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 |
|---|---|---|---|---|---|---|
Ria writes down the numbers $1,2,\cdots, 101$ in red and blue pens. The largest blue number is equal to the number of numbers written in blue and the smallest red number is equal to half the number of numbers in red. How many numbers did Ria write with red pen? | 68 | 0.9375 | 3,355.125 | 3,032.666667 | 8,192 | |
How many ways can the eight vertices of a three-dimensional cube be colored red and blue such that no two points connected by an edge are both red? Rotations and reflections of a given coloring are considered distinct. | 35 | We do casework on $R$, the number of red vertices. Let the cube be called $A B C D E F G H$, with opposite faces $A B C D$ and $E F G H$, such that $A$ is directly above $E$. - $\underline{R=0}$ : There is one such coloring, which has only blue vertices. - $\underline{R}=1$ : There are 8 ways to choose the red vertex, ... | 0 | 8,023.8125 | -1 | 8,023.8125 |
A trapezoid $ABCD$ lies on the $xy$ -plane. The slopes of lines $BC$ and $AD$ are both $\frac 13$ , and the slope of line $AB$ is $-\frac 23$ . Given that $AB=CD$ and $BC< AD$ , the absolute value of the slope of line $CD$ can be expressed as $\frac mn$ , where $m,n$ are two relatively prime positive... | 1706 | 0.375 | 7,469.6875 | 6,471.166667 | 8,068.8 | |
Given that $\sin\alpha=\frac{\sqrt{5}}{5}$ and $\sin\beta=\frac{\sqrt{10}}{10}$, where both $\alpha$ and $\beta$ are acute angles, find the value of $\alpha+\beta$. | \frac{\pi}{4} | 0.875 | 3,734.0625 | 3,480 | 5,512.5 | |
Except for the first two terms, each term of the sequence $2000, y, 2000 - y,\ldots$ is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encountered. What positive integer $y$ produces a sequence of maximum length? | 1236 | 0 | 8,192 | -1 | 8,192 | |
In rectangle $ABCD$, $AB = 4$ and $BC = 8$. The rectangle is folded so that points $B$ and $D$ coincide, forming the pentagon $ABEFC$. What is the length of segment $EF$? Express your answer in simplest radical form. | \sqrt{10} | 0 | 4,694.8125 | -1 | 4,694.8125 | |
Given a real coefficient fourth-degree polynomial with a leading coefficient of 1 that has four imaginary roots, where the product of two of the roots is \(32+\mathrm{i}\) and the sum of the other two roots is \(7+\mathrm{i}\), determine the coefficient of the quadratic term. | 114 | 0.1875 | 8,041.1875 | 7,387.666667 | 8,192 | |
A line with slope equal to $-1$ and a line with slope equal to $-2$ intersect at the point $P(2,5)$. What is the area of $\triangle PQR$? | 6.25 | 0 | 6,197.375 | -1 | 6,197.375 | |
The number $989 \cdot 1001 \cdot 1007+320$ can be written as the product of three distinct primes $p, q, r$ with $p<q<r$. Find $(p, q, r)$. | (991,997,1009) | Let $f(x)=x(x-12)(x+6)+320=x^{3}-6 x^{2}-72 x+320$, so that $f(1001)=989 \cdot 1001 \cdot 1007+320$. But $f(4)=4(-8)(10)+320=0$, so $f(x)=(x-4)\left(x^{2}-2 x-80\right)=(x-4)(x-10)(x+8)$ Thus $f(1001)=991 \cdot 997 \cdot 1009$, as desired. | 0.125 | 7,989.75 | 6,735.5 | 8,168.928571 |
Evaluate $\sqrt{2 -\!\sqrt{2 - \!\sqrt{2 - \!\sqrt{2 - \cdots}}}}$. | 1 | 1 | 2,705 | 2,705 | -1 | |
In triangle $PQR$, $\angle Q=90^\circ$, $PQ=15$ and $QR=20$. Points $S$ and $T$ are on $\overline{PR}$ and $\overline{QR}$, respectively, and $\angle PTS=90^\circ$. If $ST=12$, then what is the length of $PS$? | 15 | 0.0625 | 7,735 | 6,731 | 7,801.933333 | |
What is the smallest integer greater than $-\frac{17}{3}$? | -5 | 1 | 1,828.75 | 1,828.75 | -1 | |
Given the ellipse $C$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$, passing through point $Q(\sqrt{2}, 1)$ and having the right focus at $F(\sqrt{2}, 0)$,
(I) Find the equation of the ellipse $C$;
(II) Let line $l$: $y = k(x - 1) (k > 0)$ intersect the $x$-axis, $y$-axis, and ellipse $C$ at points $C$, ... | \frac{\sqrt{42}}{2} | 0 | 7,092.3125 | -1 | 7,092.3125 | |
Given that the sequence $\{a_n\}$ is an arithmetic sequence with a common difference of $2$, and $a_1$, $a_2$, $a_5$ form a geometric sequence, find the value of $s_8$ (the sum of the first 8 terms). | 64 | 1 | 2,103.5 | 2,103.5 | -1 | |
