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 |
|---|---|---|---|---|---|---|
The sum of the first n terms of a geometric sequence $\{a_n\}$ is $S_n$. If for any positive integer $k$, it holds that $a_k = \lim_{n\to\infty}(S_n - S_k)$, then the common ratio $q = \boxed{\frac{1}{2}}$. | \frac{1}{2} | 1 | 3,234.375 | 3,234.375 | -1 | |
For how many (not necessarily positive) integer values of $n$ is the value of $4000 \cdot \left(\frac{2}{5}\right)^n$ an integer? | 9 | 1. **Understanding the Expression**: We start by analyzing the expression $4000 \cdot \left(\frac{2}{5}\right)^n$. This expression can be rewritten as:
\[
4000 \cdot \left(\frac{2}{5}\right)^n = 4000 \cdot 2^n \cdot 5^{-n}
\]
This shows that the expression involves powers of 2 and powers of 5.
2. **Conditi... | 0.875 | 3,873 | 3,686.285714 | 5,180 |
For positive integers $n$ and $k$, let $f(n, k)$ be the remainder when $n$ is divided by $k$, and for $n > 1$ let $F(n) = \max_{\substack{1\le k\le \frac{n}{2}}} f(n, k)$. Find the remainder when $\sum\limits_{n=20}^{100} F(n)$ is divided by $1000$. | 512 | $Lemma:$ Highest remainder when $n$ is divided by $1\leq k\leq n/2$ is obtained for $k_0 = (n + (3 - n \pmod{3}))/3$ and the remainder thus obtained is $(n - k_0*2) = [(n - 6)/3 + (2/3)*n \pmod{3}]$.
$Note:$ This is the second highest remainder when $n$ is divided by $1\leq k\leq n$ and the highest remainder occurs whe... | 0 | 8,192 | -1 | 8,192 |
What is the sum of the roots of the equation $(x - 5)^2 = 9$? | 10 | 1 | 1,689.6875 | 1,689.6875 | -1 | |
Does there exist an integer \( n \) such that \( 21n \equiv 1 \mod 74 \)? | 67 | 0.8125 | 4,093.4375 | 3,147.615385 | 8,192 | |
In right triangle $XYZ$ with $\angle YXZ = 90^\circ$, we have $XY = 24$ and $YZ = 25$. Find $\tan Y$. | \frac{7}{24} | 0.9375 | 4,027.5625 | 3,749.933333 | 8,192 | |
The line $y = a$ intersects the curves $y = 2(x + 1)$ and $y = x + \ln x$ at points $A$ and $B$, respectively. Find the minimum value of $|AB|$. | \frac{3}{2} | 0.625 | 6,696.5625 | 5,799.3 | 8,192 | |
Determine the sum $25^2 - 23^2 + 21^2 - 19^2 + ... + 3^2 - 1^2.$ | 1196 | 0 | 7,329.6875 | -1 | 7,329.6875 | |
Convert $135_7$ to a base 10 integer. | 75 | 0.9375 | 2,224.5625 | 1,826.733333 | 8,192 | |
A yogurt shop sells four flavors of yogurt and has six different toppings. How many combinations of one flavor and two different toppings are available? | 60 | 1 | 1,977.3125 | 1,977.3125 | -1 | |
Compute $\left(\sqrt{625681}\right)^2$. | 625681 | 0.9375 | 3,849.9375 | 3,560.466667 | 8,192 | |
Given the graph of the function $y=\cos (x+\frac{4\pi }{3})$ is translated $\theta (\theta > 0)$ units to the right, and the resulting graph is symmetrical about the $y$-axis, determine the smallest possible value of $\theta$. | \frac{\pi }{3} | 0.6875 | 5,869.3125 | 5,210.636364 | 7,318.4 | |
Calculate $\frac{1}{4} \cdot \frac{2}{5} \cdot \frac{3}{6} \cdot \frac{4}{7} \cdots \frac{49}{52} \cdot \frac{50}{53}$. Express your answer as a common fraction. | \frac{1}{23426} | 0.9375 | 4,331.5 | 4,074.133333 | 8,192 | |
At Alpine School, there are 15 soccer players. Every player is enrolled in either physics or mathematics class, but not necessarily both. If 9 players are taking physics and 4 players are registered for both physics and mathematics, how many are taking mathematics? | 10 | 0.875 | 1,071.6875 | 1,158.928571 | 461 | |
In the convex quadrilateral \(ABCD\), the intersection point of its diagonals is \(O\). What is the minimum area of the quadrilateral if the area of triangle \(AOB\) is \(4 \mathrm{~cm}^2\) and the area of triangle \(COD\) is \(9 \mathrm{~cm}^2\)? | 25 | 0.8125 | 5,030 | 4,300.307692 | 8,192 | |
The Stromquist Comet is visible every 61 years. If the comet is visible in 2017, what is the next leap year when the comet will be visible? | 2444 | 0.3125 | 5,179.5 | 5,111.8 | 5,210.272727 | |
A notebook containing 67 pages, numbered from 1 to 67, is renumbered such that the last page becomes the first one, the second-last becomes the second one, and so on. Determine how many pages have the same units digit in their old and new numbering. | 13 | 0.625 | 4,823.375 | 3,970.4 | 6,245 | |
Let $x,$ $y,$ and $z$ be three positive real numbers whose sum is 1. If $z = 2x$ and $y = 3x$, find the minimum value of the product $xyz.$ | \frac{1}{36} | 1 | 4,493.625 | 4,493.625 | -1 | |
Given that $\{a_n\}$ is an arithmetic sequence, $a_1 > 0$, $a_{23} + a_{24} > 0$, and $a_{23} \cdot a_{24} < 0$, determine the maximum positive integer $n$ for which the sum of the first $n$ terms $S_n > 0$. | 46 | 0.1875 | 8,112.4375 | 7,767.666667 | 8,192 | |
A market survey shows that the sales volume and price of a certain product in the past 50 days are functions of the sales time t(days), and the sales volume approximately satisfies f(t)=−2t+200(1≤t≤50,t∈N). The price for the first 30 days is g(x)=12t+30(1≤t≤30,t∈N), and for the last 20 days is g(t)=45(31≤t≤50,t∈N).
