\sqrt{b^2+c^2}+\sqrt{c^2+a^2}$ . Let the volume of the tetrahedron be $V$ . Therefore $V=\frac{abc}{6}$ .
Note that $(a-b)^2\geq 0$ implies $\frac{a^2-2ab+b^2}{2}\geq 0$ , which means $\frac{a^2+b^2}{... | 0 | 7,801.125 | -1 | 7,801.125 |
There are two concentric spheres of radii 3 units and 6 units. What is the volume, in cubic units, of the region within the larger sphere and not within the smaller sphere? Express your answer in terms of $\pi$. | 252\pi | 1 | 1,296.8125 | 1,296.8125 | -1 | |
Evaluate the sum $2345 + 3452 + 4523 + 5234$ and then subtract $1234$ from the result. | 14320 | 0.9375 | 4,007.25 | 3,728.266667 | 8,192 | |
The volume of the parallelepiped generated by the vectors $\begin{pmatrix} 3 \\ 4 \\ 5 \end{pmatrix}$, $\begin{pmatrix} 2 \\ m \\ 3 \end{pmatrix}$, and $\begin{pmatrix} 2 \\ 3 \\ m \end{pmatrix}$ is 20. Find $m$, where $m > 0$. | 3 + \frac{2\sqrt{15}}{3} | 0 | 8,046.9375 | -1 | 8,046.9375 | |
A nut has the shape of a regular hexagonal prism. Each lateral face of the nut is painted in one of three colors: white, red, or blue, with adjacent faces painted in different colors. How many different nut paintings are possible? (It is not necessary to use all three colors in the painting.) | 13 | 0 | 7,525 | -1 | 7,525 | |
David has a collection of 40 rocks, 30 stones, 20 minerals and 10 gemstones. An operation consists of removing three objects, no two of the same type. What is the maximum number of operations he can possibly perform?
*Ray Li* | 30 | 0 | 8,192 | -1 | 8,192 | |
David found four sticks of different lengths that can be used to form three non-congruent convex cyclic quadrilaterals, $A,\text{ }B,\text{ }C$, which can each be inscribed in a circle with radius $1$. Let $\varphi_A$ denote the measure of the acute angle made by the diagonals of quadrilateral $A$, and define $\varphi_... | 59 | Let the sides of the quadrilaterals be $a,b,c,$ and $d$ in some order such that $A$ has $a$ opposite of $c$, $B$ has $a$ opposite of $b$, and $C$ has $a$ opposite of $d$. Then, let the diagonals of $A$ be $e$ and $f$. Similarly to solution $2$, we get that $\tfrac{2}{3}(ac+bd)=\tfrac{3}{5}(ab+cd)=\tfrac{6}{7}(ad+bc)=2K... | 0 | 8,192 | -1 | 8,192 |
The fraction
\[\frac{\left(3^{2008}\right)^2-\left(3^{2006}\right)^2}{\left(3^{2007}\right)^2-\left(3^{2005}\right)^2}\]
simplifies to what? | 9 | 1. **Rewrite the expression using the difference of squares formula**:
\[
\frac{(3^{2008})^2 - (3^{2006})^2}{(3^{2007})^2 - (3^{2005})^2} = \frac{(3^{2008} + 3^{2006})(3^{2008} - 3^{2006})}{(3^{2007} + 3^{2005})(3^{2007} - 3^{2005})}
\]
2. **Simplify each term**:
- For the numerator:
\[
3^{2008} ... | 1 | 4,045.5 | 4,045.5 | -1 |
Let \( S = \{r_1, r_2, \cdots, r_n\} \subseteq \{1, 2, 3, \cdots, 50\} \), and let any two numbers in \( S \) not be divisible by 7 when summed. Find the maximum value of \( n \). | 23 | 0.0625 | 8,056.5 | 6,024 | 8,192 | |
Determine $S$, the sum of all the real coefficients of the expansion of $(1+ix)^{2020}$, and find $\log_2(S)$. | 1010 | 0 | 8,192 | -1 | 8,192 | |
Compute $\binom{18}{6}$. | 18564 | 0.5625 | 6,746.75 | 5,622.666667 | 8,192 | |
Find the maximum value of the expression \( (\sin 2x + \sin 3y + \sin 4z)(\cos 2x + \cos 3y + \cos 4z) \). | 4.5 | 0 | 8,185.25 | -1 | 8,185.25 | |
Given $f(x)= \frac{\ln x+2^{x}}{x^{2}}$, find $f'(1)=$ ___. | 2\ln 2 - 3 | 0.3125 | 5,734.625 | 4,826.2 | 6,147.545455 | |
If 2023 were expressed as a sum of distinct powers of 2, what would be the least possible sum of the exponents of these powers? | 48 | 0.3125 | 7,421 | 5,724.8 | 8,192 | |