A tourist is learning an incorrect way to sort a permutation $(p_{1}, \ldots, p_{n})$ of the integers $(1, \ldots, n)$. We define a fix on two adjacent elements $p_{i}$ and $p_{i+1}$, to be an operation which swaps the two elements if $p_{i}>p_{i+1}$, and does nothing otherwise. The tourist performs $n-1$ rounds of fix... | 1009! \cdot 1010! | Note that the given algorithm is very similar to the well-known Bubble Sort algorithm for sorting an array. The exception is that in the $i$-th round through the array, the first $i-1$ pairs are not checked. We claim a necessary and sufficient condition for the array to be sorted after the tourist's process is: for all... | 0 | 7,864.125 | -1 | 7,864.125 |
Consider a memorable $9$-digit telephone number defined as $d_1d_2d_3d_4-d_5d_6d_7d_8d_9$. A number is memorable if the prefix sequence $d_1d_2d_3d_4$ is exactly the same as either of the sequences $d_5d_6d_7d_8$ or $d_6d_7d_8d_9$. Each digit $d_i$ can be any of the ten decimal digits $0$ through $9$. Find the number o... | 199990 | 0 | 6,251.75 | -1 | 6,251.75 | |
Using only pennies, nickels, dimes, quarters, and half-dollars, determine the smallest number of coins needed to pay any amount of money less than a dollar and a half. | 10 | 0 | 8,135.0625 | -1 | 8,135.0625 | |
The fictional country of Isoland uses an alphabet with ten unique letters: A, B, D, E, I, L, N, O, R, U. License plates in Isoland are structured with five letters. How many different license plates are possible if they must begin with A or I, end with R, cannot include the letter B, and no letters may repeat? | 420 | 0.625 | 5,373.125 | 5,360.6 | 5,394 | |
$\sqrt{8}+\sqrt{18}=$ | $5\sqrt{2}$ | 1. Simplify $\sqrt{8}$:
\[
\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}
\]
2. Simplify $\sqrt{18}$:
\[
\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2}
\]
3. Add the simplified terms:
\[
2\sqrt{2} + 3\sqrt{2} = (2+3)\sqrt{2} = 5\sqrt{2}
\]
4. Match ... | 0 | 1,599.75 | -1 | 1,599.75 |
If the square roots of a positive number are $2a+6$ and $3-a$, then the value of $a$ is ____. | -9 | 0.5 | 785.9375 | 627.375 | 944.5 | |
Find the equation of the circle that is tangent to the x-axis, has its center on the line $3x - y = 0$, and the chord cut by the line $x - y = 0$ has a length of 2. | \frac{9}{7} | 0 | 5,085.875 | -1 | 5,085.875 | |
From the numbers $1,2,3, \cdots, 2014$, select 315 different numbers (order does not matter) to form an arithmetic sequence. Among these, the number of ways to form an arithmetic sequence that includes the number 1 is ___. The total number of ways to form an arithmetic sequence is ___. | 5490 | 0.4375 | 5,972.5 | 4,398.142857 | 7,197 | |
Given $f(\alpha)=\dfrac{\sin(\alpha-3\pi)\cdot \cos(2\pi-\alpha)\cdot \sin(-\alpha+\frac{3}{2}\pi)}{\cos(-\pi-\alpha)\cdot \sin(-\pi-\alpha)}$,
(1) Simplify $f(\alpha)$;
(2) If $\sin(\alpha-\frac{3}{2}\pi)=\frac{1}{5}$, find the value of $f(\alpha)$. | -\frac{1}{5} | 0.8125 | 4,556 | 4,289.692308 | 5,710 | |
Among 6 courses, if person A and person B each choose 3 courses, the number of ways in which exactly 1 course is chosen by both A and B is \_\_\_\_\_\_. | 180 | 0.625 | 5,946.6875 | 4,599.5 | 8,192 | |
Let $m = 2^{20}5^{15}.$ How many positive integer divisors of $m^2$ are less than $m$ but do not divide $m$? | 299 | 0.125 | 6,031.875 | 3,244.5 | 6,430.071429 | |
The value of $a$ is chosen so that the number of roots of the first equation $4^{x}-4^{-x}=2 \cos(a x)$ is 2007. How many roots does the second equation $4^{x}+4^{-x}=2 \cos(a x)+4$ have for the same value of $a$? | 4014 | 0 | 8,192 | -1 | 8,192 | |
Find all triples $ (x,y,z)$ of real numbers that satisfy the system of equations
\[ \begin{cases}x^3 \equal{} 3x\minus{}12y\plus{}50, \\ y^3 \equal{} 12y\plus{}3z\minus{}2, \\ z^3 \equal{} 27z \plus{} 27x. \end{cases}\]
[i]Razvan Gelca.[/i] | (2, 4, 6) |
We are given the system of equations:
\[
\begin{cases}
x^3 = 3x - 12y + 50, \\
y^3 = 12y + 3z - 2, \\
z^3 = 27z + 27x.
