Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
12,400 |
Complex numbers $p$, $q$, and $r$ form an equilateral triangle with side length 24 in the complex plane. If $|p + q + r| = 48$, find $|pq + pr + qr|$.
|
768
| 39.0625 |
12,401 |
In the equation $\frac{1}{m} + \frac{1}{n} = \frac{1}{4}$, where $m$ and $n$ are positive integers, determine the sum of all possible values for $n$.
|
51
| 85.15625 |
12,402 |
Find all natural numbers \( n \) such that
\[
\sum_{\substack{d \mid n \\ 1 \leq d < n}} d^{2} = 5(n + 1)
\]
|
16
| 88.28125 |
12,403 |
A circle can be circumscribed around the quadrilateral $ABCD$. Additionally, $AB = 3$, $BC = 4$, $CD = 5$, and $AD = 2$.
Find $AC$.
|
\sqrt{\frac{299}{11}}
| 50 |
12,404 |
Camilla had three times as many blueberry jelly beans as cherry jelly beans. She also had some raspberry jelly beans, the number of which is not initially given. After eating 15 blueberry and 5 cherry jelly beans, she now has five times as many blueberry jelly beans as cherry jelly beans. Express the original number of blueberry jelly beans in terms of the original number of cherry jelly beans.
|
15
| 60.15625 |
12,405 |
In what ratio does the angle bisector of the acute angle divide the area of a right trapezoid inscribed in a circle?
|
1:1
| 38.28125 |
12,406 |
A chess tournament is held with the participation of boys and girls. The girls are twice as many as boys. Each player plays against each other player exactly once. By the end of the tournament, there were no draws and the ratio of girl winnings to boy winnings was $7/9$ . How many players took part at the tournament?
|
33
| 33.59375 |
12,407 |
The quadrilateral \(ABCD\) is circumscribed around a circle with a radius of \(1\). Find the greatest possible value of \(\left| \frac{1}{AC^2} + \frac{1}{BD^2} \right|\).
|
1/4
| 39.0625 |
12,408 |
The whole numbers from 1 to \( 2k \) are split into two equal-sized groups in such a way that any two numbers from the same group share no more than two distinct prime factors. What is the largest possible value of \( k \)?
|
44
| 0 |
12,409 |
It is known that $\sin y = 2 \cos x + \frac{5}{2} \sin x$ and $\cos y = 2 \sin x + \frac{5}{2} \cos x$. Find $\sin 2x$.
|
-\frac{37}{20}
| 0 |
12,410 |
Suppose you have a Viennese pretzel lying on the table. What is the maximum number of parts you can cut it into with one straight swing of the knife? In which direction should this cut be made?
|
10
| 0 |
12,411 |
Last year, the East Sea Crystal City World's business hall made a profit of 3 million yuan. At the beginning of this year, it relocated to the new Crystal City business hall, expanding its scope of operations. To achieve higher profits, it is necessary to increase advertising efforts. It is expected that starting from this year, the profit will grow at an annual rate of 26%, while on December 30th of each year, an advertising fee of x million yuan will be paid. To achieve the goal of doubling the profit after 10 years, find the maximum value of the annual advertising fee x in million yuan. (Note: $1.26^{10} \approx 10.$)
|
52
| 0 |
12,412 |
Let $ABC$ be an acute triangle with circumcircle $\omega$ . Let $D$ and $E$ be the feet of the altitudes from $B$ and $C$ onto sides $AC$ and $AB$ , respectively. Lines $BD$ and $CE$ intersect $\omega$ again at points $P \neq B$ and $Q \neq C$ . Suppose that $PD=3$ , $QE=2$ , and $AP \parallel BC$ . Compute $DE$ .