A cube-shaped container has vertices $A,$ $B,$ $C,$ and $D,$ where $\overline{AB}$ and $\overline{CD}$ are parallel edges of the cube, and $\overline{AC}$ and $\overline{BD}$ are diagonals of faces of the cube, as shown. Vertex $A$ of the cube is set on a horizontal plane $\mathcal{P}$ so that the plane of the rectangl... | 751 | Denote $h(X)$ the distance from point $X$ to $\mathcal{P}, h(A) = 0, h(B) = 2,$ $h(C) = 8, h(D) = 10, h(G) = h(I) = h(H) = 7, AB = a, AC = a \sqrt{2}.$
Let slope $AB$ to $\mathcal{P}$ be $\alpha.$ Notation is shown in the diagram. \[\tan \alpha = \frac {\sin \alpha}{\cos \alpha} = \frac {h(B)}{AB}\cdot \frac {AC}{h(C)... | 0 | 8,192 | -1 | 8,192 |
A bus ticket costs 1 yuan each. Xiaoming and 6 other children are lining up to buy tickets. Each of the 6 children has only 1 yuan, while Xiaoming has a 5-yuan note. The seller has no change. In how many ways can they line up so that the seller can give Xiaoming change when he buys a ticket? | 10800 | 0 | 7,493 | -1 | 7,493 | |
Determine all \(\alpha \in \mathbb{R}\) such that for every continuous function \(f:[0,1] \rightarrow \mathbb{R}\), differentiable on \((0,1)\), with \(f(0)=0\) and \(f(1)=1\), there exists some \(\xi \in(0,1)\) such that \(f(\xi)+\alpha=f^{\prime}(\xi)\). | \(\alpha = \frac{1}{e-1}\) | First consider the function \(h(x)=\frac{e^{x}-1}{e-1}\), which has the property that \(h^{\prime}(x)=\frac{e^{x}}{e-1}\). Note that \(h \in V\) and that \(h^{\prime}(x)-h(x)=1 /(e-1)\) is constant. As such, \(\alpha=1 /(e-1)\) is the only possible value that could possibly satisfy the condition from the problem. For \... | 0 | 8,125.375 | -1 | 8,125.375 |
Points $P$ and $Q$ are both in the line segment $AB$ and on the same side of its midpoint. $P$ divides $AB$ in the ratio $2:3$,
and $Q$ divides $AB$ in the ratio $3:4$. If $PQ=2$, then the length of $AB$ is: | 70 | 1. **Assign Variables:**
Let $AP = x$ and $PB = y$. Since $P$ divides $AB$ in the ratio $2:3$, we have:
\[
\frac{x}{y} = \frac{2}{3}
\]
This implies:
\[
3x = 2y \quad \text{(Equation 1)}
\]
2. **Express $AB$ in terms of $x$ and $y$:**
Since $P$ divides $AB$, the total length of $AB = x + y$.... | 1 | 839.5625 | 839.5625 | -1 |
A car license plate contains three letters and three digits, for example, A123BE. The allowed letters are А, В, Е, К, М, Н, О, Р, С, Т, У, Х (a total of 12 letters), and all digits except for the combination 000. Tanya considers a license plate happy if the first letter is a consonant, the second letter is also a conso... | 384000 | 0.1875 | 5,293.25 | 2,939.333333 | 5,836.461538 | |
Given seven positive integers from a list of eleven positive integers are \(3, 5, 6, 9, 10, 4, 7\). What is the largest possible value of the median of this list of eleven positive integers if no additional number in the list can exceed 10? | 10 | 0 | 7,694.5 | -1 | 7,694.5 | |
Evaluate $x^2y^3z$ if $x = \frac13$, $y = \frac23$, and $z = -9$. | -\frac{8}{27} | 1 | 3,306.125 | 3,306.125 | -1 | |
In convex quadrilateral $ABCD$, $AB=8$, $BC=4$, $CD=DA=10$, and $\angle CDA=60^\circ$. If the area of $ABCD$ can be written in the form $\sqrt{a}+b\sqrt{c}$ where $a$ and $c$ have no perfect square factors (greater than 1), what is $a+b+c$? | 259 | 0.625 | 7,397.875 | 6,921.4 | 8,192 | |
A rectangular floor is covered with congruent square tiles. If the total number of tiles that lie on the two diagonals is 57, how many tiles cover the floor. | 841 | 0.6875 | 5,791.8125 | 5,415.727273 | 6,619.2 | |