\end{cases}
\]
To find all triples \((x, y, z)\) of real numbers that satisfy these equations, we analyze the behavior of the functions involved.
1. **Case \(x > 2\):**
- For \(x > 2\), the funct... | 0 | 8,192 | -1 | 8,192 |
Find the remainder when \(5x^4 - 9x^3 + 3x^2 - 7x - 30\) is divided by \(3x - 9\). | 138 | 0.9375 | 3,634.9375 | 3,605.2 | 4,081 | |
Two digits of a number were swapped, and as a result, it increased by more than 3 times. The resulting number is 8453719. Find the original number. | 1453789 | 0 | 8,172.375 | -1 | 8,172.375 | |
Find the number of functions $f(n),$ taking the integers to the integers, such that
\[f(a + b) + f(ab) = f(a) f(b) + 1\]for all integers $a$ and $b.$ | 3 | 0 | 8,150.0625 | -1 | 8,150.0625 | |
How many whole numbers between 1 and 2000 do not contain the digits 1 or 2? | 511 | 0.375 | 6,716.5 | 5,601.666667 | 7,385.4 | |
A factory produces a certain product for the Brazilian Olympic Games with an annual fixed cost of 2.5 million yuan. For every $x$ thousand units produced, an additional cost of $C(x)$ (in ten thousand yuan) is incurred. When the annual production is less than 80 thousand units, $C(x)=\frac{1}{3}x^2+10x$; when the annua... | 100 | 0.5625 | 6,749.375 | 5,627.333333 | 8,192 | |
Given the parametric equation of curve $C\_1$ as $\begin{cases} x=2\cos \theta \ y=\sqrt{3}\sin \theta \end{cases}(\theta \text{ is the parameter})$, and curve $C\_2$ has a polar coordinate equation of $\rho=2$ with the origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis.
(1) Write the Cartesian... | 2\sqrt{7} | 0.8125 | 5,757.375 | 5,473 | 6,989.666667 | |
For each integer $n\geq 4$, let $a_n$ denote the base-$n$ number $0.\overline{133}_n$. The product $a_4a_5 \dotsm a_{99}$ can be expressed as $\frac{m}{n!}$, where $m$ and $n$ are positive integers and $n$ is as small as possible. What is the value of $m$? | 962 | 0.0625 | 7,989.3125 | 4,949 | 8,192 | |
Given $a\in\{1,3,5\}$ and $b\in\{2,4,8\}$, find the probability that the function $y=\log_{\frac{b}{a}}{\frac{1}{x}}$ is an increasing function. | \frac{1}{3} | 0.5625 | 6,255.625 | 5,533.555556 | 7,184 | |
If the set $\{1, a, \frac{b}{a}\} = \{0, a^2, a+b\}$, find the value of $a^{2015} + b^{2016}$. | -1 | 0.3125 | 6,809.3125 | 6,111.8 | 7,126.363636 | |
In triangle $ABC$, $\angle ABC = 90^\circ$ and $AD$ is an angle bisector. If $AB = 90,$ $BC = x$, and $AC = 2x - 6,$ then find the area of $\triangle ADC$. Round your answer to the nearest integer. | 1363 | 0.625 | 6,456.125 | 5,662.6 | 7,778.666667 | |
Given the function $f(x)=\cos(2x+\frac{\pi}{3})+1$, in triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively.
$(I)$ If angles $A$, $B$, and $C$ form an arithmetic sequence, find the value of $f(B)$;
$(II)$ If $f\left(\frac{B}{2}-\frac{\pi}{6}\right)=\frac{7}{4}$, and sides $a$... | 3+\sqrt{2} | 0.375 | 7,363.9375 | 5,983.833333 | 8,192 | |
Ten gangsters are standing on a flat surface, and the distances between them are all distinct. At twelve o’clock, when the church bells start chiming, each of them fatally shoots the one among the other nine gangsters who is the nearest. At least how many gangsters will be killed? | 7 |
In this problem, we need to determine the minimum number of gangsters who will be killed when each gangster shoots the nearest of the other nine gangsters. As all distances between the gangsters are distinct, each gangster has a unique nearest neighbor.
