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 |
|---|---|---|---|---|---|---|
Find all functions $f$ from the set $\mathbb{R}$ of real numbers into $\mathbb{R}$ which satisfy for all $x, y, z \in \mathbb{R}$ the identity \[f(f(x)+f(y)+f(z))=f(f(x)-f(y))+f(2xy+f(z))+2f(xz-yz).\] | f(x) = 0 \text{ and } f(x) = x^2 |
We need to find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) satisfying the given functional equation for all \( x, y, z \in \mathbb{R} \):
\[
f(f(x) + f(y) + f(z)) = f(f(x) - f(y)) + f(2xy + f(z)) + 2f(xz - yz).
\]
To solve this, we'll explore potential forms of \( f(x) \) and check if they satisfy the ... | 0 | 8,192 | -1 | 8,192 |
Two people are playing "Easter egg battle." In front of them is a large basket of eggs. They randomly pick one egg each and hit them against each other. One of the eggs breaks, the defeated player takes a new egg, and the winner keeps their egg for the next round (the outcome of each round depends only on which egg has... | 11/12 | 0 | 7,122.9375 | -1 | 7,122.9375 | |
Given the function $f(x) = \cos x \cdot \sin\left(x + \frac{\pi}{3}\right) - \sqrt{3}\cos^2x + \frac{\sqrt{3}}{4}$, where $x \in \mathbb{R}$.
(1) Find the interval of monotonic increase for $f(x)$.
(2) In an acute triangle $\triangle ABC$, where the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respec... | \frac{3\sqrt{3}}{4} | 0 | 7,888.8125 | -1 | 7,888.8125 | |
In isosceles $\vartriangle ABC, AB = AC, \angle BAC$ is obtuse, and points $E$ and $F$ lie on sides $AB$ and $AC$ , respectively, so that $AE = 10, AF = 15$ . The area of $\vartriangle AEF$ is $60$ , and the area of quadrilateral $BEFC$ is $102$ . Find $BC$ . | 36 | 0.5625 | 6,308.25 | 5,621.666667 | 7,191 | |
In a zoo, there were 200 parrots. One day, they each made a statement in turn. Starting from the second parrot, all statements were: "Among the previous statements, more than 70% are false." How many false statements did the parrots make in total? | 140 | 0 | 7,338.125 | -1 | 7,338.125 | |
In a right triangle, one of the acute angles $\beta$ satisfies \[\tan \frac{\beta}{2} = \frac{1}{\sqrt[3]{3}}.\] Let $\phi$ be the angle between the median and the angle bisector drawn from this acute angle $\beta$. Calculate $\tan \phi.$ | \frac{1}{2} | 0 | 7,951.0625 | -1 | 7,951.0625 | |
If \( x \) and \( y \) are positive integers with \( x>y \) and \( x+x y=391 \), what is the value of \( x+y \)? | 39 | Since \( x+x y=391 \), then \( x(1+y)=391 \). We note that \( 391=17 \cdot 23 \). Since 17 and 23 are both prime, then if 391 is written as the product of two positive integers, it must be \( 1 \times 391 \) or \( 17 \times 23 \) or \( 23 \times 17 \) or \( 391 \times 1 \). Matching \( x \) and \( 1+y \) to these possi... | 1 | 1,973.0625 | 1,973.0625 | -1 |
Given an arithmetic sequence $\{a\_n\}$ with a common difference $d > 0$, and $a\_2$, $a\_5-1$, $a\_{10}$ form a geometric sequence. If $a\_1=5$, and $S\_n$ represents the sum of the first $n$ terms of the sequence, find the minimum value of $\frac{2S\_n+n+32}{a\_n+1}$. | \frac{20}{3} | 0.6875 | 6,395.875 | 5,579.454545 | 8,192 | |
Given that $\cos(75^\circ + \alpha) = \frac{1}{3}$, where $\alpha$ is an angle in the third quadrant, find the value of $\cos(105^\circ - \alpha) + \sin(\alpha - 105^\circ)$. | \frac{2\sqrt{2} - 1}{3} | 0 | 6,678 | -1 | 6,678 | |
In the subtraction shown, $K, L, M$, and $N$ are digits. What is the value of $K+L+M+N$?\n$$\begin{array}{r}6 K 0 L \\ -\quad M 9 N 4 \\ \hline 2011\end{array}$$ | 17 | We work from right to left as we would if doing this calculation by hand. In the units column, we have $L-4$ giving 1. Thus, $L=5$. (There is no borrowing required.) In the tens column, we have $0-N$ giving 1. Since 1 is larger than 0, we must borrow from the hundreds column. Thus, $10-N$ gives 1, which means $N=9$. In... | 0 | 898.5 | -1 | 898.5 |
A company allocates 5 employees to 3 different departments, with each department being allocated at least one employee. Among them, employees A and B must be allocated to the same department. Calculate the number of different allocation methods. | 36 | 0.125 | 7,561.0625 | 5,366.5 | 7,874.571429 | |
Triangle $ABC$ and point $P$ in the same plane are given. Point $P$ is equidistant from $A$ and $B$, angle $APB$ is twice angle $ACB$, and $\overline{AC}$ intersects $\overline{BP}$ at point $D$. If $PB = 3$ and $PD= 2$, then $AD\cdot CD =$ | 5 | 1. **Identify the Circle and Key Points**: Since $P$ is equidistant from $A$ and $B$, $P$ lies on the perpendicular bisector of $\overline{AB}$. Given that $\angle APB = 2\angle ACB$, and $P$ is equidistant from $A$ and $B$, we can infer that $A$, $B$, and $C$ lie on a circle centered at $P$ with radius $PA = PB$.