Given the function $f(x)=3^{x}$, where $f(a+2)=27$, and the function $g(x)=λ \cdot 2^{ax}-4^{x}$ with a domain of $[0,2]$, find:
(1) The value of $a$;
(2) The value of the real number $λ$ if the maximum value of function $g(x)$ is $\frac{1}{3}$. | \frac{4}{3} | 0.3125 | 7,571.375 | 6,911.2 | 7,871.454545 | |
Given that the function $f(x) = e^x + \frac{a}{e^x}$ has a derivative $y = f'(x)$ that is an odd function and the slope of a tangent line to the curve $y = f(x)$ is $\frac{3}{2}$, determine the abscissa of the tangent point. | \ln 2 | 1 | 2,093.4375 | 2,093.4375 | -1 | |
A metal weight has a mass of 20 kg and is an alloy of four metals. The mass of the first metal in this alloy is one and a half times greater than that of the second, the mass of the second metal is in the ratio $3:4$ to the mass of the third, and the mass of the third metal is in the ratio $5:6$ to the mass of the four... | 4.91 | 0.5 | 6,277.5 | 4,763.5 | 7,791.5 | |
Pat Peano has plenty of 0's, 1's, 3's, 4's, 5's, 6's, 7's, 8's and 9's, but he has only twenty-two 2's. How far can he number the pages of his scrapbook with these digits? | 119 | To determine how far Pat Peano can number the pages of his scrapbook using only twenty-two 2's, we need to count how many times the digit '2' appears in each number from 1 onwards.
1. **Counting the '2's in one-digit numbers (1-9):**
- The only one-digit number containing '2' is 2 itself.
- Thus, there is $1$ tw... | 0 | 7,949.5625 | -1 | 7,949.5625 |
What is the radius of a circle inscribed in a rhombus with diagonals of length $10$ and $24$? | \frac{60}{13} | 1. **Identify the given values and the goal:**
- Diagonals of the rhombus are given as $d_1 = 10$ and $d_2 = 24$.
- We need to find the radius $r$ of the circle inscribed in the rhombus.
2. **Use the Pythagorean theorem to find the side length of the rhombus:**
- The diagonals of a rhombus bisect each other a... | 1 | 3,319.8125 | 3,319.8125 | -1 |
Let $A_1B_1C_1D_1$ be an arbitrary convex quadrilateral. $P$ is a point inside the quadrilateral such that each angle enclosed by one edge and one ray which starts at one vertex on that edge and passes through point $P$ is acute. We recursively define points $A_k,B_k,C_k,D_k$ symmetric to $P$ with respect to lines $A_{... | 1, 5, 9 |
Let \( A_1B_1C_1D_1 \) be an arbitrary convex quadrilateral. \( P \) is a point inside the quadrilateral such that each angle enclosed by one edge and one ray which starts at one vertex on that edge and passes through point \( P \) is acute. We recursively define points \( A_k, B_k, C_k, D_k \) symmetric to \( P \) w... | 0 | 8,192 | -1 | 8,192 |
Given that $\sin{\alpha} = -\frac{3}{5}$, $\sin{\beta} = \frac{12}{13}$, and $\alpha \in (\pi, \frac{3\pi}{2})$, $\beta \in (\frac{\pi}{2}, \pi)$, find the values of $\sin({\alpha - \beta})$, $\cos{2\alpha}$, and $\tan{\frac{\beta}{2}}$. | \frac{3}{2} | 1 | 2,611 | 2,611 | -1 | |
Let $ K$ be the curved surface obtained by rotating the parabola $ y \equal{} \frac {3}{4} \minus{} x^2$ about the $ y$ -axis.Cut $ K$ by the plane $ H$ passing through the origin and forming angle $ 45^\circ$ for the axis. Find the volume of the solid surrounded by $ K$ and $ H.$
Note that you are ... | \frac{9\pi}{256} | 0 | 8,192 | -1 | 8,192 | |
A curve C is established in the polar coordinate system with the coordinate origin O as the pole and the positive semi-axis of the x-axis as the polar axis. The polar equation of the curve C is given by $$ρ^{2}= \frac {12}{4-cos^{2}\theta }$$