Consider the following steps to determine the number of killed g... | 0 | 8,147.625 | -1 | 8,147.625 |
Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that, for any real numbers $x$ and $y$, \[ f(f(x)f(y)) + f(x+y) = f(xy). \]
[i] | f(x) = 0f(x) = 1 - xf(x) = x - 1 |
Let \(\mathbb{R}\) be the set of real numbers. We are tasked with finding all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that for any real numbers \( x \) and \( y \), the functional equation:
\[
f(f(x)f(y)) + f(x+y) = f(xy)
\]
is satisfied.
### Step 1: Checking Simple Functions
#### Case 1: Constan... | 0 | 8,044.125 | -1 | 8,044.125 |
Define the determinant $D_1$ = $|1|$ , the determinant $D_2$ = $|1 1|$ $|1 3|$
, and the determinant $D_3=$
|1 1 1|
|1 3 3|
|1 3 5|
.
In general, for positive integer n, let the determinant $D_n$ have 1s in every position of its first row and first
colu... | 12 | 0.1875 | 7,715.375 | 5,650 | 8,192 | |
What is $\left(\frac{7}{8}\right)^3 \cdot \left(\frac{7}{8}\right)^{-3}$? | 1 | 1 | 1,697.125 | 1,697.125 | -1 | |
Calculate the value of $\log_{2}9 \cdot \log_{3}5 \cdot \log_{\sqrt{5}}8 = \_\_\_\_\_\_.$ | 12 | 0.875 | 3,940.125 | 3,493.071429 | 7,069.5 | |
Andrew and John are both Beatles fans. Their respective collections share nine of the same albums. Andrew has seventeen albums in his collection. Six albums are in John's collection, but not Andrew's. How many albums are in either Andrew's or John's collection, but not both? | 14 | 0.9375 | 2,412.0625 | 2,026.733333 | 8,192 | |
For how many integer values of $n$ between 1 and 180 inclusive does the decimal representation of $\frac{n}{180}$ terminate? | 20 | 0.625 | 5,992 | 4,718.1 | 8,115.166667 | |
Given $u$ and $v$ are complex numbers such that $|u+v|=2$ and $|u^2+v^2|=8,$ find the smallest possible value of $|u^3+v^3|$. | 20 | 0.3125 | 7,687.75 | 6,578.4 | 8,192 | |
Find the number of pairs $(m, n)$ of integers which satisfy the equation $m^3 + 6m^2 + 5m = 27n^3 + 27n^2 + 9n + 1$. | 0 | We start by analyzing the given equation:
\[ m^3 + 6m^2 + 5m = 27n^3 + 27n^2 + 9n + 1. \]
We can rewrite the left-hand side as:
\[ m^3 + 6m^2 + 5m = m(m^2 + 6m + 5) = m(m+1)(m+5). \]
This factorization is achieved by recognizing that $m^2 + 6m + 5$ can be factored as $(m+1)(m+5)$.
Now, the equation becomes:
\[ m(m+1)... | 0.1875 | 7,878.6875 | 6,521 | 8,192 |
An [i]animal[/i] with $n$ [i]cells[/i] is a connected figure consisting of $n$ equal-sized cells[1].
A [i]dinosaur[/i] is an animal with at least $2007$ cells. It is said to be [i]primitive[/i] it its cells cannot be partitioned into two or more dinosaurs. Find with proof the maximum number of cells in a primitive din... | 4n-3 |
A **dinosaur** is a **polyomino** having at least 2007 cells that is also **primitive**, meaning it cannot be split into smaller dinosaurs. We need to determine the maximum number of cells in a primitive dinosaur.
To tackle this problem, let's consider a primitive dinosaur with \( n \) cells. The goal is to determine... | 0 | 6,321.5625 | -1 | 6,321.5625 |
A natural number $k > 1$ is called *good* if there exist natural numbers $$ a_1 < a_2 < \cdots < a_k $$ such that $$ \dfrac{1}{\sqrt{a_1}} + \dfrac{1}{\sqrt{a_2}} + \cdots + \dfrac{1}{\sqrt{a_k}} = 1 $$ .