*Proposed by Kyle Lee*
|
\sqrt{23}
| 0 |
12,413 |
A triangle has sides with lengths of $18$, $24$, and $30$. Calculate the length of the shortest altitude.
|
18
| 0.78125 |
12,414 |
Divide the product of the first six positive composite integers by the product of the next six composite integers. Express your answer as a common fraction.
|
\frac{1}{49}
| 0 |
12,415 |
If income of $5$ yuan is denoted as $+5$ yuan, then expenses of $5$ yuan are denoted as what?
|
-5
| 96.09375 |
12,416 |
The diagonal of a square is 10 inches, and the diameter of a circle is also 10 inches. Additionally, an equilateral triangle is inscribed within the square. Find the difference in area between the circle and the combined area of the square and the equilateral triangle. Express your answer as a decimal to the nearest tenth.
|
-14.8
| 0 |
12,417 |
Let $n$ be the largest real solution to the equation
\[\dfrac{4}{x-2} + \dfrac{6}{x-6} + \dfrac{13}{x-13} + \dfrac{15}{x-15} = x^2 - 7x - 6\]
There are positive integers $p, q,$ and $r$ such that $n = p + \sqrt{q + \sqrt{r}}$. Find $p+q+r$.
|
103
| 0 |
12,418 |
How many four-digit whole numbers are there such that the leftmost digit is a prime number, the last digit is a perfect square, and all four digits are different?
|
288
| 0 |
12,419 |
In a regular quadrilateral pyramid \(P-ABCD\) with a volume of 1, points \(E\), \(F\), \(G\), and \(H\) are the midpoints of segments \(AB\), \(CD\), \(PB\), and \(PC\), respectively. Find the volume of the polyhedron \(BEG-CFH\).
|
5/16
| 0 |
12,420 |
Given $\overrightarrow{a}=(1,-1)$ and $\overrightarrow{b}=(1,2)$, calculate the projection of $\overrightarrow{b}$ onto $\overrightarrow{a}$.
|
-\frac{\sqrt{2}}{2}
| 0 |
12,421 |
Determine the exact value of
\[
\sqrt{\left( 2 - \sin^2 \frac{\pi}{9} \right) \left( 2 - \sin^2 \frac{2 \pi}{9} \right) \left( 2 - \sin^2 \frac{4 \pi}{9} \right)}.
\]
|
\frac{\sqrt{619}}{16}
| 0 |
12,422 |
The probability of getting rain on any given day in August in Beach Town is \(\frac{1}{5}\). What is the probability that it rains on at most 3 days in the first week of August?
|
0.813
| 0 |
12,423 |
How many integers $n$ satisfy the inequality $-\frac{9\pi}{2} \leq n \leq 12\pi$?
|
53
| 2.34375 |
12,424 |
What is the smallest positive four-digit number divisible by 8 which has three odd and one even digit?
|
1032
| 73.4375 |
12,425 |
In the number $52674.1892$, calculate the ratio of the value of the place occupied by the digit 6 to the value of the place occupied by the digit 8.
|
10,000
| 0 |
12,426 |
Let \(a\) and \(b\) be real numbers such that
\[
\frac{a}{2b} + \frac{a}{(2b)^2} + \frac{a}{(2b)^3} + \dots = 6.
\]
Find
\[
\frac{a}{a + 2b} + \frac{a}{(a + 2b)^2} + \frac{a}{(a + 2b)^3} + \dots.
\]
|
\frac{3}{4}
| 0.78125 |
12,427 |
John has cut out these two polygons made out of unit squares. He joins them to each other to form a larger polygon (but they can't overlap). Find the smallest possible perimeter this larger polygon can have. He can rotate and reflect the cut out polygons.
|
18
| 0 |
12,428 |
Let \( S = \{1, 2, \ldots, 280\} \). Find the smallest natural number \( n \) such that every \( n \)-element subset of \( S \) contains 5 pairwise coprime numbers.
|
217
| 3.90625 |
12,429 |
In triangle $\triangle ABC$, where $A B = 16$, $B C = 5 \sqrt{5}$, and $C A = 9$. What is the area of the plane region covered by the set of points outside $\triangle ABC$ such that the distance to points $B$ and $C$ is less than 6?
|
54\pi + \frac{5\sqrt{95}}{4}
| 0 |
12,430 |
Find the product $abc$ for the polynomial $Q(x) = x^3 + ax^2 + bx + c$ if its roots are $\sin \frac{\pi}{6}, \sin \frac{\pi}{3},$ and $\sin \frac{5\pi}{6}$.
|
\frac{\sqrt{3}}{2}
| 0 |
12,431 |
Person A and Person B each shoot at a target once. The probability of Person A hitting the target is $\dfrac{2}{3}$, and the probability of Person B hitting the target is $\dfrac{4}{5}$. Calculate the probability that exactly one person hits the target.