2. ... | 0 | 7,526.5625 | -1 | 7,526.5625 |
Out of the digits 0 through 9, three digits are randomly chosen to form a three-digit number without repeating any digits. What is the probability that this number is not divisible by 3? | 2/3 | 0.0625 | 7,730.75 | 5,807 | 7,859 | |
Inside triangle \(ABC\), a point \(O\) is chosen such that \(\angle ABO = \angle CAO\), \(\angle BAO = \angle BCO\), and \(\angle BOC = 90^{\circ}\). Find the ratio \(AC : OC\). | \sqrt{2} | 0 | 8,192 | -1 | 8,192 | |
What is $2343_6+15325_6$? Express your answer in base $6$. | 22112_6 | 0.1875 | 7,246.6875 | 3,150.333333 | 8,192 | |
Mary divides a circle into 12 sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle? | 8 | 1. **Define the problem in terms of an arithmetic sequence:** Let the central angles of the sectors be represented by an arithmetic sequence with the first term $a_1$ and common difference $d$. The sequence will have 12 terms, and the sum of these terms must equal the total degrees in a circle, which is 360 degrees.
2... | 0.875 | 5,234.75 | 4,812.285714 | 8,192 |
Points $P, Q, R, S, T, U, V,$ and $W$ lie, in that order, on line $\overline{PW}$, dividing it into seven equal segments, each of length 1. Point $X$ is not on line $PW$. Points $Y$ and $Z$ lie on line segments $\overline{XR}$ and $\overline{XW}$ respectively. The line segments $\overline{YQ}, \overline{ZT},$ and $\ove... | \frac{7}{6} | 0.5 | 5,473.1875 | 4,043.125 | 6,903.25 | |
It is known that the numbers \( x, y, z \) form an arithmetic progression in the given order with a common difference \( \alpha = \arccos \left(-\frac{1}{3}\right) \), and the numbers \( \frac{1}{\cos x}, \frac{3}{\cos y}, \frac{1}{\cos z} \) also form an arithmetic progression in the given order. Find \( \cos^2 y \). | \frac{4}{5} | 0.25 | 7,174.1875 | 4,120.75 | 8,192 | |
How many solutions in integers $x$ and $y$ does the inequality
$$
|x| + |y| < 10
$$
have? | 181 | 0.25 | 7,689.3125 | 7,105.25 | 7,884 | |
The sum of the numerical coefficients in the expansion of the binomial $(a+b)^6$ is: | 64 | To find the sum of the numerical coefficients in the expansion of the binomial $(a+b)^6$, we can substitute $a = 1$ and $b = 1$ into the binomial. This simplifies the expression to just the sum of the coefficients, as each term in the expansion will have the form $\binom{6}{k} a^{6-k} b^k$ and substituting $a = 1$ and ... | 1 | 1,632.125 | 1,632.125 | -1 |
In a cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with a side length of 1, points $E$ and $F$ are located on $A A_{1}$ and $C C_{1}$ respectively, such that $A E = C_{1} F$. Determine the minimum area of the quadrilateral $E B F D_{1}$. | \frac{\sqrt{6}}{2} | 0 | 7,901.9375 | -1 | 7,901.9375 | |
Given the sample contains 5 individuals with values a, 0, 1, 2, 3, and the average value of the sample is 1, calculate the standard deviation of the sample. | \sqrt{2} | 0.1875 | 2,483.625 | 3,457.666667 | 2,258.846154 | |
A positive integer cannot be divisible by 2 or 3, and there do not exist non-negative integers \(a\) and \(b\) such that \(|2^a - 3^b| = n\). Find the smallest value of \(n\). | 35 | 0.0625 | 7,985.125 | 8,192 | 7,971.333333 | |
How many 3-term geometric sequences $a$ , $b$ , $c$ are there where $a$ , $b$ , and $c$ are positive integers with $a < b < c$ and $c = 8000$ ? | 39 | 0.0625 | 7,309.625 | 8,163 | 7,252.733333 | |
A total of $731$ objects are put into $n$ nonempty bags where $n$ is a positive integer. These bags can be distributed into $17$ red boxes and also into $43$ blue boxes so that each red and each blue box contain $43$ and $17$ objects, respectively. Find the minimum value of $n$ . | 17 | 0 | 7,280.8125 | -1 | 7,280.8125 | |
My grandpa has 10 pieces of art, including 3 prints by Escher. If he hangs the pieces of art in a row in a random order, what is the probability that all three pieces by Escher will be placed consecutively? | \dfrac{1}{15} | 0.9375 | 4,115.125 | 3,843.333333 | 8,192 | |