(1)... | 54600 | 0.5625 | 5,288.5 | 4,826.777778 | 5,882.142857 | |
If $k \in [-2, 2]$, find the probability that for the value of $k$, there can be two tangents drawn from the point A(1, 1) to the circle $x^2 + y^2 + kx - 2y - \frac{5}{4}k = 0$. | \frac{1}{4} | 0.5625 | 6,953.875 | 6,496 | 7,542.571429 | |
What is the sum of the distinct prime factors of 315? | 15 | 1 | 1,458.125 | 1,458.125 | -1 | |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $(\sin A + \sin B)(a-b) = c(\sin C - \sqrt{3}\sin B)$.
$(1)$ Find the measure of angle $A$;
$(2)$ If $\cos \angle ABC = -\frac{1}{7}$, $D$ is a point on segment $AC$, $\angle ABD = \angle CBD$, $BD = ... | 7\sqrt{3} | 0.75 | 5,878.0625 | 5,411 | 7,279.25 | |
The number $2.29^{\star \star} N$ is an integer. Its representation in base $b$ is 777. Find the smallest positive integer $b$ such that $N$ is a perfect fourth power. | 18 | 0.25 | 7,156.25 | 4,049 | 8,192 | |
Anastasia is taking a walk in the plane, starting from $(1,0)$. Each second, if she is at $(x, y)$, she moves to one of the points $(x-1, y),(x+1, y),(x, y-1)$, and $(x, y+1)$, each with $\frac{1}{4}$ probability. She stops as soon as she hits a point of the form $(k, k)$. What is the probability that $k$ is divisible ... | \frac{3-\sqrt{3}}{3} | The key idea is to consider $(a+b, a-b)$, where $(a, b)$ is where Anastasia walks on. Then, the first and second coordinates are independent random walks starting at 1, and we want to find the probability that the first is divisible by 3 when the second reaches 0 for the first time. Let $C_{n}$ be the $n$th Catalan num... | 0 | 8,048.3125 | -1 | 8,048.3125 |
The length of the interval of solutions of the inequality $a \le 2x + 3 \le b$ is $10$. What is $b - a$? | 20 | 1. **Rewrite the inequality in terms of $x$:**
Given the inequality $a \le 2x + 3 \le b$, we first isolate $x$. Subtract $3$ from all parts of the inequality:
\[
a - 3 \le 2x \le b - 3
\]
2. **Divide by $2$ to solve for $x$:**
Next, divide the entire inequality by $2$ to solve for $x$:
\[
\... | 1 | 1,234.25 | 1,234.25 | -1 |
Let $\mathcal{P}$ be the set of all polynomials $p(x)=x^4+2x^2+mx+n$ , where $m$ and $n$ range over the positive reals. There exists a unique $p(x) \in \mathcal{P}$ such that $p(x)$ has a real root, $m$ is minimized, and $p(1)=99$ . Find $n$ .