|
\dfrac{86}{225}
| 0 |
12,432 |
This puzzle features a unique kind of problem where only one digit is known. It appears to have a single solution and, surprisingly, filling in the missing digits is not very difficult. Given that a divisor multiplied by 7 results in a three-digit number, we conclude that the first digit of the divisor is 1. Additionally, it can be shown that the first digit of the dividend is also 1. Since two digits of the dividend are brought down, the second last digit of the quotient is 0. Finally, the first and last digits of the quotient are greater than 7, as they result in four-digit products when multiplied by the divisor, and so on.
|
124
| 0 |
12,433 |
Given an arithmetic sequence $\{a_n\}$ where $a_1=1$ and $a_n=70$ (for $n\geq3$), find all possible values of $n$ if the common difference is a natural number.
|
70
| 0 |
12,434 |
On the Cartesian plane, the midpoint between two points $P(p,q)$ and $Q(r,s)$ is $N(x,y)$. If $P$ is moved vertically upwards 10 units and horizontally to the right 5 units, and $Q$ is moved vertically downwards 5 units and horizontally to the left 15 units, find the new midpoint $N'$ between $P$ and $Q$ and the distance between $N$ and $N'$.
|
5.59
| 0 |
12,435 |
The area of triangle \(ABC\) is 1. Points \(B'\), \(C'\), and \(A'\) are placed respectively on the rays \(AB\), \(BC\), and \(CA\) such that:
\[ BB' = 2 AB, \quad CC' = 3 BC, \quad AA' = 4 CA. \]
Calculate the area of triangle \(A'B'C'\).
|
39
| 0 |
12,436 |
A brand of orange juice is available in shop $A$ and shop $B$ at an original price of $\$2.00$ per bottle. Shop $A$ provides a "buy 4 get 1 free" promotion and shop $B$ provides a $15\%$ discount if one buys 4 bottles or more. Find the minimum cost (in cents) if one wants to buy 13 bottles of the orange juice.
|
2160
| 0 |
12,437 |
Given a fixed circle \\(F_{1}:(x+2)^{2}+y^{2}=24\\) and a moving circle \\(N\\) passing through point \\(F_{2}(2,0)\\) and tangent to circle \\(F_{1}\\), denote the locus of the center of circle \\(N\\) as \\(E\\).
\\((I)\\) Find the equation of the locus \\(E\\);
\\((II)\\) If a line \\(l\\) not coincident with the x-axis passes through point \\(F_{2}(2,0)\\) and intersects the locus \\(E\\) at points \\(A\\) and \\(B\\), is there a fixed point \\(M\\) on the x-axis such that \\(\overrightarrow{MA}^{2}+ \overrightarrow{MA}· \overrightarrow{AB}\\) is a constant? If it exists, find the coordinates of point \\(M\\) and the constant value; if not, explain why.
|
-\frac{5}{9}
| 0 |
12,438 |
Given $X \sim N(5, 4)$, find $P(1 < X \leq 7)$.
|
0.9759
| 0 |
12,439 |
Among the five-digit numbers formed using the digits 0, 1, 2, 3, 4, how many have the first and last digits the same, and the three middle digits all different?
|
240
| 22.65625 |
12,440 |
As shown in the diagram, \(E, F, G, H\) are the midpoints of the sides \(AB, BC, CD, DA\) of the quadrilateral \(ABCD\). The intersection of \(BH\) and \(DE\) is \(M\), and the intersection of \(BG\) and \(DF\) is \(N\). What is \(\frac{S_{\mathrm{BMND}}}{S_{\mathrm{ABCD}}}\)?
|
1/3
| 6.25 |
12,441 |
A regular triangular pyramid \(SABC\) is given, with the edge of its base equal to 1. Medians of the lateral faces are drawn from the vertices \(A\) and \(B\) of the base \(ABC\), and these medians do not intersect. It is known that the edges of a certain cube lie on the lines containing these medians. Find the length of the lateral edge of the pyramid.
|
\frac{\sqrt{6}}{2}
| 9.375 |
12,442 |
The minimum positive period and maximum value of the function $f\left(x\right)=\sin \frac{x}{3}+\cos \frac{x}{3}$ are respectively $3\pi$ and $\sqrt{2}$.