A palindromic number is a number that reads the same when the order of its digits is reversed. What is the difference between the largest and smallest five-digit palindromic numbers that are both multiples of 45? | 9090 | 0.5625 | 4,931.5625 | 4,413.666667 | 5,597.428571 | |
A person can commute by train or car. If he takes the train to work in the morning, he takes the car in the afternoon; if he takes the train home in the afternoon, he takes the car in the morning. Over $x$ days, this person took the train 9 times, took the car in the morning 8 times, and took the car in the afternoon 1... | 16 | 0.375 | 6,158.75 | 4,394.833333 | 7,217.1 | |
Let \( n = 1990 \), then evaluate the expression \(\frac{1}{2^{n}}\left(1 - 3 C_{n}^{2} + 3^{2} C_{n}^{4} - 3^{3} C_{n}^{6} + \cdots + 3^{994} C_{n}^{1988} - 3^{995} C_{n}^{1990}\right)\). | -\frac{1}{2} | 0.3125 | 7,205.9375 | 5,949.2 | 7,777.181818 | |
Let $\mathbb{R}$ denote the set of real numbers. Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
\[f(xf(y)+y)+f(-f(x))=f(yf(x)-y)+y\]
for all $x,y\in\mathbb{R}$ | f(x) = x + 1 |
We are tasked with finding all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy the functional equation:
\[
f(xf(y)+y)+f(-f(x))=f(yf(x)-y)+y
\]
for all \( x, y \in \mathbb{R} \).
### Step 1: Explore the Equation by Substituting Special Values
First, we substitute \( y = 0 \) into the equation. Thi... | 0 | 8,148.25 | -1 | 8,148.25 |
289. A remarkable number. Find a number such that its fractional part, its integer part, and the number itself form a geometric progression. | \frac{1+\sqrt{5}}{2} | 0 | 4,984.5625 | -1 | 4,984.5625 | |
Determine the number of real number $a$ , such that for every $a$ , equation $x^3=ax+a+1$ has a root $x_0$ satisfying following conditions:
(a) $x_0$ is an even integer;
(b) $|x_0|<1000$ . | 999 | 0.4375 | 7,221.1875 | 5,973 | 8,192 | |
A mail carrier delivers mail to the nineteen houses on the east side of Elm Street. The carrier notices that no two adjacent houses ever get mail on the same day, but that there are never more than two houses in a row that get no mail on the same day. How many different patterns of mail delivery are possible? | 351 | Let $a_n$ be the number of ways if the first house has mail, and let $b_n$ be the number of ways if the first house does not get mail.
$a_n=a_{n-2}+a_{n-3}$ because if the first house gets mail, the next house that gets mail must either be the third or fourth house.
$b_n=a_{n-1}+a_{n-2}$ because if the first house does... | 0.0625 | 8,192 | 8,192 | 8,192 |
The gas tank in Catherine's car is $\frac{1}{8}$ full. When 30 litres of gas are added, the tank becomes $\frac{3}{4}$ full. If the gas costs Catherine $\$ 1.38$ per litre, how much will it cost her to fill the remaining quarter of the tank? | \$16.56 | When Catherine adds 30 litres of gasoline, the tank goes from $\frac{1}{8}$ full to $\frac{3}{4}$ full. Since $\frac{3}{4}-\frac{1}{8}=\frac{6}{8}-\frac{1}{8}=\frac{5}{8}$, then $\frac{5}{8}$ of the capacity of the tank is 30 litres. Thus, $\frac{1}{8}$ of the capacity of the tank is $30 \div 5=6$ litres. Also, the ful... | 0.9375 | 2,940.8125 | 2,590.733333 | 8,192 |
Eight spheres of radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the origin, that contains these eight spheres? | 1+\sqrt{3} | 1. **Understanding the setup**: Each of the eight spheres is tangent to the three coordinate planes in its respective octant. This means each sphere's center is at a distance of 1 (its radius) from each of the coordinate planes. Thus, the coordinates of the centers of these spheres are permutations of $(1, 1, 1)$ with ... | 0.875 | 5,245.0625 | 4,824.071429 | 8,192 |
The sum of seven integers is $-1$. What is the maximum number of the seven integers that can be larger than $13$? | 6 | To solve this problem, we need to determine the maximum number of integers among the seven that can exceed 13 while still achieving a total sum of $-1$.