*Proposed by **AOPS12142015*** | 56 | 0.375 | 7,494.5 | 6,332 | 8,192 | |
A toy store manager received a large order of Mr. Slinkums just in time for the holidays. The manager places $20\%$ of them on the shelves, leaving the other 120 Mr. Slinkums in storage. How many Mr. Slinkums were in this order? | 150 | 1 | 1,314.125 | 1,314.125 | -1 | |
Given a function $f(x)=\log _{a}\left(\sqrt {x^{2}+1}+x\right)+\dfrac{1}{a^{x}-1}+\dfrac{3}{2}$, where $a > 0$ and $a \neq 1$. If $f\left(\log _{3}b\right)=5$ for $b > 0$ and $b \neq 1$, find the value of $f\left(\log _{\frac{1}{3}}b\right)$. | -3 | 0.8125 | 4,539.0625 | 4,059.615385 | 6,616.666667 | |
The function $f$ defined by $f(x)= \frac{ax+b}{cx+d}$, where $a$,$b$,$c$ and $d$ are nonzero real numbers, has the properties $f(19)=19$, $f(97)=97$ and $f(f(x))=x$ for all values except $\frac{-d}{c}$. Find the unique number that is not in the range of $f$. | 58 | 0.75 | 5,315 | 4,569.666667 | 7,551 | |
Anton ran down a moving escalator and counted 30 steps. Then he decided to run up the same escalator at the same speed relative to the escalator and counted 150 steps. How many steps did he count while descending with a policeman on a stationary escalator? | 50 | 0.25 | 7,437.8125 | 5,727.75 | 8,007.833333 | |
In right triangle $PQR$, $PQ=15$, $QR=8$, and angle $R$ is a right angle. A semicircle is inscribed in the triangle such that it touches $PQ$ and $QR$ at their midpoints and the hypotenuse $PR$. What is the radius of the semicircle?
A) $\frac{24}{5}$
B) $\frac{12}{5}$
C) $\frac{17}{4}$
D) $\frac{15}{3}$ | \frac{24}{5} | 0 | 7,775.125 | -1 | 7,775.125 | |
Let $r$ be the speed in miles per hour at which a wheel, $15$ feet in circumference, travels. If the time for a complete rotation of the wheel is shortened by $\frac{1}{3}$ of a second, the speed $r$ is increased by $4$ miles per hour. Determine the original speed $r$.
A) 9
B) 10
C) 11
D) 12
E) 13 | 12 | 0 | 8,192 | -1 | 8,192 | |
A function \(f(x)\) is defined for all real numbers \(x\). For all non-zero values \(x\), we have
\[3f\left(x\right) + f\left(\frac{1}{x}\right) = 15x + 8.\]
Let \(S\) denote the sum of all of the values of \(x\) for which \(f(x) = 2004\). Compute the integer nearest to \(S\). | 356 | 0.75 | 6,456.375 | 5,877.833333 | 8,192 | |
The sum of the first n terms of the sequence {a_n} is S_n = n^2 + n + 1, and b_n = (-1)^n a_n (n ∈ N^*). Determine the sum of the first 50 terms of the sequence {b_n}. | 49 | 0.25 | 7,398.75 | 6,218.5 | 7,792.166667 | |
How many whole numbers between 1 and 2000 do not contain the digits 1 or 2? | 511 | 0.5 | 6,562.125 | 6,063.375 | 7,060.875 | |
Find all real solutions to $x^{4}+(2-x)^{4}=34$. | 1 \pm \sqrt{2} | Let $y=2-x$, so $x+y=2$ and $x^{4}+y^{4}=34$. We know $$(x+y)^{4}=x^{4}+4 x^{3} y+6 x^{2} y^{2}+4 x y^{3}+y^{4}=x^{4}+y^{4}+2 x y(2 x^{2}+2 y^{2}+3 x y) .$$ Moreover, $x^{2}+y^{2}=(x+y)^{2}-2 x y$, so the preceding equation becomes $2^{4}=34+2 x y(2. 2^{2}-x y)$, or $(x y)^{2}-8 x y-9=0$. Hence $x y=9$ or -1 . Solving ... | 0 | 3,826.4375 | -1 | 3,826.4375 |
In the triangle $ABC$, the side lengths are given as $AB=\sqrt{2}$, $BC=\sqrt{5}$, and $AC=3$. Compare the measure of the angle $\angle BOC$ to $112.5^{\circ}$, where $O$ is the center of the circle inscribed in triangle $ABC$. | 112.5 | 0.75 | 5,423.625 | 4,500.833333 | 8,192 | |
If $n$ is a positive integer, let $s(n)$ denote the sum of the digits of $n$. We say that $n$ is zesty if there exist positive integers $x$ and $y$ greater than 1 such that $x y=n$ and $s(x) s(y)=s(n)$. How many zesty two-digit numbers are there? | 34 | Let $n$ be a zesty two-digit number, and let $x$ and $y$ be as in the problem statement. Clearly if both $x$ and $y$ are one-digit numbers, then $s(x) s(y)=n \neq s(n)$. Thus either $x$ is a two-digit number or $y$ is. Assume without loss of generality that it is $x$. If $x=10 a+b, 1 \leq a \leq 9$ and $0 \leq b \leq 9... | 0 | 8,191.6875 | -1 | 8,191.6875 |
A cyclist traveled from point A to point B, stayed there for 30 minutes, and then returned to A. On the way to B, he overtook a pedestrian, and met him again 2 hours later on his way back. The pedestrian arrived at point B at the same time the cyclist returned to point A. How much time did it take the pedestrian to tra... | 10 | 0 | 8,192 | -1 | 8,192 | |
Suppose $n^{*}$ means $\frac{1}{n}$, the reciprocal of $n$. For example, $5^{*}=\frac{1}{5}$. How many of the following statements are true?
i) $3^*+6^*=9^*$
ii) $6^*-4^*=2^*$
iii) $2^*\cdot 6^*=12^*$
iv) $10^*\div 2^* =5^*$ | 2 | We will evaluate each statement one by one using the definition $n^* = \frac{1}{n}$.