|
\sqrt{2}
| 10.15625 |
12,443 |
Let \( N_{0} \) be the set of non-negative integers, and \( f: N_{0} \rightarrow N_{0} \) be a function such that \( f(0)=0 \) and for any \( n \in N_{0} \), \( [f(2n+1)]^{2} - [f(2n)]^{2} = 6f(n) + 1 \) and \( f(2n) > f(n) \). Determine how many elements in \( f(N_{0}) \) are less than 2004.
|
128
| 0 |
12,444 |
Find $x$ if
\[2 + 7x + 12x^2 + 17x^3 + \dotsb = 100.\]
|
\frac{2}{25}
| 0 |
12,445 |
Let $d$ be a number such that when 229 is divided by $d$, the remainder is 4. Compute the sum of all possible two-digit values of $d$.
|
135
| 1.5625 |
12,446 |
Find the number of positive integers $n$ that satisfy
\[(n - 1)(n - 5)(n - 9) \dotsm (n - 101) < 0.\]
|
25
| 16.40625 |
12,447 |
Find the positive real number $x$ such that $\lfloor x \rfloor \cdot x = 54$. Express $x$ as a decimal.
|
7.714285714285714
| 91.40625 |
12,448 |
There are 2016 points arranged on a circle. We are allowed to jump 2 or 3 points clockwise at will.
How many jumps must we make at least to reach all the points and return to the starting point again?
|
2017
| 7.03125 |
12,449 |
Given the cubic equation
\[
x^3 + Ax^2 + Bx + C = 0 \quad (A, B, C \in \mathbb{R})
\]
with roots \(\alpha, \beta, \gamma\), find the minimum value of \(\frac{1 + |A| + |B| + |C|}{|\alpha| + |\beta| + |\gamma|}\).
|
\frac{\sqrt[3]{2}}{2}
| 0 |
12,450 |
Find $x$, given that $x$ is neither zero nor one and the numbers $\{x\}$, $\lfloor x \rfloor$, and $x$ form a geometric sequence in that order. (Recall that $\{x\} = x - \lfloor x\rfloor$).
|
1.618
| 0 |
12,451 |
In the quadrilateral \(ABCD\), the lengths of the sides \(BC\) and \(CD\) are 2 and 6, respectively. The points of intersection of the medians of triangles \(ABC\), \(BCD\), and \(ACD\) form an equilateral triangle. What is the maximum possible area of quadrilateral \(ABCD\)? If necessary, round the answer to the nearest 0.01.
|
29.32
| 0 |
12,452 |
What is the largest possible length of an arithmetic progression formed of positive primes less than $1,000,000$?
|
12
| 0 |
12,453 |
Given that $\alpha$ is an angle in the third quadrant, $f\left( \alpha \right)=\dfrac{\sin (\alpha -\dfrac{\pi }{2})\cos (\dfrac{3\pi }{2}+\alpha )\tan (\pi -\alpha )}{\tan (-\alpha -\pi )\sin (-\alpha -\pi )}$.
(1) Simplify $f\left( \alpha \right)$
(2) If $\cos (\alpha -\dfrac{3\pi }{2})=\dfrac{1}{5}$, find the value of $f\left( \alpha \right)$
|
-\dfrac{2\sqrt{6}}{5}
| 35.9375 |
12,454 |
On a $300 \times 300$ board, several rooks are placed that beat the entire board. Within this case, each rook beats no more than one other rook. At what least $k$ , it is possible to state that there is at least one rook in each $k\times k$ square ?
|
201
| 0 |
12,455 |
Let \( a \) and \( b \) be positive integers such that \( 79 \mid (a + 77b) \) and \( 77 \mid (a + 79b) \). Find the smallest possible value of \( a + b \).
|
193
| 2.34375 |
12,456 |
The base of the pyramid \( P A B C D \) is a trapezoid \( A B C D \), with the base \( A D \) being twice as large as the base \( B C \). The segment \( M N \) is the midline of the triangle \( A B P \), parallel to the side \( A B \). Find the ratio of the volumes of the two solids into which the plane \( D M N \) divides this pyramid.
|
\frac{5}{13}
| 0 |
12,457 |
An array of integers is arranged in a grid of 7 rows and 1 column with eight additional squares forming a separate column to the right. The sequence of integers in the main column of squares and in each of the two rows form three distinct arithmetic sequences. Find the value of $Q$ if the sequence in the additional columns only has one number given.