1. **Assume the maximum number of integers greater than 13**: Let's denote these integers that are greater than 13 as $x_1, x_2, \ldots, x_k$ where $k$ is the number ... | 0.875 | 5,737.625 | 5,387 | 8,192 |
Given $$\frac {1}{3} \leq a \leq 1$$, if the function $f(x) = ax^2 - 2x + 1$ has a domain of $[1, 3]$.
(1) Find the minimum value of $f(x)$ in its domain (expressed in terms of $a$);
(2) Let the maximum value of $f(x)$ in its domain be $M(a)$, and the minimum value be $N(a)$. Find the minimum value of $M(a) - N(a)$... | \frac {1}{2} | 0.5 | 6,980.9375 | 6,402.125 | 7,559.75 | |
On the AMC 8 contest Billy answers 13 questions correctly, answers 7 questions incorrectly and doesn't answer the last 5. What is his score? | 13 | The AMC 8 scoring system is structured as follows:
- Each correct answer awards 1 point.
- Each incorrect answer awards 0 points.
- Questions left unanswered also award 0 points.
Given the problem statement:
- Billy answers 13 questions correctly.
- Billy answers 7 questions incorrectly.
- Billy does not answer 5 ques... | 0.875 | 1,230.8125 | 872.357143 | 3,740 |
$x$ is a real number with the property that $x+\tfrac1x = 3$. Let $S_m = x^m + \tfrac{1}{x^m}$. Determine the value of $S_7$.
| 843 | 0.9375 | 4,843 | 4,619.733333 | 8,192 | |
How many paths are there from the starting point $C$ to the end point $D$, if every step must be up or to the right in a grid of 8 columns and 7 rows? | 6435 | 0.25 | 3,947.6875 | 3,669.25 | 4,040.5 | |
The hour and minute hands of a clock move continuously and at constant speeds. A moment of time $X$ is called interesting if there exists such a moment $Y$ (the moments $X$ and $Y$ do not necessarily have to be different), so that the hour hand at moment $Y$ will be where the minute hand is at moment $X$, and the minu... | 143 | 0 | 8,192 | -1 | 8,192 | |
How many perfect cubes are between 100 and 900? | 5 | 1 | 2,689.5625 | 2,689.5625 | -1 | |
What is the arithmetic mean of the integers from -4 through 5, inclusive? Express your answer as a decimal to the nearest tenth. | 0.5 | 0.9375 | 3,778.25 | 3,484 | 8,192 | |
A box contains 4 cards, each with one of the following functions defined on \\(R\\): \\(f_{1}(x)={x}^{3}\\), \\(f_{2}(x)=|x|\\), \\(f_{3}(x)=\sin x\\), \\(f_{4}(x)=\cos x\\). Now, if we randomly pick 2 cards from the box and multiply the functions on the cards to get a new function, the probability that the resulting f... | \dfrac{2}{3} | 0.8125 | 4,818 | 4,203.230769 | 7,482 | |
Juan, Carlos and Manu take turns flipping a coin in their respective order. The first one to flip heads wins. What is the probability that Manu will win? Express your answer as a common fraction. | \frac{1}{7} | 0.8125 | 4,976.8125 | 4,234.846154 | 8,192 | |
Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is 1/19 of the original integer. | 95 | 0.8125 | 6,518.875 | 6,227.307692 | 7,782.333333 | |
Two skaters, Allie and Billie, are at points $A$ and $B$, respectively, on a flat, frozen lake. The distance between $A$ and $B$ is $100$ meters. Allie leaves $A$ and skates at a speed of $8$ meters per second on a straight line that makes a $60^\circ$ angle with $AB$. At the same time Allie leaves $A$, Billie leaves $... | 160 | Label the point of intersection as $C$. Since $d = rt$, $AC = 8t$ and $BC = 7t$. According to the law of cosines,
[asy] pointpen=black; pathpen=black+linewidth(0.7); pair A=(0,0),B=(10,0),C=16*expi(pi/3); D(B--A); D(A--C); D(B--C,dashed); MP("A",A,SW);MP("B",B,SE);MP("C",C,N);MP("60^{\circ}",A+(0.3,0),NE);MP("100",(A+B... | 0.4375 | 5,006.0625 | 3,576.714286 | 6,117.777778 |
Consider all 1000-element subsets of the set $\{1, 2, 3, ... , 2015\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.
Hint
Use the Hockey Stick Identity in the form
\[\binom{a}... | 431 | Let $p$ be the size of the large set and $q$ be the size of the subset (i.e. in this problem, $p = 2015$ and $q = 1000$). We can easily find the answers for smaller values of $p$ and $q$:
For $p = 2$ and $q = 2$, the answer is $1$.