1. Find the rectangular coordinate equation of the curve C.
2. Suppose a lin... | \frac{4}{3} | 0.8125 | 4,970.8125 | 4,966.692308 | 4,988.666667 | |
In \(\triangle ABC\), \(DC = 2BD\), \(\angle ABC = 45^\circ\), and \(\angle ADC = 60^\circ\). Find \(\angle ACB\) in degrees. | 75 | 0.5 | 7,225.875 | 6,259.75 | 8,192 | |
Let $A B C D$ be a convex quadrilateral whose diagonals $A C$ and $B D$ meet at $P$. Let the area of triangle $A P B$ be 24 and let the area of triangle $C P D$ be 25 . What is the minimum possible area of quadrilateral $A B C D ?$ | 49+20 \sqrt{6} | Note that $\angle A P B=180^{\circ}-\angle B P C=\angle C P D=180^{\circ}-\angle D P A$ so $4[B P C][D P A]=(P B \cdot P C \cdot \sin B P C)(P D \cdot P A \cdot \sin D P A)=(P A \cdot P B \cdot \sin A P B)(P C \cdot P D \cdot \sin C P D)=4[A P B][C P D]=2400 \Longrightarrow$ $[B P C][D P A]=600$. Hence by AM-GM we have... | 0.3125 | 7,297.375 | 6,263.8 | 7,767.181818 |
A quadrilateral is inscribed in a circle of radius $200\sqrt{2}$. Three of the sides of this quadrilateral have length $200$. What is the length of the fourth side? | 500 | 1. **Setup and Diagram**: Let quadrilateral $ABCD$ be inscribed in a circle with center $O$ and radius $200\sqrt{2}$. Assume $AD$ is the side of unknown length $x$. The other three sides $AB$, $BC$, and $CD$ each have length $200$.
2. **Using the Pythagorean Theorem in $\triangle BOC$**: Draw the altitude from $O$ to ... | 0.625 | 6,981.6875 | 6,255.5 | 8,192 |
How many of the natural numbers from 1 to 800, inclusive, contain the digit 7 at least once? | 62 | 0 | 8,192 | -1 | 8,192 | |
A rectangular floor measures $a$ by $b$ feet, where $a$ and $b$ are positive integers and $b > a$. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the floor. The unpainted part of the floor forms a border of width $1$ foot around the painted rectangle and occupies half the area of ... | 2 | 1. **Identify the area of the painted and unpainted regions**: Given that the unpainted border is 1 foot wide and surrounds the entire painted rectangle, the dimensions of the painted rectangle are $(a-2)$ by $(b-2)$. The total area of the floor is $ab$, and the area of the painted rectangle is $(a-2)(b-2)$.
2. **Set ... | 0.875 | 4,569.1875 | 4,083.642857 | 7,968 |
Let $\{x\}$ denote the smallest integer not less than the real number $x$. Then, find the value of the following expression:
$$
\left\{\log _{2} 1\right\}+\left\{\log _{2} 2\right\}+\left\{\log _{2} 3\right\}+\cdots+\left\{\log _{2} 1991\right\}
$$ | 19854 | 0 | 7,967.75 | -1 | 7,967.75 | |
Compute the largest integer $k$ such that $2025^k$ divides $(2025!)^2$. | 505 | 0.3125 | 3,908.375 | 3,032.4 | 4,306.545455 | |
Points $P$ and $R$ are located at (2, 1) and (12, 15) respectively. Point $M$ is the midpoint of segment $\overline{PR}$. Segment $\overline{PR}$ is reflected over the $x$-axis. What is the sum of the coordinates of the image of point $M$ (the midpoint of the reflected segment)? | -1 | 1 | 1,573.5625 | 1,573.5625 | -1 | |
What is the largest integer that must divide the product of any $5$ consecutive integers? | 60 | 0 | 7,321.6875 | -1 | 7,321.6875 | |
Find all pairs of natural numbers $ (a, b)$ such that $ 7^a \minus{} 3^b$ divides $ a^4 \plus{} b^2$.
[i]Author: Stephan Wagner, Austria[/i] | (2, 4) |
To find all pairs of natural numbers \((a, b)\) such that \(7^a - 3^b\) divides \(a^4 + b^2\), we proceed as follows:
1. Let \(d = 7^a - 3^b\). We need to ensure \(d \mid a^4 + b^2\). This implies that \(a^4 + b^2 = k \cdot (7^a - 3^b)\) for some integer \(k\).