[asy]
unitsize(0.35inch);
draw((0,0)--(0,7)--(1,7)--(1,0)--cycle);
draw((0,1)--(1,1));
draw((0,2)--(1,2));
draw((0,3)--(1,3));
draw((0,4)--(1,4));
draw((0,5)--(1,5));
draw((0,6)--(1,6));
draw((1,5)--(2,5)--(2,0)--(1,0)--cycle);
draw((1,1)--(2,1));
draw((1,2)--(2,2));
draw((1,3)--(2,3));
draw((1,4)--(2,4));
label("-9",(0.5,6.5),S);
label("56",(0.5,2.5),S);
label("$Q$",(1.5,4.5),S);
label("16",(1.5,0.5),S);
[/asy]
|
\frac{-851}{3}
| 0 |
12,458 |
The distance from the center \( O \) of a sphere with radius 12, which is circumscribed around a regular quadrangular pyramid, to a lateral edge is \( 4 \sqrt{2} \). Find:
1) the height of the pyramid;
2) the distance from point \( O \) to the lateral face of the pyramid;
3) the radius of the sphere inscribed in the pyramid.
|
\frac{8}{3}\left(2 \sqrt{2} - 1\right)
| 0 |
12,459 |
Let $M$ be the number of positive integers that are less than or equal to $2050$ and whose base-$2$ representation has more $1$'s than $0$'s. Find the remainder when $M$ is divided by $1000$.
|
374
| 0.78125 |
12,460 |
Let $a$ and $b$ be real numbers such that $a + b = 4.$ Find the maximum value of
\[a^4 b + a^3 b + a^2 b + ab + ab^2 + ab^3 + ab^4.\]
|
\frac{7225}{56}
| 0.78125 |
12,461 |
In triangle $ABC$, angle $C$ is a right angle and the altitude from $C$ meets $\overline{AB}$ at $D$. The lengths of the sides of $\triangle ABC$ are integers, $BD=29^2$, and $\sin B = p/q$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
|
17
| 0 |
12,462 |
Let $P$ be a point chosen on the interior of side $\overline{BC}$ of triangle $\triangle ABC$ with side lengths $\overline{AB} = 10, \overline{BC} = 10, \overline{AC} = 12$ . If $X$ and $Y$ are the feet of the perpendiculars from $P$ to the sides $AB$ and $AC$ , then the minimum possible value of $PX^2 + PY^2$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .
*Proposed by Andrew Wen*
|
1936
| 0 |
12,463 |
If $\cos 2^{\circ} - \sin 4^{\circ} -\cos 6^{\circ} + \sin 8^{\circ} \ldots + \sin 88^{\circ}=\sec \theta - \tan \theta$ , compute $\theta$ in degrees.
*2015 CCA Math Bonanza Team Round #10*
|
94
| 0 |
12,464 |
What real number is equal to the expression $3 + \frac{5}{2 + \frac{5}{3 + \frac{5}{2 + \cdots}}}$, where the $2$s and $3$s alternate?
|
\frac{5}{3}
| 0 |
12,465 |
Compute
\[
\sum_{n = 1}^\infty \frac{1}{n(n + 3)}.
\]
|
\frac{1}{3}
| 0 |
12,466 |
Points are marked on a circle, dividing it into 2012 equal arcs. From these, $k$ points are chosen to construct a convex $k$-gon with vertices at the chosen points. What is the maximum possible value of $k$ such that this polygon has no parallel sides?
|
1509
| 0 |
12,467 |
A 20-step path is to go from $(-5,-5)$ to $(5,5)$ with each step increasing either the $x$-coordinate or the $y$-coordinate by 1. How many such paths stay outside or on the boundary of the rectangle $-3 \le x \le 3$, $-1 \le y \le 1$ at each step?
|
2126
| 0 |
12,468 |
Given that Fox wants to ensure he has 20 coins left after crossing the bridge four times, and paying a $50$-coin toll each time, determine the number of coins that Fox had at the beginning.
|
25
| 0 |
12,469 |
The product of positive integers $a$, $b$, and $c$ equals 2450. What is the minimum possible value of the sum $a + b + c$?
|
76
| 3.125 |
12,470 |
What is the largest four-digit number that is divisible by 6?