**Statement i) $3^* + 6^* = 9^*$**
\[
3^* + 6^* = \frac{1}{3} + \frac{1}{6} = \frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}
\]
\[
9^* = \frac{1}{9}
\]
Since $\frac{1}{2} \neq \frac{1}{9}$, statement i) is **false**.
**Stateme... | 0.9375 | 1,632.375 | 1,582.933333 | 2,374 |
Given the volume of the right prism $ABCD-A_{1}B_{1}C_{1}D_{1}$ is equal to the volume of the cylinder with the circumscribed circle of square $ABCD$ as its base, calculate the ratio of the lateral area of the right prism to that of the cylinder. | \sqrt{2} | 0.5625 | 4,571.4375 | 3,160 | 6,386.142857 | |
Everyday at school, Jo climbs a flight of $6$ stairs. Jo can take the stairs $1$, $2$, or $3$ at a time. For example, Jo could climb $3$, then $1$, then $2$. In how many ways can Jo climb the stairs? | 24 | 1. **Define the recursive function**: Let $f(n)$ represent the number of ways Jo can climb to the $n$-th step. We start by defining the base cases:
- $f(0) = 1$: There is only one way to be on the ground (by starting there).
- $f(1) = 1$: There is only one way to reach the first step, which is by taking a single ... | 0.875 | 5,164.875 | 4,732.428571 | 8,192 |
The number $10!$ ($10$ is written in base $10$), when written in the base $12$ system, ends with exactly $k$ zeros. The value of $k$ is | 4 | 1. **Understanding the problem**: We need to find how many zeros are at the end of $10!$ when it is expressed in base $12$. This means we need to determine the highest power of $12$ that divides $10!$.
2. **Factorizing $12$**: Since $12 = 2^2 \cdot 3$, we need to find the number of times $2$ and $3$ appear as factors ... | 0.9375 | 3,420.0625 | 3,101.933333 | 8,192 |
A $\text{palindrome}$, such as $83438$, is a number that remains the same when its digits are reversed. The numbers $x$ and $x+32$ are three-digit and four-digit palindromes, respectively. What is the sum of the digits of $x$? | 24 |
1. **Identify the range of $x$ and $x+32$:**
- Since $x$ is a three-digit palindrome, the maximum value of $x$ is $999$.
- Consequently, the maximum value of $x+32$ is $999 + 32 = 1031$.
- The minimum value of $x+32$ is $1000$ (since it is a four-digit palindrome).
2. **Determine the possible values for $x+3... | 1 | 3,335.3125 | 3,335.3125 | -1 |
Let $f(x)$ and $g(x)$ be two monic cubic polynomials, and let $r$ be a real number. Two of the roots of $f(x)$ are $r + 2$ and $r + 8$. Two of the roots of $g(x)$ are $r + 5$ and $r + 11$, and
\[f(x) - g(x) = 2r\] for all real numbers $x$. Find $r$. | 20.25 | 0 | 7,477.3125 | -1 | 7,477.3125 | |
Given the function \( y = \sqrt{2x^2 + 2} \) with the graph represented by the curve \( G \). The curve \( G \) has a focus at \( F \). A line \( l_1 \) passing through \( F \) intersects the curve \( G \) at points \( A \) and \( C \), and another line \( l_2 \) passing through \( F \) intersects the curve \( G \) at ... | 16 | 0 | 8,192 | -1 | 8,192 | |
Given that $15^{-1} \equiv 31 \pmod{53}$, find $38^{-1} \pmod{53}$, as a residue modulo 53. | 22 | 0 | 6,497.4375 | -1 | 6,497.4375 | |
If $\alpha$ and $\beta$ are acute angles, and $\sin \alpha = \frac{\sqrt{5}}{5}$, $\cos \beta = \frac{3\sqrt{10}}{10}$, then $\sin (\alpha + \beta) =$____, $\alpha + \beta =$____. | \frac{\pi}{4} | 0.4375 | 4,326.1875 | 4,036 | 4,551.888889 | |
Given that $a$ and $b$ are coefficients, and the difference between $ax^2+2xy-x$ and $3x^2-2bxy+3y$ does not contain a quadratic term, find the value of $a^2-4b$. | 13 | 0.9375 | 2,279.25 | 1,885.066667 | 8,192 | |
