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 |
|---|---|---|---|---|---|---|
Define
\[
d_k = k + \cfrac{1}{3k + \cfrac{1}{3k + \cfrac{1}{3k + \dotsb}}}.
\]
Calculate $\sum_{k = 1}^{10} (d_k^2 + 2)$. | 405 | 0 | 8,139.875 | -1 | 8,139.875 | |
There are 36 students in a club. If any 33 of them attend a session, girls will always be in the majority. However, if 31 students attend, it might happen that boys are in the majority. How many girls are in the club? | 20 | 0.25 | 7,085.9375 | 4,056 | 8,095.916667 | |
100 people participated in a quick calculation test consisting of 10 questions. The number of people who answered each question correctly is given in the table below:
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Problem Number & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline
Number of Correct Answers & 93 & 90 ... | 62 | 0 | 5,359.5 | -1 | 5,359.5 | |
There are two pairs $(x,y)$ of real numbers that satisfy the equation $x+y = 3xy = 4$. Given that the solutions $x$ are in the form $x = \frac{a \pm b\sqrt{c}}{d}$ where $a$, $b$, $c$, and $d$ are positive integers and the expression is completely simplified, what is the value of $a + b + c + d$? | 17 | 0.9375 | 4,824.125 | 4,599.6 | 8,192 | |
Given the sequence $\{a_n\}$ satisfies $a_1=3$, and for any integer $n$, always has $(a_{n+1}-1)(1-a_n)=2a_n$, calculate the sum of the first 2018 terms of the sequence $\{a_n\}$. | 589 | 1 | 3,069.3125 | 3,069.3125 | -1 | |
In an isosceles trapezoid \(ABCD\), the side \(AB\) and the shorter base \(BC\) are both equal to 2, and \(BD\) is perpendicular to \(AB\). Find the area of this trapezoid. | 3\sqrt{3} | 0 | 8,123.625 | -1 | 8,123.625 | |
Compute the sum \[\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor + \cdots + \lfloor \sqrt{25} \rfloor.\] | 75 | 0.8125 | 5,531.75 | 4,917.846154 | 8,192 | |
Suppose I have 6 shirts and 5 ties. How many shirt-and-tie outfits can I make? | 30 | 1 | 609.0625 | 609.0625 | -1 | |
In the diagram, the area of triangle $ABC$ is 27 square units. What is the area of triangle $BCD$?
[asy]
draw((0,0)--(32,0)--(9,15)--(0,0));
dot((0,0));
label("$A$",(0,0),SW);
label("6",(3,0),S);
dot((6,0));
label("$C$",(6,0),S);
label("26",(19,0),S);
dot((32,0));
label("$D$",(32,0),SE);
dot((9,15));
label("$B$",(9,... | 117 | 0.6875 | 6,740.5 | 6,080.727273 | 8,192 | |
All of the roots of $x^3+ax^2+bx+c$ are positive integers greater than $2$ , and the coefficients satisfy $a+b+c+1=-2009$ . Find $a$ | -58 | 0.6875 | 5,332.125 | 4,032.181818 | 8,192 | |
What is the number of units in the area of the circle with center at $P$ and passing through $Q$? Express your answer in terms of $\pi$.
[asy]
size(150); pair P = (-3,4), Q=(9,-3); string stringpair(pair p){return "$("+string(p.x)+", "+string(p.y)+"$)";}
draw((-15,0)--(15,0),Arrows(4)); draw((0,-15)--(0,15),Arrows(4))... | 193\pi | 0.9375 | 1,301.125 | 1,297.533333 | 1,355 | |
Tracy had a bag of candies, and none of the candies could be broken into pieces. She ate $\frac{1}{3}$ of them and then gave $\frac{1}{4}$ of what remained to her friend Rachel. Tracy and her mom then each ate 15 candies from what Tracy had left. Finally, Tracy's brother took somewhere from one to five candies, leaving... | 72 | 1 | 2,950.9375 | 2,950.9375 | -1 | |
A parallelogram in the coordinate plane has vertices at points (2,1), (7,1), (5,6), and (10,6). Calculate the sum of the perimeter and the area of the parallelogram. | 35 + 2\sqrt{34} | 0.875 | 5,692.125 | 5,335 | 8,192 | |
There are 10 cities in a state, and some pairs of cities are connected by roads. There are 40 roads altogether. A city is called a "hub" if it is directly connected to every other city. What is the largest possible number of hubs? | 6 | If there are $h$ hubs, then $\binom{h}{2}$ roads connect the hubs to each other, and each hub is connected to the other $10-h$ cities; we thus get $\binom{h}{2}+h(10-h)$ distinct roads. So, $40 \geq\binom{ h}{2}+h(10-h)=-h^{2} / 2+19 h / 2$, or $80 \geq h(19-h)$. The largest $h \leq 10$ satisfying this condition is $h=... | 0.1875 | 7,975.1875 | 7,035.666667 | 8,192 |