For $p = 3$ and $q = 2$, the answer is $\frac43$.
For $p = 4$ and $q = 2$, the answer... | 0.125 | 7,546.8125 | 5,962 | 7,773.214286 |
If the sum of the first $3n$ positive integers is $150$ more than the sum of the first $n$ positive integers, then the sum of the first $4n$ positive integers is | 300 | 1. **Set up the equation based on the problem statement:**
The sum of the first $3n$ positive integers is given by the formula $\frac{3n(3n+1)}{2}$, and the sum of the first $n$ positive integers is given by $\frac{n(n+1)}{2}$. According to the problem, the sum of the first $3n$ integers is $150$ more than the sum o... | 1 | 2,347.1875 | 2,347.1875 | -1 |
Given that \( m \) and \( n \) are two distinct positive integers and the last four digits of \( 2019^{m} \) and \( 2019^{n} \) are the same, find the minimum value of \( m+n \). | 502 | 0.125 | 7,970 | 6,510 | 8,178.571429 | |
The number of books issued from the library to readers constitutes $\frac{1}{16}$ of the number of books on the shelves. After transferring 2000 books from the library to the reading room, the number of books absent from the shelves became $\frac{1}{15}$ of the number of books remaining on the shelves. How many books d... | 544000 | 0 | 3,617.8125 | -1 | 3,617.8125 | |
Out of 500 participants in a remote math olympiad, exactly 30 did not like the problem conditions, exactly 40 did not like the organization of the event, and exactly 50 did not like the method used to determine the winners. A participant is called "significantly dissatisfied" if they were dissatisfied with at least two... | 60 | 0 | 8,192 | -1 | 8,192 | |
Let \( A(2,0) \) be a fixed point on the plane, \( P\left(\sin \left(2 t-60^{\circ}\right), \cos \left(2 t-60^{\circ}\right)\right) \) be a moving point. When \( t \) changes from \( 15^{\circ} \) to \( 45^{\circ} \), the area swept by the line segment \( AP \) is ______. | \frac{\pi}{6} | 0.1875 | 7,590.1875 | 6,262.333333 | 7,896.615385 | |
A steak initially at a temperature of 5°C is put into an oven. After 15 minutes, its temperature reaches 45°C. After another 15 minutes, its temperature is 77°C. The oven maintains a constant temperature. The steak changes temperature at a rate proportional to the difference between its temperature and that of the oven... | 205 | 0.6875 | 6,833.625 | 6,216.181818 | 8,192 | |
Given that $f(x)$ is an odd function defined on $\mathbb{R}$, and for $x \geqslant 0$, $f(x) = \begin{cases} \log_{\frac{1}{2}}(x+1), & 0 \leqslant x < 1 \\ 1-|x-3|, & x \geqslant 1 \end{cases}$, determine the sum of all zeros of the function $y = f(x) + \frac{1}{2}$. | \sqrt{2} - 1 | 0.4375 | 6,145.75 | 5,639.857143 | 6,539.222222 | |
Claudia has 12 coins, each of which is a 5-cent coin or a 10-cent coin. There are exactly 17 different values that can be obtained as combinations of one or more of her coins. How many 10-cent coins does Claudia have? | 6 | 1. **Define Variables:**
Let $x$ be the number of 5-cent coins Claudia has, and let $y$ be the number of 10-cent coins. Since Claudia has 12 coins in total, we have:
\[
x + y = 12
\]
2. **Express Total Value Range:**
The smallest value Claudia can make is 5 cents (using one 5-cent coin) and the largest ... | 0 | 7,945.375 | -1 | 7,945.375 |
Let $p,$ $q,$ $r,$ $s$ be distinct real numbers such that the roots of $x^2 - 12px - 13q = 0$ are $r$ and $s,$ and the roots of $x^2 - 12rx - 13s = 0$ are $p$ and $q.$ Find the value of $p + q + r + s.$ | 1716 | 0 | 8,130.625 | -1 | 8,130.625 | |
In triangle $PQR,$ $S$ is on $\overline{PQ}$ such that $PS:SQ = 4:1,$ and $T$ is on $\overline{QR}$ such that $QT:TR = 4:1.$ If lines $ST$ and $PR$ intersect at $U,$ then find $\frac{ST}{TU}.$ | \frac{1}{3} | 0 | 5,992.8125 | -1 | 5,992.8125 | |
Let $A = (-3, 0),$ $B=(-2,1),$ $C=(2,1),$ and $D=(3,0).$ Suppose that point $P$ satisfies \[PA + PD = PB + PC = 8.\]Then the $y-$coordinate of $P,$ when simplified, can be expressed in the form $\frac{-a + b \sqrt{c}}{d},$ where $a,$ $b,$ $c,$ $d$ are positive integers. Find $a + b + c + d.$ | 35 | 0.5625 | 6,882.5 | 5,929.444444 | 8,107.857143 | |
A rectangle with a diagonal of length $x$ is twice as long as it is wide. What is the area of the rectangle? | \frac{2}{5}x^2 | 1. **Define the dimensions of the rectangle**: Let the width of the rectangle be $l$ and the length be $2l$ (since the length is twice the width).