Let $P=\{1,2,\ldots,6\}$, and let $A$ and $B$ be two non-empty subsets of $P$. Find the number of pairs of sets $(A,B)$ such that the maximum number in $A$ is less than the minimum number in $B$. | 129 | 0 | 7,994.5 | -1 | 7,994.5 | |
What is the value of $\sqrt{3^3 + 3^3 + 3^3}$? | 9 | 1 | 1,272.4375 | 1,272.4375 | -1 | |
A club consists of five leaders and some regular members. Each year, all leaders leave the club and each regular member recruits three new people to join as regular members. Subsequently, five new leaders are elected from outside the club to join. Initially, there are eighteen people in total in the club. How many peop... | 3164 | 0.625 | 4,668.8125 | 4,886 | 4,306.833333 | |
The sequence $a_{1}, a_{2}, a_{3}, \ldots$ of real numbers satisfies the recurrence $a_{n+1}=\frac{a_{n}^{2}-a_{n-1}+2 a_{n}}{a_{n-1}+1}$. Given that $a_{1}=1$ and $a_{9}=7$, find $a_{5}$. | 3 | Let $b_{n}=a_{n}+1$. Then the recurrence becomes $b_{n+1}-1=\left(b_{n}^{2}-b_{n-1}\right) / b_{n-1}=b_{n}^{2} / b_{n-1}-1$, so $b_{n+1}=b_{n}^{2} / b_{n-1}$. It follows that the sequence $\left(b_{n}\right)$ is a geometric progression, from which $b_{5}^{2}=b_{1} b_{9}=2 \cdot 8=16 \Rightarrow b_{5}= \pm 4$. However, ... | 0.0625 | 7,997.0625 | 5,073 | 8,192 |
Given: In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$, and it is known that $\frac {\cos A-2\cos C}{\cos B}= \frac {2c-a}{b}$.
$(1)$ Find the value of $\frac {\sin C}{\sin A}$;
$(2)$ If $\cos B= \frac {1}{4}$ and $b=2$, find the area $S$ of $\triangle ABC$. | \frac { \sqrt {15}}{4} | 0 | 5,843.0625 | -1 | 5,843.0625 | |
A certain coin is weighted such that the chance of flipping heads is $\frac{1}{3}$ and the chance of flipping tails is $\frac{2}{3}$. Suppose that we win $\$3$ if we flip a heads on a coin toss, but lose $\$2$ if we flip tails. What is the expected value, in dollars, of our winnings after one flip? Express your answ... | -\frac{1}{3} | 1 | 1,612.6875 | 1,612.6875 | -1 | |
Find the length of the common chord of the circle $x^{2}+y^{2}=50$ and $x^{2}+y^{2}-12x-6y+40=0$. | 2\sqrt{5} | 1 | 3,911.8125 | 3,911.8125 | -1 | |
In a large 15 by 20 rectangular region, one quarter area of the rectangle is shaded. If the shaded quarter region itself represents one fourth of its quarter area, calculate the fraction of the total area that is shaded.
A) $\frac{1}{16}$
B) $\frac{1}{12}$
C) $\frac{1}{4}$
D) $\frac{3}{20}$
E) $\frac{1}{5}$ | \frac{1}{16} | 0 | 4,157.8125 | -1 | 4,157.8125 | |
Convert the binary number \(11111011111_2\) to its decimal representation. | 2015 | 0.6875 | 6,146.8125 | 5,217.181818 | 8,192 | |
In the arithmetic sequence $\{a_n\}$, $a_{10} < 0$, $a_{11} > 0$ and $a_{11} > |a_{10}|$. If the sum of the first $n$ terms of $\{a_n\}$, denoted as $S_n$, is less than $0$, the maximum value of $n$ is ____. | 19 | 0.5625 | 7,015.125 | 6,099.777778 | 8,192 | |
A five-digit number \(abcde\) satisfies:
\[ a < b, \, b > c > d, \, d < e, \, \text{and} \, a > d, \, b > e. \]
For example, 34 201, 49 412. If the digit order's pattern follows a variation similar to the monotonicity of a sine function over one period, then the five-digit number is said to follow the "sine rule." Fin... | 2892 | 0 | 8,131.75 | -1 | 8,131.75 | |
A line initially 1 inch long grows according to the following law, where the first term is the initial length.