Given any point $P$ on the line $l: x-y+4=0$, two tangent lines $AB$ are drawn to the circle $O: x^{2}+y^{2}=4$ with tangent points $A$ and $B$. The line $AB$ passes through a fixed point ______; let the midpoint of segment $AB$ be $Q$. The minimum distance from point $Q$ to the line $l$ is ______. | \sqrt{2} | 0.6875 | 6,738.9375 | 6,466.181818 | 7,339 | |
Let $p$ be a prime. We arrange the numbers in ${\{1,2,\ldots ,p^2} \}$ as a $p \times p$ matrix $A = ( a_{ij} )$. Next we can select any row or column and add $1$ to every number in it, or subtract $1$ from every number in it. We call the arrangement [i]good[/i] if we can change every number of the matrix to $0$ in a f... | 2(p!)^2 |
Let \( p \) be a prime. We arrange the numbers in \( \{1, 2, \ldots, p^2\} \) as a \( p \times p \) matrix \( A = (a_{ij}) \). We can select any row or column and add 1 to every number in it, or subtract 1 from every number in it. We call the arrangement "good" if we can change every number of the matrix to 0 in a fin... | 0 | 8,192 | -1 | 8,192 |
The number of students in Jakob's graduating class is more than 100 and fewer than 200 and is 2 less than a multiple of 4, 3 less than a multiple of 5, and 4 less than a multiple of 6. How many students are in Jakob's graduating class? | 122 | 0.125 | 8,082 | 7,774 | 8,126 | |
Let \( a \) and \( b \) be positive integers such that \( 15a + 16b \) and \( 16a - 15b \) are both perfect squares. Find the smallest possible value among these squares. | 481^2 | 0 | 8,192 | -1 | 8,192 | |
What is the $87^{\mathrm{th}}$ odd positive integer? | 173 | 1 | 2,162.75 | 2,162.75 | -1 | |
Determine the number of intersection points of 10 lines, given that only two of them are parallel and exactly three of these lines intersect at one point. | 42 | 0.75 | 5,628.1875 | 4,773.583333 | 8,192 | |
Given the line $y=x+1$ intersects with the ellipse $mx^2+my^2=1(m > n > 0)$ at points $A$ and $B$, where the x-coordinate of the midpoint of the chord $AB$ is equal to $-\frac{1}{3}$, find the eccentricity of the hyperbola $\frac{y^2}{m^2}-\frac{x^2}{n^2}=1$. | \frac{\sqrt{5}}{2} | 0 | 6,451.4375 | -1 | 6,451.4375 | |
Given the function f(x) = a^x (a > 0, a ≠ 1).
(I) If $f(1) + f(-1) = \frac{5}{2}$, find the value of f(2) + f(-2).
(II) If the difference between the maximum and minimum values of the function f(x) on [-1, 1] is $\frac{8}{3}$, find the value of the real number a. | \frac{1}{3} | 0.625 | 5,086.5625 | 5,294.2 | 4,740.5 | |
Let $N$ be the number of consecutive $0$'s at the right end of the decimal representation of the product $1!2!3!4!\cdots99!100!.$ Find the remainder when $N$ is divided by $1000$. | 124 | A number in decimal notation ends in a zero for each power of ten which divides it. Thus, we need to count both the number of 5s and the number of 2s dividing into our given expression. Since there are clearly more 2s than 5s, it is sufficient to count the number of 5s.
One way to do this is as follows: $96$ of the nu... | 0.5 | 7,427.4375 | 6,662.875 | 8,192 |
The amplitude, period, frequency, phase, and initial phase of the function $y=3\sin \left( \frac {1}{2}x- \frac {\pi}{6}\right)$ are ______, ______, ______, ______, ______, respectively. | - \frac {\pi}{6} | 1 | 3,094.4375 | 3,094.4375 | -1 | |
On a blackboard lies $50$ magnets in a line numbered from $1$ to $50$ , with different magnets containing different numbers. David walks up to the blackboard and rearranges the magnets into some arbitrary order. He then writes underneath each pair of consecutive magnets the positive difference between the number... | 4925 | 0.5 | 6,710.8125 | 5,242.75 | 8,178.875 | |
How many ways are there to choose distinct positive integers $a, b, c, d$ dividing $15^6$ such that none of $a, b, c,$ or $d$ divide each other? (Order does not matter.)
*Proposed by Miles Yamner and Andrew Wu*
(Note: wording changed from original to clarify) | 1225 | 0 | 8,192 | -1 | 8,192 | |
For the numbers \(1000^{2}, 1001^{2}, 1002^{2}, \ldots\), the last two digits are discarded. How many of the first terms in the resulting sequence form an arithmetic progression? | 10 | 0.0625 | 8,192 | 8,192 | 8,192 | |
Express $0.3\overline{45}$ as a common fraction. | \frac{83}{110} | 0 | 3,890.9375 | -1 | 3,890.9375 | |
Given vectors $a=(\cos α, \sin α)$ and $b=(\cos β, \sin β)$, with $|a-b|= \frac{2 \sqrt{5}}{5}$, find the value of $\cos (α-β)$.