|
9960
| 0 |
12,471 |
Three squares \( GQOP, HJNO \), and \( RKMN \) have vertices which sit on the sides of triangle \( FIL \) as shown. The squares have areas of 10, 90, and 40 respectively. What is the area of triangle \( FIL \)?
|
220.5
| 0 |
12,472 |
Given vectors $a=(1,1)$ and $b=(2,t)$, find the value of $t$ such that $|a-b|=a·b$.
|
\frac{-5 - \sqrt{13}}{2}
| 0 |
12,473 |
The sequence consists of 19 ones and 49 zeros arranged in a random order. A group is defined as the maximal subsequence of identical symbols. For example, in the sequence 110001001111, there are five groups: two ones, then three zeros, then one one, then two zeros, and finally four ones. Find the expected value of the length of the first group.
|
2.83
| 0 |
12,474 |
Given a triangular pyramid $S-ABC$ whose base is an equilateral triangle, the projection of point $A$ on the face $SBC$ is the orthocenter $H$ of triangle $\triangle SBC$. The dihedral angle between the planes $H-AB-C$ is $30^{\circ}$, and $SA = 2\sqrt{3}$. Find the volume of the triangular pyramid $S-ABC$.
|
\frac{9\sqrt{3}}{4}
| 7.8125 |
12,475 |
Moe's rectangular lawn measures 100 feet by 160 feet. He uses a mower with a swath that is 30 inches wide, but overlaps each pass by 6 inches to ensure no grass is missed. He mows at a speed of 0.75 miles per hour. What is the approximate time it will take Moe to mow the entire lawn?
|
2.02
| 26.5625 |
12,476 |
There are 3 boys and 3 girls, making a total of 6 students standing in a row.
(1) If the three girls must stand together, find the total number of different arrangements.
(2) If boy A cannot stand at either end, and among the 3 girls, exactly two girls stand together, find the number of different arrangements.
|
288
| 7.03125 |
12,477 |
Given the function f(x) = 3^x with a range of M, where x < -1, find the probability that a number x randomly chosen from the interval (-1, 1) belongs to M.
|
\frac{1}{6}
| 25 |
12,478 |
What is the sum of all the solutions of \( x = |2x - |50-2x|| \)?
|
\frac{170}{3}
| 0 |
12,479 |
Given that the quiz consists of 4 multiple-choice questions, each with 3 choices, calculate the probability that the contestant wins the quiz.
|
\frac{1}{9}
| 25.78125 |
12,480 |
Given that A is a moving point on the ray $x+y=0$ (where $x \leq 0$), and B is a moving point on the positive half of the x-axis, if line AB is tangent to the circle $x^2+y^2=1$, the minimum value of $|AB|$ is _______ .
|
2 + 2\sqrt{2}
| 0 |
12,481 |
If $g(x) = 3x^2 + 4$ and $h(x) = -2x^3 + 2$, what is the value of $g(h(2))$?
|
592
| 97.65625 |
12,482 |
Given an isosceles triangle \(ABC\) with \(AB = AC\) and \(\angle ABC = 53^\circ\), find the measure of \(\angle BAM\). Point \(K\) is such that \(C\) is the midpoint of segment \(AK\). Point \(M\) is chosen such that:
- \(B\) and \(M\) are on the same side of line \(AC\);
- \(KM = AB\);
- \(\angle MAK\) is the maximum possible.
How many degrees is \(\angle BAM\)?
|
44
| 0 |
12,483 |
A pentagon is formed by placing an equilateral triangle on top of a square. Calculate the percentage of the pentagon's total area that is made up by the equilateral triangle.
|
25.4551\%
| 0 |
12,484 |
How many of the natural numbers from 1 to 700, inclusive, contain the digit 0 at least once?
|
123
| 1.5625 |
12,485 |
A hexagon inscribed in a circle has three consecutive sides, each of length 4, and three consecutive sides, each of length 7. The chord of the circle that divides the hexagon into two trapezoids, one with three sides each of length 4, and the other with three sides each of length 7, has length equal to $p/q$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.
|
1017
| 0 |
12,486 |
In three-dimensional space, find the number of lattice points that have a distance of 4 from the origin.
|
42
| 0 |
12,487 |
In three-dimensional space, find the number of lattice points that have a distance of 5 from the origin.