\[1+\frac{1}{4}\sqrt{2}+\frac{1}{4}+\frac{1}{16}\sqrt{2}+\frac{1}{16}+\frac{1}{64}\sqrt{2}+\frac{1}{64}+\cdots\]
If the growth process continues forever, the limit of the length of the line is: | \frac{1}{3}(4+\sqrt{2}) | 1. **Identify the pattern and separate the terms**:
The given series is:
\[
1 + \frac{1}{4}\sqrt{2} + \frac{1}{4} + \frac{1}{16}\sqrt{2} + \frac{1}{16} + \frac{1}{64}\sqrt{2} + \frac{1}{64} + \cdots
\]
We can observe that this series can be split into two separate series: one involving powers of $\frac{1... | 0 | 6,582.5 | -1 | 6,582.5 |
The distance a dog covers in 3 steps is the same as the distance a fox covers in 4 steps and the distance a rabbit covers in 12 steps. In the time it takes the rabbit to run 10 steps, the dog runs 4 steps and the fox runs 5 steps. Initially, the distances between the dog, fox, and rabbit are as shown in the diagram. Wh... | 40 | 0 | 8,067.0625 | -1 | 8,067.0625 | |
The circle having $(0,0)$ and $(8,6)$ as the endpoints of a diameter intersects the $x$-axis at a second point. What is the $x$-coordinate of this point? | 8 | 1. **Identify the center and radius of the circle:**
Given points $(0,0)$ and $(8,6)$ are endpoints of a diameter. The center of the circle, $(h,k)$, is the midpoint of the diameter. Using the midpoint formula:
\[
h = \frac{0+8}{2} = 4, \quad k = \frac{0+6}{2} = 3
\]
Therefore, the center of the circle i... | 1 | 3,818.5 | 3,818.5 | -1 |
Calculate the product of the base nine numbers $35_9$ and $47_9$, express it in base nine, and find the base nine sum of the digits of this product. Additionally, subtract $2_9$ from the sum of the digits. | 22_9 | 0 | 4,642.25 | -1 | 4,642.25 | |
Two balls are randomly chosen from a box containing 20 balls numbered from 1 to 20. Calculate the probability that the sum of the numbers on the two balls is divisible by 3. | \frac{32}{95} | 0.6875 | 5,024.6875 | 3,585 | 8,192 | |
Let $m$ be the smallest positive, three-digit integer congruent to 6 (mod 13). Let $n$ be the smallest positive, four-digit integer congruent to 7 (mod 17). What is $n-m$? | 900 | 1 | 3,510.9375 | 3,510.9375 | -1 | |
In a workshop, each participant has a 1 in 40 chance of being late. What is the probability that out of any three participants chosen at random, exactly one will be late? Express your answer as a percent rounded to the nearest tenth. | 7.1\% | 0.9375 | 3,276.875 | 2,949.2 | 8,192 | |
The three-digit positive integer $N$ has a ones digit of 3. What is the probability that $N$ is divisible by 3? Express your answer as a common fraction. | \frac{1}{3} | 1 | 5,184 | 5,184 | -1 | |
What is the greatest possible sum of two consecutive integers whose product is less than 400? | 39 | 0.9375 | 5,135.8125 | 4,932.066667 | 8,192 | |
Let $\omega$ be a complex number such that $\omega^7 = 1$ and $\omega \ne 1.$ Compute
\[\omega^{16} + \omega^{18} + \omega^{20} + \dots + \omega^{54}.\] | -1 | 0.125 | 7,753.4375 | 7,460.5 | 7,795.285714 | |
There exist constants $b_1, b_2, b_3, b_4, b_5, b_6, b_7$ such that
\[
\cos^7 \theta = b_1 \cos \theta + b_2 \cos 2 \theta + b_3 \cos 3 \theta + b_4 \cos 4 \theta + b_5 \cos 5 \theta + b_6 \cos 6 \theta + b_7 \cos 7 \theta
\]
for all angles $\theta$. Find $b_1^2 + b_2^2 + b_3^2 + b_4^2 + b_5^2 + b_6^2 + b_7^2$. | \frac{429}{1024} | 0.3125 | 6,957.6875 | 5,188.6 | 7,761.818182 | |
A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet?