Let $f(n)$ be the sum of the first $n$ *[good* numbers, $n \geq$ 1. Find the sum of all values of $n... | 18 | 0.0625 | 8,124.6875 | 7,635 | 8,157.333333 | |
The area of the triangular region bounded by the $x$-axis, the $y$-axis and the line with equation $y=2x-6$ is one-quarter of the area of the triangular region bounded by the $x$-axis, the line with equation $y=2x-6$ and the line with equation $x=d$, where $d>0$. What is the value of $d$? | 9 | The line with equation $y=2x-6$ has $y$-intercept -6. Also, the $x$-intercept of $y=2x-6$ occurs when $y=0$, which gives $0=2x-6$ or $2x=6$ which gives $x=3$. Therefore, the triangle bounded by the $x$-axis, the $y$-axis, and the line with equation $y=2x-6$ has base of length 3 and height of length 6, and so has area $... | 0.3125 | 5,487.125 | 3,872.4 | 6,221.090909 |
Two dice appear to be normal dice with their faces numbered from $1$ to $6$, but each die is weighted so that the probability of rolling the number $k$ is directly proportional to $k$. The probability of rolling a $7$ with this pair of dice is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Fin... | 71 | It is easier to think of the dice as $21$ sided dice with $6$ sixes, $5$ fives, etc. Then there are $21^2=441$ possible rolls. There are $2\cdot(1\cdot 6+2\cdot 5+3\cdot 4)=56$ rolls that will result in a seven. The odds are therefore $\frac{56}{441}=\frac{8}{63}$. The answer is $8+63=\boxed{071}$
See also 2006 AMC 12... | 0.9375 | 2,886.5625 | 2,532.866667 | 8,192 |
Calculate the sum of the following fractions: $\frac{1}{12} + \frac{2}{12} + \frac{3}{12} + \frac{4}{12} + \frac{5}{12} + \frac{6}{12} + \frac{7}{12} + \frac{8}{12} + \frac{9}{12} + \frac{65}{12} + \frac{3}{4}$.
A) $\frac{119}{12}$
B) $9$
C) $\frac{113}{12}$
D) $10$ | \frac{119}{12} | 0.1875 | 3,642.5 | 806.666667 | 4,296.923077 | |
Points $A$ and $B$ are 10 units apart. Points $B$ and $C$ are 4 units apart. Points $C$ and $D$ are 3 units apart. If $A$ and $D$ are as close as possible, then the number of units between them is | 3 | 1. **Understanding the Problem:**
- We are given four points $A$, $B$, $C$, and $D$ with specific distances between them: $AB = 10$, $BC = 4$, and $CD = 3$.
- We need to find the minimum possible distance between $A$ and $D$.
2. **Applying the Triangle Inequality to $ABC$:**
- By the triangle inequality, for ... | 0.75 | 5,804.4375 | 5,008.583333 | 8,192 |
Given $a \gt 0$, $b \gt 0$, and $a+2b=1$, find the minimum value of $\frac{{b}^{2}+a+1}{ab}$. | 2\sqrt{10} + 6 | 0 | 7,775.375 | -1 | 7,775.375 | |
How many pairs of positive integers \( (m, n) \) satisfy \( m^2 \cdot n < 30 \)? | 41 | 0.4375 | 7,104 | 5,705.142857 | 8,192 | |
Given that $n\in N^{*}$, select $k(k\in N, k\geqslant 2)$ numbers $j\_1$, $j\_2$, $...$, $j\_k$ from the set ${1,2,3,...,n}$ such that they simultaneously satisfy the following two conditions: $①1\leqslant j\_1 < j\_2 < ...j\_k\leqslant n$; $②j_{i+1}-j_{i}\geqslant m(i=1,2,…,k-1)$. Then the array $(j\_1, j\_2, ..., j\_... | 10 | 0.625 | 6,547.5625 | 5,560.9 | 8,192 | |
Find the number of non-congruent scalene triangles whose sides all have integral length, and the longest side has length $11$ . | 20 | 0.5625 | 7,368.6875 | 6,728.333333 | 8,192 | |
How many whole numbers between 1 and 2000 do not contain the digits 1 or 2? | 1535 | 0 | 7,181.8125 | -1 | 7,181.8125 | |
George is planning a dinner party for three other couples, his wife, and himself. He plans to seat the four couples around a circular table for 8, and wants each husband to be seated opposite his wife. How many seating arrangements can he make, if rotations and reflections of each seating arrangement are not considered... | 24 | 0.75 | 5,625 | 5,141.583333 | 7,075.25 | |
Anton colors a cell in a \(4 \times 50\) rectangle. He then repeatedly chooses an uncolored cell that is adjacent to at most one already colored cell. What is the maximum number of cells that can be colored? | 150 | 0 | 8,192 | -1 | 8,192 | |