Note: A point is a lattice point if all its coordinates are integers.
|
54
| 0 |
12,488 |
A train moves at a speed of 60 kilometers per hour, making stops every 48 kilometers. The duration of each stop, except for the fifth stop, is 10 minutes. The fifth stop lasts half an hour. What distance did the train travel if it departed at noon on September 29 and arrived at its destination on October 1 at 22:00?
|
2870
| 0 |
12,489 |
Define a function $f$ by $f(1)=1$, $f(2)=2$, and for all integers $n \geq 3$,
\[ f(n) = f(n-1) + f(n-2) + n. \]
Determine $f(10)$.
|
420
| 0 |
12,490 |
In Flower Town, there are $99^{2}$ residents, some of whom are knights (who always tell the truth) and others are liars (who always lie). The houses in the town are arranged in the cells of a $99 \times 99$ square grid (totaling $99^{2}$ houses, arranged on 99 vertical and 99 horizontal streets). Each house is inhabited by exactly one resident. The house number is denoted by a pair of numbers $(x ; y)$, where $1 \leq x \leq 99$ is the number of the vertical street (numbers increase from left to right), and $1 \leq y \leq 99$ is the number of the horizontal street (numbers increase from bottom to top). The flower distance between two houses numbered $\left(x_{1} ; y_{1}\right)$ and $\left(x_{2} ; y_{2}\right)$ is defined as the number $\rho=\left|x_{1}-x_{2}\right|+\left|y_{1}-y_{2}\right|$. It is known that on every vertical or horizontal street, at least $k$ residents are knights. Additionally, all residents know which house Knight Znayka lives in, but you do not know what Znayka looks like. You want to find Znayka's house and you can approach any house and ask the resident: "What is the flower distance from your house to Znayka’s house?". What is the smallest value of $k$ that allows you to guarantee finding Znayka’s house?
|
75
| 0 |
12,491 |
Determine the common rational root \( k \) of the following polynomial equations which is not integral:
\[45x^4 + ax^3 + bx^2 + cx + 8 = 0\]
\[8x^5 + dx^4 + ex^3 + fx^2 + gx + 45 = 0\]
This root \( k \) is assumed to be a negative non-integer.
|
-\frac{1}{3}
| 46.09375 |
12,492 |
Initially, there is a rook on each square of a chessboard. Each move, you can remove a rook from the board which attacks an odd number of rooks. What is the maximum number of rooks that can be removed? (Rooks attack each other if they are in the same row or column and there are no other rooks between them.)
|
59
| 0 |
12,493 |
Given that in triangle PQR, side PR = 6 cm and side PQ = 10 cm, point S is the midpoint of QR, and the length of the altitude from P to QR is 4 cm, calculate the length of QR.
|
4\sqrt{5}
| 2.34375 |
12,494 |
Point \((x,y)\) is randomly picked from the rectangular region with vertices at \((0,0), (3000,0), (3000,4500),\) and \((0,4500)\). What is the probability that \(x < 3y\)? Express your answer as a common fraction.
|
\frac{11}{18}
| 8.59375 |
12,495 |
Determine the sum $25^2 - 23^2 + 21^2 - 19^2 + ... + 3^2 - 1^2.$
|
1196
| 0 |
12,496 |
What is the value of $(-1)^1+(-1)^2+\cdots+(-1)^{2007}$?
|
-1
| 71.875 |
12,497 |
Bus stop \(B\) is located on a straight highway between stops \(A\) and \(C\). After some time driving from \(A\), the bus finds itself at a point on the highway where the distance to one of the three stops is equal to the sum of the distances to the other two stops. After the same amount of time, the bus again finds itself at a point with this property, and 25 minutes later it arrives at \(B\). How much time does the bus need for the entire journey from \(A\) to \(C\) if its speed is constant and it stops at \(B\) for 5 minutes?
|
180
| 0 |
12,498 |
What is the smallest possible area, in square units, of a right triangle with side lengths $7$ units and $10$ units?
|
35
| 24.21875 |
12,499 |
The fictional country of Isoland uses a 6-letter license plate system using the same 12-letter alphabet as the Rotokas of Papua New Guinea (A, E, G, I, K, O, P, R, T, U, V). Design a license plate that starts with a vowel (A, E, I, O, U), ends with a consonant (G, K, P, R, T, V), contains no repeated letters and does not include the letter S.
|
151200
| 10.15625 |
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