(2) Suppose $α∈(0,\frac{π}{2})$, $β∈(-\frac{π}{2},0)$, and $\cos (\frac{5π}{2}-β) = -\frac{5}{13}$, find the value of $\sin α$. | \frac{33}{65} | 0.4375 | 7,655.5 | 7,132.571429 | 8,062.222222 | |
The equation $a^7xy-a^6y-a^5x=a^4(b^4-1)$ is equivalent to the equation $(a^mx-a^n)(a^py-a^2)=a^4b^4$ for some integers $m$, $n$, and $p$. Find $mnp$. | 24 | 0.125 | 7,662.1875 | 3,953.5 | 8,192 | |
Determine the number of angles between 0 and $2 \pi,$ other than integer multiples of $\frac{\pi}{2},$ such that $\sin \theta,$ $\cos \theta$, and $\tan \theta$ form a geometric sequence in some order. | 4 | 0 | 8,192 | -1 | 8,192 | |
How many of the natural numbers from 1 to 700, inclusive, contain the digit 0 at least once? | 123 | 0 | 8,038.1875 | -1 | 8,038.1875 | |
A school library purchased 17 identical books. How much do they cost if they paid more than 11 rubles 30 kopecks, but less than 11 rubles 40 kopecks for 9 of these books? | 2142 | 0.0625 | 5,453.4375 | 7,798 | 5,297.133333 | |
What is the area of the shaded figure shown below? | 6 | To find the area of the shaded figure with vertices given as $(1,0)$, $(3,2)$, $(5,0)$, and $(3,5)$, we will use the Shoelace Theorem. The Shoelace Theorem states that the area of a polygon with vertices $(x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)$ is given by:
\[
\text{Area} = \frac{1}{2} \left| \sum_{i=1}^{n-1} (x_i... | 0 | 8,192 | -1 | 8,192 |
Convert $110_{(5)}$ to binary. | 11110_{(2)} | 0.125 | 3,536.75 | 3,475 | 3,545.571429 | |
Positive numbers $x$, $y$, and $z$ satisfy $xyz = 10^{81}$ and $(\log_{10}x)(\log_{10} yz) + (\log_{10}y) (\log_{10}z) = 468$. Find $\sqrt {(\log_{10}x)^2 + (\log_{10}y)^2 + (\log_{10}z)^2}$. | 75 | Let $a=\log_{10}x$, $b=\log_{10}y,$ and $c=\log_{10}z$.
We have $a+b+c=81$ and $a(b+c)+bc=ab+ac+bc=468$. Since these two equations look a lot like Vieta's for a cubic, create the polynomial $x^3-81x^2+468x=0$ (leave the constant term as $0$ to make things easy). Dividing by $x$ yields $x^2-81x+468=0$.
Now we use the ... | 1 | 2,245 | 2,245 | -1 |
Let $y = ax^{2} + x - b$ where $a \in \mathbb{R}$ and $b \in \mathbb{R}$.
$(1)$ If $b = 1$ and the set $\{x | y = 0\}$ has exactly one element, find the set of possible values for the real number $a$.
$(2)$ Solve the inequality with respect to $x$: $y < (a-1)x^{2} + (b+2)x - 2b$.
$(3)$ When $a > 0$ and $b > 1$, l... | \frac{1}{2} | 0.0625 | 8,128.875 | 7,182 | 8,192 | |
For all positive integers $n > 1$ , let $f(n)$ denote the largest odd proper divisor of $n$ (a proper divisor of $n$ is a positive divisor of $n$ except for $n$ itself). Given that $N=20^{23}\cdot23^{20}$ , compute
\[\frac{f(N)}{f(f(f(N)))}.\] | 25 | 0.8125 | 4,331.9375 | 3,881.076923 | 6,285.666667 | |
Juan rolls a fair regular decagonal die marked with the numbers 1 through 10. Then Amal rolls a fair eight-sided die. What is the probability that the product of the two rolls is a multiple of 4? | \frac{2}{5} | 0 | 7,819.5 | -1 | 7,819.5 | |
Let $C=\{ z \in \mathbb{C} : |z|=1 \}$ be the unit circle on the complex plane. Let $z_1, z_2, \ldots, z_{240} \in C$ (not necessarily different) be $240$ complex numbers, satisfying the following two conditions:
(1) For any open arc $\Gamma$ of length $\pi$ on $C$, there are at most $200$ of $j ~(1 \le j \le 240)$ suc... | 80 + 40\sqrt{3} |
Let \( C = \{ z \in \mathbb{C} : |z| = 1 \} \) be the unit circle on the complex plane. Let \( z_1, z_2, \ldots, z_{240} \in C \) (not necessarily different) be 240 complex numbers satisfying the following two conditions:
1. For any open arc \(\Gamma\) of length \(\pi\) on \(C\), there are at most 200 of \( j ~(1 \le ... | 0 | 8,192 | -1 | 8,192 |
Consider the integer\[N = 9 + 99 + 999 + 9999 + \cdots + \underbrace{99\ldots 99}_\text{321 digits}.\]Find the sum of the digits of $N$.