2. **Expression for the area of the rectangle**: The area $A$ of the rectangle is given by the product of its length and width. Thus,
\[
A = \text{length} \times \tex... | 0.0625 | 1,977.0625 | 1,866 | 1,984.466667 |
Given a sequence $\{a_n\}$ that satisfies $a_{n+1} = a_n - a_{n-1}$ ($n \in N^*, n \geqslant 2$), with $a_1 = 2018$ and $a_2 = 2017$. Let $S_n$ be the sum of the first $n$ terms of the sequence $\{a_n\}$. The value of $S_{100}$ is ______. | 2016 | 0.875 | 4,675.75 | 4,173.428571 | 8,192 | |
The sides of a regular polygon of $n$ sides, $n>4$, are extended to form a star. The number of degrees at each point of the star is: | \frac{(n-2)180}{n} | 1. **Identify the internal angle of the regular polygon**:
The internal angle of a regular polygon with $n$ sides can be calculated using the formula:
\[
\text{Internal angle} = \frac{(n-2) \times 180^\circ}{n}
\]
2. **Calculate the external angle of the polygon**:
The external angle is the supplementa... | 0 | 5,707.9375 | -1 | 5,707.9375 |
Two points $A, B$ are randomly chosen on a circle with radius $100.$ For a positive integer $x$ , denote $P(x)$ as the probability that the length of $AB$ is less than $x$ . Find the minimum possible integer value of $x$ such that $\text{P}(x) > \frac{2}{3}$ . | 174 | 0.625 | 6,190 | 4,988.8 | 8,192 | |
Compute
$$\sum_{k=1}^{2000} k(\lceil \log_{2}{k}\rceil- \lfloor\log_{2}{k} \rfloor).$$ | 1998953 | 0.8125 | 3,353.0625 | 3,187.846154 | 4,069 | |
An equilateral triangle is inscribed in a circle. A smaller equilateral triangle has one vertex coinciding with a vertex of the larger triangle and another vertex on the midpoint of a side of the larger triangle. What percent of the area of the larger triangle is the area of the smaller triangle? | 25\% | 0.0625 | 7,390.25 | 4,679 | 7,571 | |
A store arranges a decorative tower of balls where the top level has 2 balls and each lower level has 3 more balls than the level above. The display uses 225 balls. What is the number of levels in the tower? | 12 | 0 | 8,192 | -1 | 8,192 | |
Compute
\[\cos^6 0^\circ + \cos^6 1^\circ + \cos^6 2^\circ + \dots + \cos^6 90^\circ.\] | \frac{229}{8} | 0.25 | 7,709.5625 | 7,197 | 7,880.416667 | |
A circle with radius 1 is tangent to a circle with radius 3 at point \( C \). A line passing through point \( C \) intersects the smaller circle at point \( A \) and the larger circle at point \( B \). Find \( AC \), given that \( AB = 2\sqrt{5} \). | \frac{\sqrt{5}}{2} | 0 | 6,476.6875 | -1 | 6,476.6875 | |
Simplify $21 \cdot \frac{8}{15} \cdot \frac{1}{14}$. | \frac{4}{5} | 1 | 2,497 | 2,497 | -1 | |
Let $ABC$ be a triangle with $AB=9$ , $BC=10$ , $CA=11$ , and orthocenter $H$ . Suppose point $D$ is placed on $\overline{BC}$ such that $AH=HD$ . Compute $AD$ . | \sqrt{102} | 0.75 | 6,380.5625 | 5,776.75 | 8,192 | |
A rectangle with dimensions $8 \times 2 \sqrt{2}$ and a circle with a radius of 2 have a common center. Find the area of their overlapping region. | 2 \pi + 4 | 0 | 7,915.8125 | -1 | 7,915.8125 | |
Let \( p, q, r, s \) be distinct real numbers such that the roots of \( x^2 - 12px - 13q = 0 \) are \( r \) and \( s \), and the roots of \( x^2 - 12rx - 13s = 0 \) are \( p \) and \( q \). Find the value of \( p + q + r + s \). | -13 | 0 | 7,681.125 | -1 | 7,681.125 | |
The polynomial $P(x)$ is a monic, quartic polynomial with real coefficients, and two of its roots are $\cos \theta + i \sin \theta$ and $\sin \theta + i \cos \theta,$ where $0 < \theta < \frac{\pi}{4}.$ When the four roots of $P(x)$ are plotted in the complex plane, they form a quadrilateral whose area is equal to hal... | 1 + \sqrt{3} | 0.25 | 7,305.25 | 6,222 | 7,666.333333 | |
On the game show $\text{\emph{Wheel of Fraction}}$, you see the following spinner. Given that each region is the same area, what is the probability that you will earn exactly $\$1700$ in your first three spins? Express your answer as a common fraction. [asy]
import olympiad; import geometry; import graph; size(150); de... | \frac{6}{125} | 0.6875 | 5,823.875 | 5,419.545455 | 6,713.4 | |
The pentagon \( A B C D E \) is inscribed around a circle.