Vasya wrote a set of distinct natural numbers on the board, each of which does not exceed 2023. It turned out that for any two written numbers \(a\) and \(b\), the number \(a + b\) is not divisible by the number \(a - b\). What is the maximum number of numbers Vasya might have written? | 675 | 0.0625 | 8,034 | 5,664 | 8,192 | |
Given that $\binom{17}{10}=19448$, $\binom{17}{11}=12376$ and $\binom{19}{12}=50388$, find $\binom{18}{12}$. | 18564 | 0.625 | 5,737.875 | 4,265.4 | 8,192 | |
Given the ellipse \( x^{2}+ \frac {y^{2}}{b^{2}+1}=1(b > 0) \) has an eccentricity of \( \frac {\sqrt {10}}{10} \), determine the value of \( b \). | \frac{1}{3} | 0.6875 | 4,116.1875 | 3,828.363636 | 4,749.4 | |
23 people attend a party. Each person shakes hands with at least two other people. What is the minimum possible number of handshakes? | 23 | 0.9375 | 3,575.0625 | 3,267.266667 | 8,192 | |
Find the maximum value of the expression \((\sqrt{8-4 \sqrt{3}} \sin x - 3 \sqrt{2(1+\cos 2x)} - 2) \cdot (3 + 2 \sqrt{11 - \sqrt{3}} \cos y - \cos 2y)\). If the answer is a non-integer, round it to the nearest whole number. | 33 | 0.3125 | 7,261.625 | 5,681.2 | 7,980 | |
The difference between the cube and the square of a number has the form $a b c a b c$ (in the decimal system). What is this number? | 78 | 0.25 | 7,647.6875 | 6,014.75 | 8,192 | |
On a plane with 100 seats, there are 100 passengers, each with an assigned seat. The first passenger ignores the assigned seat and randomly sits in one of the 100 seats. After that, each subsequent passenger either sits in their assigned seat if it is available or chooses a random seat if their assigned seat is taken. ... | \frac{1}{2} | 0.375 | 7,279.75 | 5,759.333333 | 8,192 | |
In the right triangle \( \triangle ABC \),
\[
\angle A = 90^\circ, \, AB = AC
\]
\( M \) and \( N \) are the midpoints of \( AB \) and \( AC \) respectively. \( D \) is an arbitrary point on the segment \( MN \) (excluding points \( M \) and \( N \)). The extensions of \( BD \) and \( CD \) intersect \( AC \) and \( ... | 4\sqrt{2} | 0.3125 | 7,007.6875 | 4,402.2 | 8,192 | |
Given \( x, y, z > 0 \) and \( x + y + z = 1 \), find the maximum value of
$$
f(x, y, z) = \sum \frac{x(2y - z)}{1 + x + 3y}.
$$ | 1/7 | 0 | 8,192 | -1 | 8,192 | |
Let $\mathbf{M}$ be a matrix, and let $\mathbf{v}$ and $\mathbf{w}$ be vectors, such that
\[\mathbf{M} \mathbf{v} = \begin{pmatrix} 1 \\ -5 \end{pmatrix} \quad \text{and} \quad \mathbf{M} \mathbf{w} = \begin{pmatrix} 7 \\ 2 \end{pmatrix}.\]Compute $\mathbf{M} (-2 \mathbf{v} + \mathbf{w}).$ | \begin{pmatrix} 5 \\ 12 \end{pmatrix} | 1 | 1,528.625 | 1,528.625 | -1 | |
Four circles are inscribed such that each circle touches the midpoint of each side of a square. The side of the square is 10 cm, and the radius of each circle is 5 cm. Determine the area of the square not covered by any circle. | 100 - 50\pi | 0 | 7,355.125 | -1 | 7,355.125 | |
Let $f(x)$ be an odd function. Is $f(f(x))$ even, odd, or neither?
Enter "odd", "even", or "neither". | \text{odd} | 0.5 | 2,073.3125 | 2,076 | 2,070.625 | |
A cube is painted so that one side is blue, two sides are red, and three sides are green. How many different such cubes can be painted? Two cubes are considered the same if one cube can be rotated in any way to match the second cube. | 3 | 0 | 8,192 | -1 | 8,192 | |
Small lights are hung on a string $6$ inches apart in the order red, red, green, green, green, red, red, green, green, green, and so on continuing this pattern of $2$ red lights followed by $3$ green lights. How many feet separate the 3rd red light and the 21st red light? | 22.5 | 1. **Identify the pattern and the position of red lights**: The pattern of lights is 2 red followed by 3 green. This pattern repeats every 5 lights. The red lights occur at positions 1, 2, 6, 7, 11, 12, and so on in each repeating group.