| 342 | 0.0625 | 8,013.75 | 7,612 | 8,040.533333 | |
Alex, Bonnie, and Chris each have $3$ blocks, colored red, blue, and green; and there are $3$ empty boxes. Each person independently places one of their blocks into each box. Each block placement by Bonnie and Chris is picked such that there is a 50% chance that the color matches the color previously placed by Alex or ... | \frac{37}{64} | 0 | 7,978.4375 | -1 | 7,978.4375 | |
$\alpha$ and $\beta$ are two parallel planes. Four points are taken within plane $\alpha$, and five points are taken within plane $\beta$.
(1) What is the maximum number of lines and planes that can be determined by these points?
(2) What is the maximum number of tetrahedrons that can be formed with these points as ver... | 120 | 0.8125 | 5,025.1875 | 4,701.615385 | 6,427.333333 | |
A semicircle is inscribed in a quarter circle. What fraction of the quarter circle is shaded? | $\frac{2}{3}$ | 0 | 7,306.625 | -1 | 7,306.625 | |
Calculate the value of $\frac{2 \times 3 + 4}{2 + 3}$. | 2 | Evaluating, $\frac{2 \times 3+4}{2+3}=\frac{6+4}{5}=\frac{10}{5}=2$. | 1 | 665.4375 | 665.4375 | -1 |
Find $B^2$, where $B$ is the sum of the absolute values of all roots of the equation:
\[x = \sqrt{26} + \frac{119}{{\sqrt{26}+\frac{119}{{\sqrt{26}+\frac{119}{{\sqrt{26}+\frac{119}{{\sqrt{26}+\frac{119}{x}}}}}}}}}.\] | 502 | 0.1875 | 7,619.3125 | 5,137.666667 | 8,192 | |
Xiaopang arranges the 50 integers from 1 to 50 in ascending order without any spaces in between. Then, he inserts a "+" sign between each pair of adjacent digits, resulting in an addition expression: \(1+2+3+4+5+6+7+8+9+1+0+1+1+\cdots+4+9+5+0\). Please calculate the sum of this addition expression. The result is ______... | 330 | 0.6875 | 6,517.625 | 5,763.363636 | 8,177 | |
Let $P_1$ be a regular $r~\mbox{gon}$ and $P_2$ be a regular $s~\mbox{gon}$ $(r\geq s\geq 3)$ such that each interior angle of $P_1$ is $\frac{59}{58}$ as large as each interior angle of $P_2$. What's the largest possible value of $s$?
| 117 | 0.3125 | 7,673.9375 | 6,534.2 | 8,192 | |
How many non-empty subsets $S$ of $\{1,2,3,\ldots,20\}$ have the following three properties?
$(1)$ No two consecutive integers belong to $S$.
$(2)$ If $S$ contains $k$ elements, then $S$ contains no number less than $k$.
$(3)$ $S$ contains at least one number greater than $10$. | 2526 | 0 | 8,140.6875 | -1 | 8,140.6875 | |
Three boys and two girls are to stand in a row according to the following requirements. How many different arrangements are there? (Answer with numbers)
(Ⅰ) The two girls stand next to each other;
(Ⅱ) Girls cannot stand at the ends;
(Ⅲ) Girls are arranged from left to right from tallest to shortest;
(Ⅳ) Girl A ... | 78 | 0.6875 | 7,038 | 6,513.454545 | 8,192 | |
In triangle \( \triangle ABC \), \( \angle BAC = 30^\circ \), \( \angle ABC = 70^\circ \), and \( M \) is a point inside the triangle such that \( \angle MAB = \angle MCA = 20^\circ \). Find the degree measure of \( \angle MBA \). | 30 | 0.25 | 7,429.5625 | 6,323.5 | 7,798.25 | |
If $ P(x)$ denotes a polynomial of degree $ n$ such that $ P(k)\equal{}\frac{k}{k\plus{}1}$ for $ k\equal{}0,1,2,\ldots,n$, determine $ P(n\plus{}1)$. | \frac{(-1)^{n+1} + (n+1)}{n+2} | To solve this problem, we need to determine the value of \( P(n+1) \) for the given polynomial \( P(x) \) of degree \( n \) such that
\[
P(k) = \frac{k}{k+1} \quad \text{for } k = 0, 1, 2, \ldots, n.
\]
Our goal is to express \( P(x) \) as:
\[
P(x) = x - \frac{x(x-1)\cdots(x-n)}{n+1}.