[asy]
size(250);defaultpen(linewidth(0.8));
draw(ellipse(origin, 3, 1)... | 240 | 0.9375 | 4,216.875 | 3,951.866667 | 8,192 | |
Find the length of the chord that the line given by the parametric equations
$$\begin{cases} x=1+ \frac {4}{5}t \\ y=-1- \frac {3}{5}t \end{cases}$$
(where t is the parameter) cuts off from the curve whose polar equation is $\rho= \sqrt {2}\cos\left(\theta+ \frac {\pi}{4}\right)$. | \frac {7}{5} | 0.8125 | 4,982.9375 | 4,581.538462 | 6,722.333333 | |
A 10-digit arrangement $ 0,1,2,3,4,5,6,7,8,9$ is called *beautiful* if (i) when read left to right, $ 0,1,2,3,4$ form an increasing sequence, and $ 5,6,7,8,9$ form a decreasing sequence, and (ii) $ 0$ is not the leftmost digit. For example, $ 9807123654$ is a beautiful arrangement. Determine the number of bea... | 126 | 0.5625 | 6,911.875 | 5,916.222222 | 8,192 | |
If $y=\frac{12x^4+4x^3+9x^2+5x+3}{3x^4+2x^3+8x^2+3x+1}$, at what value of $y$ will there be a horizontal asymptote? | 4 | 1 | 1,598.875 | 1,598.875 | -1 | |
Find the sum of all angles $x \in [0^\circ, 360^\circ]$ that satisfy
\[\sin^5 x - \cos^5 x = \frac{1}{\cos x} - \frac{1}{\sin x}.\] | 270^\circ | 0.4375 | 7,309.625 | 6,175.142857 | 8,192 | |
Let \\(\alpha\\) be an acute angle. If \\(\cos (\alpha+ \dfrac {\pi}{6})= \dfrac {3}{5}\\), then \\(\sin (\alpha- \dfrac {\pi}{6})=\\) \_\_\_\_\_\_. | \dfrac {4-3 \sqrt {3}}{10} | 0 | 7,000 | -1 | 7,000 | |
If we count by $3\text{'s}$ starting with $1,$ the following sequence is obtained: $1,$ $4,$ $7,$ $10,$ $\dots.$ What is the $100^\text{th}$ number in the sequence? | 298 | 1 | 2,184.1875 | 2,184.1875 | -1 | |
Triangle $ABC$ is isosceles with $AC = BC$ and $\angle ACB = 106^\circ.$ Point $M$ is in the interior of the triangle so that $\angle MAC = 7^\circ$ and $\angle MCA = 23^\circ.$ Find the number of degrees in $\angle CMB.$
[asy] pointpen = black; pathpen = black+linewidth(0.7); size(220); /* We will WLOG AB = 2 to draw ... | 83 | Without loss of generality, let $AC = BC = 1$. Then, using the Law of Sines in triangle $AMC$, we get $\frac {1}{\sin 150} = \frac {MC}{\sin 7}$, and using the sine addition formula to evaluate $\sin 150 = \sin (90 + 60)$, we get $MC = 2 \sin 7$.
Then, using the Law of Cosines in triangle $MCB$, we get $MB^2 = 4\sin^2... | 0.3125 | 7,749.375 | 6,775.6 | 8,192 |
Suppose $a, b, c$, and $d$ are pairwise distinct positive perfect squares such that $a^{b}=c^{d}$. Compute the smallest possible value of $a+b+c+d$. | 305 | Note that if $a$ and $c$ are divisible by more than one distinct prime, then we can just take the prime powers of a specific prime. Thus, assume $a$ and $c$ are powers of a prime $p$. Assume $a=4^{x}$ and $c=4^{y}$. Then $x b=y d$. Because $b$ and $d$ are squares, the ratio of $x$ to $y$ is a square, so assume $x=1$ an... | 0 | 8,192 | -1 | 8,192 |
Given positive integer $n$ and $r$ pairwise distinct primes $p_1,p_2,\cdots,p_r.$ Initially, there are $(n+1)^r$ numbers written on the blackboard: $p_1^{i_1}p_2^{i_2}\cdots p_r^{i_r} (0 \le i_1,i_2,\cdots,i_r \le n).$
Alice and Bob play a game by making a move by turns, with Alice going first. In Alice's round, she e... | M^{\lfloor \frac{n}{2} \rfloor} |
Given positive integer \( n \) and \( r \) pairwise distinct primes \( p_1, p_2, \cdots, p_r \). Initially, there are \( (n+1)^r \) numbers written on the blackboard: \( p_1^{i_1} p_2^{i_2} \cdots p_r^{i_r} \) where \( 0 \le i_1, i_2, \cdots, i_r \le n \).
Alice and Bob play a game by making a move by turns, with Ali... | 0 | 8,192 | -1 | 8,192 |
Find the units digit of the sum, $$ 1! + 2! + 3! + \cdots + 2006!. $$ | 3 | 1 | 1,824.6875 | 1,824.6875 | -1 | |
In the rectangular prism \(ABCD-A_1B_1C_1D_1\), \(AB=2\), \(AA_1=AD=1\). Points \(E\), \(F\), and \(G\) are the midpoints of edges \(AA_1\), \(C_1D_1\), and \(BC\) respectively. What is the volume of the tetrahedron \(B_1-EFG\)? | \frac{3}{8} | 0.875 | 5,651.625 | 5,288.714286 | 8,192 | |
A sequence $ (S_n), n \geq 1$ of sets of natural numbers with $ S_1 = \{1\}, S_2 = \{2\}$ and
\[{ S_{n + 1} = \{k \in }\mathbb{N}|k - 1 \in S_n \text{ XOR } k \in S_{n - 1}\}.