2. We know that for any potential solution, \(7^a > 3^b... | 0.0625 | 8,191.75 | 8,188 | 8,192 |
To enhance and beautify the city, all seven streetlights on a road are to be changed to colored lights. If there are three colors available for the colored lights - red, yellow, and blue - and the installation requires that no two adjacent streetlights are of the same color, with at least two lights of each color, ther... | 114 | 0 | 8,192 | -1 | 8,192 | |
The union of sets \( A \) and \( B \) is \( A \cup B = \left\{a_{1}, a_{2}, a_{3}\right\} \). When \( A \neq B \), the pairs \((A, B)\) and \((B, A)\) are considered different. How many such pairs \((A, B)\) are there? | 27 | 0.6875 | 4,854.1875 | 4,989.909091 | 4,555.6 | |
Expand and simplify the expression $-(4-d)(d+3(4-d))$. What is the sum of the coefficients of the expanded form? | -30 | 1 | 2,558.875 | 2,558.875 | -1 | |
In the diagram below, $AB = 30$ and $\angle ADB = 90^\circ$. If $\sin A = \frac{3}{5}$ and $\sin C = \frac{1}{4}$, what is the length of $DC$? | 18\sqrt{15} | 0 | 7,682.6875 | -1 | 7,682.6875 | |
Let $ABCD$ be a parallelogram such that $\angle BAD = 60^{\circ}.$ Let $K$ and $L$ be the midpoints of $BC$ and $CD,$ respectively. Assuming that $ABKL$ is a cyclic quadrilateral, find $\angle ABD.$ | 75 | 0.1875 | 7,913.5625 | 7,023 | 8,119.076923 | |
Suppose $ABC$ is a scalene right triangle, and $P$ is the point on hypotenuse $\overline{AC}$ such that $\angle{ABP} =
45^{\circ}$. Given that $AP = 1$ and $CP = 2$, compute the area of $ABC$. | \frac{9}{5} | 0.9375 | 4,427.4375 | 4,176.466667 | 8,192 | |
Given that the polynomial $x^2-5x+t$ has only positive integer roots, find the average of all distinct possible values of $t$. | 5 | 1 | 1,607.5625 | 1,607.5625 | -1 | |
I planned to work 20 hours a week for 12 weeks this summer to earn $3000 to buy a used car. Unfortunately, due to unforeseen events, I wasn't able to work any hours during the first three weeks of the summer. How many hours per week do I need to work for the remaining summer to achieve my financial goal? | 27 | 0 | 583.3125 | -1 | 583.3125 | |
How many integers are between $(11.2)^3$ and $(11.3)^3$? | 38 | 1 | 4,296.8125 | 4,296.8125 | -1 | |
In the Cartesian coordinate system $(xOy)$, the parametric equations of the curve $C$ are given by $\begin{cases} x=3\cos \alpha \\ y=\sin \alpha \end{cases}$ ($\alpha$ is the parameter). In the polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, the polar equation of the li... | \dfrac{18\sqrt{2}}{5} | 0 | 5,781.25 | -1 | 5,781.25 | |
Given the origin of the rectangular coordinate system xOy as the pole and the positive semi-axis of the x-axis as the polar axis, establish a polar coordinate system with the same unit length. The parametric equation of the line l is $$\begin{cases} \overset{x=2+t}{y=1+t}\end{cases}$$ (t is the parameter), and the pola... | \sqrt{2} | 0.875 | 5,176.9375 | 4,973.714286 | 6,599.5 | |
On January 15 in the stormy town of Stormville, there is a $50\%$ chance of rain. Every day, the probability of it raining has a $50\%$ chance of being $\frac{2017}{2016}$ times that of the previous day (or $100\%$ if this new quantity is over $100\%$ ) and a $50\%$ chance of being $\frac{1007}{2016}$ time... | 243/2048 | 0.125 | 7,923.4375 | 6,327 | 8,151.5 | |
The base of the quadrilateral pyramid \( S A B C D \) is a rhombus \( A B C D \) with an acute angle at vertex \( A \). The height of the rhombus is 4, and the point of intersection of its diagonals is the orthogonal projection of vertex \( S \) onto the plane of the base. A sphere with radius 2 touches the planes of a... | 8\sqrt{2} | 0 | 8,192 | -1 | 8,192 |
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