\]
This assumes \( P(x) \) be... | 0 | 6,528.875 | -1 | 6,528.875 |
The tangent value of the angle between the slant height and the base is when the lateral area of the cone with volume $\frac{\pi}{6}$ is minimum. | \sqrt{2} | 0.9375 | 3,723.25 | 3,789.933333 | 2,723 | |
If $x=t^{\frac{1}{t-1}}$ and $y=t^{\frac{t}{t-1}},t>0,t \ne 1$, a relation between $x$ and $y$ is: | $y^x=x^y$ | 1. **Define the expressions for $x$ and $y$:**
Given $x = t^{1/(t-1)}$ and $y = t^{t/(t-1)}$ where $t > 0$ and $t \neq 1$.
2. **Substitute $x$ and $y$ into the expressions in option (C):**
We need to verify if $y^x = x^y$ holds true.
3. **Calculate $x^y$:**
\[
x^y = \left(t^{\frac{1}{t-1}}\right)^{t^{\fra... | 0 | 5,126.9375 | -1 | 5,126.9375 |
Points A and B are on a circle of radius 7 and AB = 8. Point C is the midpoint of the minor arc AB. What is the length of the line segment AC? | \sqrt{98 - 14\sqrt{33}} | 0 | 7,007.875 | -1 | 7,007.875 | |
In the plane quadrilateral \(ABCD\), points \(E\) and \(F\) are the midpoints of sides \(AD\) and \(BC\) respectively. Given that \(AB = 1\), \(EF = \sqrt{2}\), and \(CD = 3\), and that \(\overrightarrow{AD} \cdot \overrightarrow{BC} = 15\), find \(\overrightarrow{AC} \cdot \overrightarrow{BD}\). | 16 | 0.25 | 7,426.6875 | 5,961.25 | 7,915.166667 | |
A moving particle starts at the point $(4,4)$ and moves until it hits one of the coordinate axes for the first time. When the particle is at the point $(a,b)$, it moves at random to one of the points $(a-1,b)$, $(a,b-1)$, or $(a-1,b-1)$, each with probability $\frac{1}{3}$, independently of its previous moves. The prob... | 252 | 0.25 | 6,414.3125 | 5,710.75 | 6,648.833333 | |
A rectangle having integer length and width has a perimeter of 100 units. What is the number of square units in the least possible area? | 49 | 1 | 2,662.375 | 2,662.375 | -1 | |
Find the maximum $y$-coordinate of a point on the graph of $r = \sin 2 \theta.$ | \frac{4 \sqrt{3}}{9} | 0 | 6,191.5625 | -1 | 6,191.5625 | |
How many different integral solutions \((x, y)\) does \(3|x| + 5|y| = 100\) have? | 26 | 0.5 | 6,696.5 | 5,201 | 8,192 | |
If the tangent line of the curve $y=\ln x$ at point $P(x_{1}, y_{1})$ is tangent to the curve $y=e^{x}$ at point $Q(x_{2}, y_{2})$, then $\frac{2}{{x_1}-1}+x_{2}=$____. | -1 | 0.3125 | 7,364.1875 | 5,543 | 8,192 | |
Given the ellipse \(\frac{x^2}{2} + y^2 = 1\) with two foci \(F_1\) and \(F_2\), a chord \(AB\) passing through the right focus \(F_2\) has an inclination of \(45^\circ\). Determine the area of the triangle \(\triangle ABF_1\). | \frac{4}{3} | 1 | 4,683.9375 | 4,683.9375 | -1 | |
Since the beginning of the school year, Andrey has been recording his math grades. Upon receiving a new grade (2, 3, 4, or 5), he called it unexpected if, up to that point, it had occurred less frequently than each of the other possible grades. (For example, if he received the grades 3, 4, 2, 5, 5, 5, 2, 3, 4, 3 in ord... | 10 | 0.1875 | 7,809.0625 | 7,415 | 7,900 | |
The $25$ integers from $-10$ to $14,$ inclusive, can be arranged to form a $5$-by-$5$ square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum? | 10 | To solve this problem, we need to find the common sum of the numbers in each row, column, and diagonal of a $5 \times 5$ square matrix using the integers from $-10$ to $14$ inclusive.
1. **Calculate the Total Sum of All Numbers:**
The integers from $-10$ to $14$ form an arithmetic sequence with the first term $a = ... | 0.875 | 3,177.1875 | 2,460.785714 | 8,192 |
Two identical smaller cubes are stacked next to a larger cube. Each of the two smaller cubes has a volume of 8. What is the volume of the larger cube? | 64 | Since $2 \times 2 \times 2=8$, a cube with edge length 2 has volume 8. Therefore, each of the cubes with volume 8 have a height of 2. This means that the larger cube has a height of $2+2=4$, which means that its volume is $4^{3}=4 \times 4 \times 4=64$. | 0.5625 | 6,748.1875 | 5,971 | 7,747.428571 |
Let $A$ be a $n\times n$ matrix such that $A_{ij} = i+j$. Find the rank of $A$.