\]
Determine $ S_{1024}.$ | 1024 | 0 | 7,999.5625 | -1 | 7,999.5625 | |
If the positive real numbers \(a\) and \(b\) satisfy \(\frac{1}{a} + \frac{1}{b} \leq 2 \sqrt{2}\) and \((a - b)^2 = 4 (ab)^3\), then \(\log_a b =\) ? | -1 | 0.125 | 8,047.75 | 7,038 | 8,192 | |
Given that \\(\theta\\) is an angle in the fourth quadrant, and \\(\sin (\theta+ \frac {\pi}{4})= \frac {3}{5}\\), then \\(\tan (\theta- \frac {\pi}{4})=\\) \_\_\_\_\_\_ . | - \frac {4}{3} | 0.875 | 5,434.75 | 5,040.857143 | 8,192 | |
A bear is in the center of the left down corner of a $100*100$ square .we call a cycle in this grid a bear cycle if it visits each square exactly ones and gets back to the place it started.Removing a row or column with compose the bear cycle into number of pathes.Find the minimum $k$ so that in any bear cycle we ca... | 5000 | 0.0625 | 8,010.8125 | 8,192 | 7,998.733333 | |
The complex number $2+i$ and the complex number $\frac{10}{3+i}$ correspond to points $A$ and $B$ on the complex plane, calculate the angle $\angle AOB$. | \frac{\pi}{4} | 0.75 | 3,535.3125 | 3,613.25 | 3,301.5 | |
A class has 50 students, and their scores in a math test $\xi$ follow a normal distribution $N(100, 10^2)$. It is known that $P(90 \leq \xi \leq 100) = 0.3$. Estimate the number of students scoring above 110. | 10 | 0.25 | 7,406.9375 | 6,598.25 | 7,676.5 | |
Let $M$ be the second smallest positive integer that is divisible by every positive integer less than 10 and includes at least one prime number greater than 10. Find the sum of the digits of $M$. | 18 | 0.9375 | 5,308.4375 | 5,116.2 | 8,192 | |
An amoeba is placed in a puddle one day, and on that same day, it splits into two amoebas with a probability of 0.8. Each subsequent day, every amoeba in the puddle has a probability of 0.8 to split into two new amoebas. After one week, assuming no amoebas die, how many amoebas are there in the puddle on average? (Assu... | 26.8435456 | 0 | 6,643.375 | -1 | 6,643.375 | |
A bag contains 4 identical balls, numbered 0, 1, 2, and 2. Player A draws a ball and puts it back, then player B draws a ball. If the number on the drawn ball is larger, that player wins (if the numbers are the same, it's a tie). What is the probability that player B draws the ball numbered 1, given that player A wins? | \frac{2}{5} | 0.0625 | 6,634.25 | 4,202 | 6,796.4 | |
Given that $\tbinom{n}{k}=\tfrac{n!}{k!(n-k)!}$ , the value of $$ \sum_{n=3}^{10}\frac{\binom{n}{2}}{\binom{n}{3}\binom{n+1}{3}} $$ can be written in the form $\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m+n$ . | 329 | 0.0625 | 7,654.375 | 5,158 | 7,820.8 | |
Let $z$ be a complex number such that
\[ |z - 2i| + |z - 5| = 7. \]
Find the minimum value of $|z|$. | \sqrt{\frac{100}{29}} | 0 | 8,192 | -1 | 8,192 | |
Given the function $f(x)=(\sqrt{3}\cos{x}-\sin{x})\sin{x}$, where $x \in \mathbb{R}$.
(Ⅰ) Find the smallest positive period of $f(x)$ and the intervals on which $f(x)$ is monotonic increasing;
(Ⅱ) Find the maximum and minimum values of $f(x)$ on the interval $\left[0, \frac{\pi}{4}\right]$. | \frac{1}{2} | 0 | 5,550.75 | -1 | 5,550.75 | |
What is the largest $2$-digit prime factor of the integer $n = {300\choose 150}$? | 97 | 0.625 | 6,640.5 | 5,869.3 | 7,925.833333 | |
What is the maximum number of rooks that can be placed in an $8 \times 8 \times 8$ cube so that no two rooks attack each other? | 64 | 0.3125 | 6,724.625 | 6,267.4 | 6,932.454545 | |
Let \(x, y, z\) be nonzero real numbers such that \(x + y + z = 0\) and \(xy + xz + yz \neq 0\). Find all possible values of
\[
\frac{x^7 + y^7 + z^7}{xyz (xy + xz + yz)}.
\] | -7 | 0 | 7,942.5 | -1 | 7,942.5 | |
How many solutions in natural numbers \(x, y\) does the system of equations have?
$$
\left\{\begin{array}{l}
\text{GCD}(x, y) = 20! \\
\text{LCM}(x, y) = 30!
\end{array}\right.
$$
(where \(n! = 1 \cdot 2 \cdot 3 \cdot \ldots \cdot n\)) | 1024 | 0.5 | 6,408.8125 | 6,055 | 6,762.625 |
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