The angles \( \angle A B C \), \( \angle B A E \), and \( \angle C D E \) each measure \( 104^\circ \). Find \( \angle A D B \). Provide the answer in degrees (only the number, without units). | 38 | 0 | 8,192 | -1 | 8,192 | |
Find the area bounded by the graph of $y = \arcsin(\cos x)$ and the $x$-axis on the interval $\frac{\pi}{4} \le x \le \frac{9\pi}{4}.$ | 2\pi^2 | 0 | 8,178.25 | -1 | 8,178.25 | |
In a math competition, 5 problems were assigned. There were no two contestants who solved exactly the same problems. However, for any problem that is disregarded, for each contestant there is another contestant who solved the same set of the remaining 4 problems. How many contestants participated in the competition? | 32 | 0.0625 | 7,506.4375 | 7,604 | 7,499.933333 | |
$n$ coins are simultaneously flipped. The probability that two or fewer of them show tails is $\frac{1}{4}$. Find $n$. | n = 5 | 0 | 8,192 | -1 | 8,192 | |
If $x$ is a positive integer, what is the value of $x$ for the equation $(x!-(x-3)!) \div 23 = 1$? | 4 | 1 | 2,111.8125 | 2,111.8125 | -1 | |
Six positive integers are written on the faces of a cube. Each vertex is labeled with the product of the three numbers on the faces adjacent to the vertex. If the sum of the numbers on the vertices is equal to $1386$, then what is the sum of the numbers written on the faces? | 38 | 0 | 7,486.75 | -1 | 7,486.75 | |
Given the function $f(x)=\frac{ax^{2}+bx+c}{e^{x}} (a > 0)$ whose derivative $y=f′(x)$ has two zeros at $-3$ and $0$.
1. Find the monotonic intervals of $f(x)$;
2. If the minimum value of $f(x)$ is $-e^{3}$, find the maximum value of $f(x)$ on the interval $[-5,+\infty)$. | 5e^{5} | 0 | 6,063.4375 | -1 | 6,063.4375 | |
In $\triangle ABC$, with $AB=3$, $AC=4$, $BC=5$, let $I$ be the incenter of $\triangle ABC$ and $P$ be a point inside $\triangle IBC$ (including the boundary). If $\overrightarrow{AP}=\lambda \overrightarrow{AB} + \mu \overrightarrow{AC}$ (where $\lambda, \mu \in \mathbf{R}$), find the minimum value of $\lambda + \mu$. | 7/12 | 0.5 | 6,825.5 | 5,683.625 | 7,967.375 | |
Triangle $ABC$ has an inradius of $5$ and a circumradius of $16$. If $2\cos{B} = \cos{A} + \cos{C}$, then the area of triangle $ABC$ can be expressed as $\frac{a\sqrt{b}}{c}$, where $a, b,$ and $c$ are positive integers such that $a$ and $c$ are relatively prime and $b$ is not divisible by the square of any prime. Comp... | 141 | 0 | 8,105.5625 | -1 | 8,105.5625 | |
A boss plans a business meeting at Starbucks with the two engineers below him. However, he fails to set a time, and all three arrive at Starbucks at a random time between 2:00 and 4:00 p.m. When the boss shows up, if both engineers are not already there, he storms out and cancels the meeting. Each engineer is willing t... | \frac{7}{24} | 0 | 7,856.125 | -1 | 7,856.125 | |
Experts and Viewers play "What? Where? When?" until one side wins six rounds. The probability of Experts winning a single round is 0.6, and there are no ties. Currently, the Experts are losing with a score of 3 to 4. Find the probability that the Experts will eventually win. | 0.4752 | 0 | 8,048.9375 | -1 | 8,048.9375 | |
Calculate the lengths of the arcs of curves given by the equations in the rectangular coordinate system.