[hide="Remark"]Not asked in the contest: $A$ is diagonalisable since real symetric matrix it is not difficult to find its eigenvalues.[/hide] | 2 |
Let \( A \) be an \( n \times n \) matrix where each entry \( A_{ij} = i + j \). We aim to find the rank of this matrix.
**Step 1: Analyze the Structure of Matrix \( A \)**
The entry \( A_{ij} \) depends linearly on the indices \( i \) and \( j \):
\[
A = \begin{bmatrix}
2 & 3 & 4 & \cdots & n+1 \\
3 & 4 & 5 & \cdot... | 0.4375 | 7,247 | 6,032 | 8,192 |
For each positive integer $n > 1$, let $P(n)$ denote the greatest prime factor of $n$. For how many positive integers $n$ is it true that both $P(n) = \sqrt{n}$ and $P(n+48) = \sqrt{n+48}$? | 1 | 1. **Understanding the conditions**: We are given two conditions:
- $P(n) = \sqrt{n}$, which implies $n = p_1^2$ where $p_1$ is a prime number.
- $P(n+48) = \sqrt{n+48}$, which implies $n+48 = p_2^2$ where $p_2$ is a different prime number.
2. **Setting up the equation**:
- From $n = p_1^2$ and $n+48 = p_2^2$... | 0.875 | 5,965.0625 | 5,646.928571 | 8,192 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively. If the area of $\triangle ABC$ equals $8$, $a=5$, and $\tan B= -\frac{4}{3}$, then find the value of $\frac{a+b+c}{\sin A+\sin B+\sin C}$. | \frac{5 \sqrt{65}}{4} | 0 | 5,998.75 | -1 | 5,998.75 | |
A card is secretly removed from a standard deck of 52 cards. Then two cards are dealt at random from the now 51-card deck. What is the probability that both cards drawn are from the same suit and the second card is a face card? | \frac{3}{50} | 0 | 8,192 | -1 | 8,192 | |
Find the value of $m$ for which the quadratic equation $3x^2 + mx + 36 = 0$ has exactly one solution in $x$. | 12\sqrt{3} | 0 | 2,669 | -1 | 2,669 | |
The polynomial \( f(x)=x^{2007}+17 x^{2006}+1 \) has distinct zeroes \( r_{1}, \ldots, r_{2007} \). A polynomial \( P \) of degree 2007 has the property that \( P\left(r_{j}+\frac{1}{r_{j}}\right)=0 \) for \( j=1, \ldots, 2007 \). Determine the value of \( P(1) / P(-1) \). | 289/259 | For some constant \( k \), we have \( P(z)=k \prod_{j=1}^{2007}\left(z-\left(r_{j}+\frac{1}{r_{j}}\right)\right) \). Now writing \( \omega^{3}=1 \) with \( \omega \neq 1 \), we have \( \omega^{2}+\omega=-1 \). Then \( P(1) / P(-1)=\frac{k \prod_{j=1}^{2007}\left(1-\left(r_{j}+\frac{1}{r_{j}}\right)\right)}{k \prod_{j=1... | 0 | 8,192 | -1 | 8,192 |
One half of the water is poured out of a full container. Then one third of the remainder is poured out. Continue the process: one fourth of the remainder for the third pouring, one fifth of the remainder for the fourth pouring, etc. After how many pourings does exactly one tenth of the original water remain? | 9 | 1. **Model the Remaining Water**: Let's denote the initial amount of water in the container as 1 (representing 100%). We will calculate the remaining fraction of water after each pouring.
2. **Calculate the Remaining Fraction After Each Pouring**:
- After the first pouring, half of the water is poured out, so $\fra... | 1 | 2,964.875 | 2,964.875 | -1 |
What is the constant term in the expansion of $\left(\sqrt{x}+\dfrac5x\right)^{9}$? | 10500 | 1 | 2,162.5625 | 2,162.5625 | -1 | |
Each of 100 students sends messages to 50 different students. What is the least number of pairs of students who send messages to each other? | 50 | 0.125 | 7,645.5625 | 6,852.5 | 7,758.857143 | |
Two different natural numbers are selected from the set $\{1, 2, 3, \ldots, 8\}$. What is the probability that the greatest common factor of these two numbers is one? Express your answer as a common fraction. | \frac{11}{14} | 0 | 7,224.75 | -1 | 7,224.75 | |
Moe, Loki, Nick, and Ott are friends. Thor initially had no money, while the other friends did have money. Moe gave Thor one-sixth of his money, Loki gave Thor one-fifth of his money, Nick gave Thor one-fourth of his money, and Ott gave Thor one-third of his money. Each friend gave Thor $2. Determine the fractional par... | \frac{2}{9} | 0.4375 | 2,774.875 | 2,664.857143 | 2,860.444444 | |
A rectangular field is 300 feet wide and 400 feet long. Random sampling indicates that there are, on the average, three ants per square inch through out the field. [12 inches = 1 foot.] Of the following, the number that most closely approximates the number of ants in the field is | 50000000 | 1. **Convert dimensions from feet to inches**:
- The width of the field in inches is:
\[
300 \text{ feet} \times \frac{12 \text{ inches}}{1 \text{ foot}} = 3600 \text{ inches}
\]
- The length of the field in inches is:
\[
400 \text{ feet} \times \frac{12 \text{ inches}}{1 \text{ foot}} =... | 0 | 6,522.5625 | -1 | 6,522.5625 |
Determine the sum of all positive even factors of the number $720$. | 2340 | 0.6875 | 5,835.5 | 4,764.363636 | 8,192 | |
Given the function $f(x)=3x^3+2$, find the value of $x$ so that $f^{-1}(x)=4$. | 194 | 1 | 2,366.625 | 2,366.625 | -1 | |
The diagonal of a square is 10 inches and the diameter of a circle is also 10 inches. Calculate:
1. By how many square inches is the area of the circle greater than the area of the square?
2. By how many inches is the circumference of the circle greater than the perimeter of the square? Express your answers as decimals... | 3.1 | 1 | 2,066.3125 | 2,066.3125 | -1 | |
Let $EFGH$, $EFDC$, and $EHBC$ be three adjacent square faces of a cube, for which $EC = 8$, and let $A$ be the eighth vertex of the cube. Let $I$, $J$, and $K$, be the points on $\overline{EF}$, $\overline{EH}$, and $\overline{EC}$, respectively, so that $EI = EJ = EK = 2$. A solid $S$ is obtained by drilling a tunnel... | 417 | [asy] import three; currentprojection = orthographic(camera=(1/4,2,3/4)); defaultpen(linewidth(0.7)); pen l = linewidth(0.5) + linetype("10 2"); triple S=(1,0,0), T=(2,0,2), U=(8,6,8), V=(8,8,6), W=(2,2,0), X=(6,8,8); draw((1,0,0)--(8,0,0)--(8,0,8)--(0,0,8)--(0,0,1)); draw((1,0,0)--(8,0,0)--(8,8,0)--(0,8,0)--(0,1,0),l)... | 0 | 8,192 | -1 | 8,192 |
One of the following four-digit numbers is not divisible by 4: 3544, 3554, 3564, 3572, 3576. What is the product of the units digit and the tens digit of that number? | 20 | 1 | 1,222.5625 | 1,222.5625 | -1 | |
If the perimeter of a square is 28, what is the side length of the square? | 7 | Since a square has four equal sides, the side length of a square equals one-quarter of the perimeter of the square. Thus, the side length of a square with perimeter 28 is $28 \div 4 = 7$. | 1 | 919 | 919 | -1 |
For what value of $x$ is the expression $\frac{2x^3+3}{x^2-20x+100}$ not defined? | 10 | 1 | 1,240.375 | 1,240.375 | -1 | |
What is the number of degrees in $x$? [asy]
import olympiad; import geometry; defaultpen(linewidth(0.8)); size(150);
pair A = origin, B = (Cos(50),Sin(50)), C = (Cos(50) + Sin(50)*sqrt(3),0);
draw(A--B--C--cycle);
label("$30^\circ$",C-(0.3,-.03),NW);
label("$2x$",B + (-.10,-0.1),SE);
label("$x$",A+(0.15,.03),NE);
[/asy... | 50^\circ | 0.5625 | 6,334 | 4,888.888889 | 8,192 | |
How many integer side lengths are possible to complete a triangle in which the other sides measure 6 units and 3 units? | 5 | 0.9375 | 1,937.1875 | 1,942.066667 | 1,864 | |
The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers $x,y$ and $r$ with $|x|>|y|$,
\[(x+y)^r=x^r+rx^{r-1}y+\dfrac{r(r-1)}{2}x^{r-2}y^2+\dfrac{r(r-1)(r-2)}{3!}x^{r-3}y^3 \cdots\]
What are the first three digits to the right of the decimal point in the decimal representation... | 428 | $1^n$ will always be 1, so we can ignore those terms, and using the definition ($2002 / 7 = 286$):
\[(10^{2002} + 1)^{\frac {10}7} = 10^{2860}+\dfrac{10}{7}10^{858}+\dfrac{15}{49}10^{-1144}+\cdots\]
Since the exponent of the $10$ goes down extremely fast, it suffices to consider the first few terms. Also, the $10^{286... | 0.1875 | 8,090.875 | 7,652.666667 | 8,192 |
In $\triangle ABC$, if $BC=4$, $\cos B= \frac{1}{4}$, then $\sin B=$ _______, the minimum value of $\overrightarrow{AB} \cdot \overrightarrow{AC}$ is: _______. | -\frac{1}{4} | 0.625 | 6,322.6875 | 5,443.8 | 7,787.5 | |
Given that $f(x)$ is an even function defined on $\mathbb{R}$ and satisfies $f(x+2)=- \frac{1}{f(x)}$. When $1 \leq x \leq 2$, $f(x)=x-2$. Find $f(6.5)$. | -0.5 | 0 | 7,118.0625 | -1 | 7,118.0625 | |
Consider the line parameterized by
\begin{align*}
x&= 4t + 2,\\
y& = t+2.