2. **Determine the position of the 3rd and 21st red lights**:
- The 3rd red li... | 0.1875 | 7,450.75 | 7,674.333333 | 7,399.153846 |
In trapezoid $ABCD$, the parallel sides $AB$ and $CD$ have lengths of 8 and 20 units, respectively, and the altitude is 12 units. Points $E$ and $F$ are the midpoints of sides $AD$ and $BC$, respectively. What is the area of quadrilateral $EFCD$ in square units? | 102 | 0.8125 | 5,672.9375 | 5,091.615385 | 8,192 | |
In the expansion of $((x^2+1)^2(x-1)^6)$, find the coefficient of the $x^3$ term. | -32 | 0.8125 | 6,132.875 | 5,657.692308 | 8,192 | |
Frederik wants to make a special seven-digit password. Each digit of his password occurs exactly as many times as its digit value. The digits with equal values always occur consecutively, e.g., 4444333 or 1666666. How many possible passwords can he make?
A) 6
B) 7
C) 10
D) 12
E) 13 | 13 | 0 | 8,192 | -1 | 8,192 | |
How many whole numbers from $1$ through $46$ are divisible by either $3$ or $5$ or both? | 21 | 1. **Count numbers divisible by 3**:
To find how many numbers between 1 and 46 are divisible by 3, we calculate:
\[
\left\lfloor \frac{46}{3} \right\rfloor = 15
\]
This means there are 15 numbers divisible by 3.
2. **Count numbers divisible by 5**:
Similarly, to find how many numbers between 1 and ... | 1 | 2,782.125 | 2,782.125 | -1 |
Given that $x > 0$ and $y > 0$, and $\frac{4}{x} + \frac{3}{y} = 1$.
(I) Find the minimum value of $xy$ and the values of $x$ and $y$ when the minimum value is obtained.
(II) Find the minimum value of $x + y$ and the values of $x$ and $y$ when the minimum value is obtained. | 7 + 4\sqrt{3} | 0.1875 | 5,079.875 | 5,628.666667 | 4,953.230769 | |
Ryosuke is picking up his friend from work. The odometer reads 74,568 when he picks his friend up, and it reads 74,592 when he drops his friend off at his house. Ryosuke's car gets 28 miles per gallon and the price of one gallon of gas is $\$4.05$. What was the cost of the gas that was used for Ryosuke to drive his fri... | \$3.47 | 1 | 3,090.75 | 3,090.75 | -1 | |
What is the sum of all three-digit numbers \( n \) for which \(\frac{3n+2}{5n+1}\) is not in its simplest form? | 70950 | 0.875 | 4,566.0625 | 4,048.071429 | 8,192 | |
20. Given an ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \left( a > b > 0 \right)$ passes through point $M\left( 1,\frac{3}{2} \right)$, $F_1$ and $F_2$ are the two foci of ellipse $C$, and $\left| MF_1 \right|+\left| MF_2 \right|=4$, $O$ is the center of ellipse $C$.
(1) Find the equation of ellipse $C$;
(2)... | \frac{9}{2} | 0.9375 | 6,055.25 | 5,912.8 | 8,192 | |
Let $ABCD$ be a unit square. Let $Q_1$ be the midpoint of $\overline{CD}$. For $i=1,2,\dots,$ let $P_i$ be the intersection of $\overline{AQ_i}$ and $\overline{BD}$, and let $Q_{i+1}$ be the foot of the perpendicular from $P_i$ to $\overline{CD}$. What is
\[\sum_{i=1}^{\infty} \text{Area of } \triangle DQ_i P_i \, ?\] | \frac{1}{4} |
#### Step 1: Understanding the Problem
We are given a unit square $ABCD$ and a sequence of points $Q_i$ and $P_i$ defined recursively. We need to find the sum of the areas of triangles $\triangle DQ_iP_i$ for all $i \geq 1$.
#### Step 2: Analyzing the First Triangle $\triangle DQ_1P_1$
- $Q_1$ is the midpoint of $\ov... | 0.75 | 5,334.125 | 4,708 | 7,212.5 |
Solve for $n$, if $8^n\cdot8^n\cdot8^n=64^3$. | 2 | 1 | 1,844.5 | 1,844.5 | -1 | |
A game wheel is divided into six regions labeled $A$, $B$, $C$, $D$, $E$, and $F$. The probability of the wheel stopping on region $A$ is $\frac{1}{3}$, the probability it stops on region $B$ is $\frac{1}{6}$, and the probability of it stopping on regions $C$ and $D$ are equal, as are the probabilities for regions $E$ ... | \frac{1}{8} | 0.8125 | 4,319 | 4,070.769231 | 5,394.666667 | |
If a pot can hold 2 cakes at a time and it takes 5 minutes to cook both sides of a cake, calculate the minimum time it will take to cook 3 cakes thoroughly. | 15 | 0.3125 | 498.8125 | 493 | 501.454545 | |
Let $R$ be a unit square region and $n \geq 4$ an integer. A point $X$ in the interior of $R$ is called n-ray partitional if there are $n$ rays emanating from $X$ that divide $R$ into $n$ triangles of equal area. How many points are $100$-ray partitional but not $60$-ray partitional?