$$
y=\ln x, \sqrt{3} \leq x \leq \sqrt{15}
$$ | \frac{1}{2} \ln \frac{9}{5} + 2 | 0 | 6,692.0625 | -1 | 6,692.0625 | |
Given that $\cos (\alpha - \frac{\pi }{3}) - \cos \alpha = \frac{1}{3}$, find the value of $\sin (\alpha - \frac{\pi }{6})$. | \frac{1}{3} | 0.875 | 3,990.875 | 3,390.714286 | 8,192 | |
If $g(x) = 3x^2 + 4$ and $h(x) = -2x^3 + 2$, what is the value of $g(h(2))$? | 592 | 1 | 1,685.125 | 1,685.125 | -1 | |
Tom is searching for the $6$ books he needs in a random pile of $30$ books. What is the expected number of books must he examine before finding all $6$ books he needs? | 14.7 | 0 | 7,468.1875 | -1 | 7,468.1875 | |
Consider three coins where two are fair and a third coin lands on heads with a probability of $\frac{3}{5}$. Alice flips the three coins, and then Bob flips the same three coins. Let $\frac{p}{q}$ be the probability that Alice and Bob get the same number of heads, where $p$ and $q$ are coprime integers. Find $p + q$. | 263 | 0.6875 | 5,922.25 | 4,890.545455 | 8,192 | |
How many different right-angled triangles exist, one of the legs of which is \(\sqrt{2016}\), and the other leg and hypotenuse are expressed in natural numbers? | 12 | 0.8125 | 6,681.8125 | 6,333.307692 | 8,192 | |
Given that vectors $\overrightarrow{α}$ and $\overrightarrow{β}$ are two mutually perpendicular unit vectors in a plane, and $(5\overrightarrow{α} - 2\overrightarrow{γ}) \cdot (12\overrightarrow{β} - 2\overrightarrow{γ}) = 0$, find the maximum value of $|\overrightarrow{γ}|$. | \frac{13}{2} | 0.8125 | 5,232.375 | 4,549.384615 | 8,192 | |
Let $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are integers. Suppose that $f(1)=0$, $50<f(7)<60$, $70<f(8)<80$, $5000k<f(100)<5000(k+1)$ for some integer $k$. What is $k$? | 3 | 0.875 | 4,632.625 | 4,335.357143 | 6,713.5 | |
The function \[f(x) = \left\{ \begin{aligned} 2x + 1 & \quad \text{ if } x < 3 \\ x^2 & \quad \text{ if } x \ge 3 \end{aligned} \right.\] has an inverse $f^{-1}.$ Compute the value of $f^{-1}(-3) + f^{-1}(0) + \dots + f^{-1}(4) + f^{-1}(9).$ | 3.5 | 0 | 4,660.4375 | -1 | 4,660.4375 | |
Evaluate the expression $$\frac{\sin 10°}{1 - \sqrt{3}\tan 10°}.$$ | \frac{1}{2} | 0 | 4,208.25 | -1 | 4,208.25 | |
Let $x,$ $y,$ and $z$ be nonnegative real numbers such that $x + y + z = 8.$ Find the maximum value of
\[\sqrt{3x + 2} + \sqrt{3y + 2} + \sqrt{3z + 2}.\] | 3\sqrt{10} | 1 | 4,118.625 | 4,118.625 | -1 | |
The seventh and tenth terms of a geometric sequence are $7$ and $21$, respectively. What is the $13$th term of this progression? | 63 | 1 | 1,807.3125 | 1,807.3125 | -1 | |
A linear function \( f(x) \) is given. It is known that the distance between the points of intersection of the graphs \( y = x^{2} \) and \( y = f(x) \) is \( 2 \sqrt{3} \), and the distance between the points of intersection of the graphs \( y = x^{2}-2 \) and \( y = f(x)+1 \) is \( \sqrt{60} \). Find the distance be... | 2 \sqrt{11} | 0.6875 | 6,984.0625 | 6,435 | 8,192 | |
A real number $a$ is chosen randomly and uniformly from the interval $[-20, 18]$. The probability that the roots of the polynomial
$x^4 + 2ax^3 + (2a - 2)x^2 + (-4a + 3)x - 2$
are all real can be written in the form $\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | 37 | The polynomial we are given is rather complicated, so we could use Rational Root Theorem to turn the given polynomial into a degree-2 polynomial. With Rational Root Theorem, $x = 1, -1, 2, -2$ are all possible rational roots. Upon plugging these roots into the polynomial, $x = -2$ and $x = 1$ make the polynomial equal ... | 0.9375 | 5,006.4375 | 4,794.066667 | 8,192 |
A clothing retailer offered a discount of $\frac{1}{4}$ on all jackets tagged at a specific price. If the cost of the jackets was $\frac{2}{3}$ of the price they were actually sold for and considering this price included a sales tax of $\frac{1}{10}$, what would be the ratio of the cost to the tagged price?
**A)** $\fr... | \frac{11}{30} | 0.3125 | 4,768.125 | 917.8 | 6,518.272727 |
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