\end{align*}Find a vector $\begin{pmatrix}a \\ b \end{pmatrix}$ pointing from the origin to this line that is parallel to $\begin{pmatrix}2 \\1 \end{pmatrix}$. | \begin{pmatrix}6\\3\end{pmatrix} | 1 | 3,122.625 | 3,122.625 | -1 | |
Set $B = \{55, 57, 58, 59, 61\}.$ Which number in set B has the smallest prime factor? | 58 | 1 | 1,539.5625 | 1,539.5625 | -1 | |
Given equations of the form $x^2 + bx + c = 0$, determine the number of such equations that have real roots and have coefficients $b$ and $c$ selected from the set of integers $\{1, 2, 3, 4, 5, 7\}$. | 18 | 0 | 6,982.9375 | -1 | 6,982.9375 | |
Suppose Xiao Ming's family subscribes to a newspaper. The delivery person may deliver the newspaper to Xiao Ming's home between 6:30 and 7:30 in the morning. Xiao Ming's father leaves for work between 7:00 and 8:00 in the morning. What is the probability that Xiao Ming's father can get the newspaper before leaving home... | \frac{7}{8} | 0 | 7,611.4375 | -1 | 7,611.4375 | |
Given a linear function \( f(x) \). It is known that the distance between the points of intersection of the graphs \( y = x^2 + 1 \) and \( y = f(x) \) is \( 3\sqrt{2} \), and the distance between the points of intersection of the graphs \( y = x^2 \) and \( y = f(x) - 2 \) is \( \sqrt{10} \). Find the distance between... | \sqrt{26} | 0.125 | 7,263.8125 | 5,080 | 7,575.785714 | |
Meredith drives 5 miles to the northeast, then 15 miles to the southeast, then 25 miles to the southwest, then 35 miles to the northwest, and finally 20 miles to the northeast. How many miles is Meredith from where she started? | 20 | 0.625 | 5,295 | 4,918.5 | 5,922.5 | |
A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths 3, 4, and 5. What is the area of the triangle?
$\mathrm{(A) \ 6 } \qquad \mathrm{(B) \frac{18}{\pi^2} } \qquad \mathrm{(C) \frac{9}{\pi^2}(\sqrt{3}-1) } \qquad \mathrm{(D) \frac{9}{\pi^2}(\sqrt{3}-1) } \qquad ... | \frac{9}{\pi^2}(\sqrt{3}+3) | 0 | 7,802.875 | -1 | 7,802.875 | |
Let $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be three vectors such that $\mathbf{a}$ is a unit vector, $\mathbf{b}$ and $\mathbf{c}$ have magnitudes of 2, and $\mathbf{a} \cdot \mathbf{b} = 0$, $\mathbf{b} \cdot \mathbf{c} = 4$. Given that
\[
\mathbf{a} = p(\mathbf{a} \times \mathbf{b}) + q(\mathbf{b} \times \mathb... | \frac{1}{2} | 0 | 8,192 | -1 | 8,192 | |
There is a set of points \( M \) on a plane and seven different circles \( C_{1}, C_{2}, \dots, C_{7} \). Circle \( C_{7} \) passes through exactly 7 points in \( M \); circle \( C_{6} \) passes through exactly 6 points in \( M \); ..., circle \( C_{1} \) passes through exactly 1 point in \( M \). Determine the minimum... | 12 | 0 | 7,965.9375 | -1 | 7,965.9375 | |
Given that it is currently between 4:00 and 5:00 o'clock, and eight minutes from now, the minute hand of a clock will be exactly opposite to the position where the hour hand was six minutes ago, determine the exact time now. | 4:45\frac{3}{11} | 0.375 | 7,514 | 7,255.833333 | 7,668.9 | |
There is a unique positive real number $x$ such that the three numbers $\log_8{2x}$, $\log_4{x}$, and $\log_2{x}$, in that order, form a geometric progression with positive common ratio. The number $x$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | 17 | If we set $x=2^y$, we can obtain three terms of a geometric sequence through logarithm properties. The three terms are \[\frac{y+1}{3}, \frac{y}{2}, y.\] In a three-term geometric sequence, the middle term squared is equal to the product of the other two terms, so we obtain the following: \[\frac{y^2+y}{3} = \frac{y^2}... | 0.875 | 3,932.875 | 3,324.428571 | 8,192 |
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