$\textbf{(A)}\ 1500 \qquad \textbf{... | 2320 | 0 | 8,081.8125 | -1 | 8,081.8125 | |
Remove all perfect squares and perfect cubes from the set
$$
A=\left\{n \mid n \leqslant 10000, n \in \mathbf{Z}_{+}\right\}
$$
and arrange the remaining elements in ascending order. What is the 2014th element of this sequence? | 2068 | 0.6875 | 6,977.0625 | 6,865.454545 | 7,222.6 | |
Define $m(n)$ to be the greatest proper natural divisor of $n\in \mathbb{N}$ . Find all $n \in \mathbb{N} $ such that $n+m(n) $ is a power of $10$ .
*N. Agakhanov* | 75 | 0 | 8,192 | -1 | 8,192 | |
In space, there are 3 planes and a sphere. How many distinct ways can a second sphere be placed in space so that it touches the three given planes and the first sphere? (In this problem, sphere touching is considered, i.e., it is not assumed that the spheres can only touch externally.) | 16 | 0 | 7,841.625 | -1 | 7,841.625 | |
Given $f(x)=\sqrt{3}\cos^2{\omega}x+\sin{\omega}x\cos{\omega}x (\omega>0)$, if there exists a real number $x_{0}$ such that for any real number $x$, $f(x_{0})\leq f(x)\leq f(x_{0}+2022\pi)$ holds, then the minimum value of $\omega$ is ____. | \frac{1}{4044} | 0.1875 | 6,997.8125 | 6,159.333333 | 7,191.307692 | |
On an island, there are only knights, who always tell the truth, and liars, who always lie. There are at least two knights and at least two liars. One day, each islander pointed to each of the others in turn and said either "You are a knight!" or "You are a liar!". The phrase "You are a liar!" was said exactly 230 time... | 526 | 0.1875 | 7,457.5625 | 4,917.666667 | 8,043.692308 | |
No math tournament exam is complete without a self referencing question. What is the product of
the smallest prime factor of the number of words in this problem times the largest prime factor of the
number of words in this problem | 1681 | 0.0625 | 4,203.25 | 3,019 | 4,282.2 | |
Let $N$ be the smallest positive integer $N$ such that $2008N$ is a perfect square and $2007N$ is a perfect cube. Find the remainder when $N$ is divided by $25$ . | 17 | 0.6875 | 5,412.0625 | 4,892.272727 | 6,555.6 | |
For what value of the parameter \( p \) will the sum of the squares of the roots of the equation
\[
x^{2}+(3 p-2) x-7 p-1=0
\]
be minimized? What is this minimum value? | \frac{53}{9} | 1 | 2,678.5625 | 2,678.5625 | -1 | |
The two digits in Jack's age are the same as the digits in Bill's age, but in reverse order. In five years Jack will be twice as old as Bill will be then. What is the difference in their current ages? | 18 | 1. **Define the ages**: Let Jack's age be represented as $\overline{ab} = 10a + b$ and Bill's age as $\overline{ba} = 10b + a$, where $a$ and $b$ are the tens and units digits of their ages, respectively.
2. **Age in five years**: In five years, Jack's age will be $10a + b + 5$ and Bill's age will be $10b + a + 5$.
3... | 1 | 3,185.0625 | 3,185.0625 | -1 |
Let \(\alpha\) and \(\beta\) be the two real roots of the quadratic equation \(x^{2} - 2kx + k + 20 = 0\). Find the minimum value of \((\alpha+1)^{2} + (\beta+1)^{2}\), and determine the value of \(k\) for which this minimum value is achieved. | 18 | 0 | 5,371.4375 | -1 | 5,371.4375 | |
For how many integer values of $x$ is $x^2 < 7x$? | 6 | 1 | 1,881.875 | 1,881.875 | -1 | |
How many distinct four letter arrangements can be formed by rearranging the letters found in the word **FLUFFY**? For example, FLYF and ULFY are two possible arrangements. | 72 | 0.3125 | 7,132.3125 | 6,741.2 | 7,310.090909 | |
Given the function $y=\sin (3x+ \frac {\pi}{3})\cos (x- \frac {\pi}{6})+\cos (3x+ \frac {\pi}{3})\sin (x- \frac {\pi}{6})$, find the equation of one of the axes of symmetry. | \frac {\pi}{12} | 0.4375 | 7,301.0625 | 6,155.571429 | 8,192 |
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