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 |
|---|---|---|---|---|---|---|
Two different natural numbers are selected from the set $\{1, 2, 3, \ldots, 7\}$. What is the probability that the greatest common divisor of these two numbers is one? Express your answer as a common fraction. | \frac{17}{21} | 0.5 | 7,138.375 | 6,084.75 | 8,192 | |
In response to the national medical and health system reform and the "Three Visits to the Countryside" cultural and scientific activities nationwide in 2023, to truly implement the concept of "putting the people first" and promote the transfer and decentralization of high-quality medical resources, continuously enhance... | \frac{2}{5} | 0.75 | 5,232.0625 | 4,245.416667 | 8,192 | |
The product of all integers whose absolute value is greater than 3 but not greater than 6 is ____. | -14400 | 0.0625 | 481.75 | 556 | 476.8 | |
Each of the integers $1,2, \ldots, 729$ is written in its base-3 representation without leading zeroes. The numbers are then joined together in that order to form a continuous string of digits: $12101112202122 \ldots \ldots$ How many times in this string does the substring 012 appear? | 148 | Ignore $729=3^{6}=1000000_{3}$ since it will not contribute to a 012 substring. Break into cases on how 012 appears: (i) when an individual integer contains the string 012 ; (ii) when 01 are the last two digits of an integer and 2 is the first digit of the next integer; and (iii) when 0 is the last digit of an integer ... | 0 | 8,192 | -1 | 8,192 |
A circle and two distinct lines are drawn on a sheet of paper. What is the largest possible number of points of intersection of these figures? | 5 | To determine the largest possible number of points of intersection involving a circle and two distinct lines, we need to consider the maximum number of intersections each pair of figures can have:
1. **Intersections between a line and a circle**: A line can intersect a circle at most at two points. This is because the... | 0.9375 | 3,974.9375 | 3,693.8 | 8,192 |
The sides of rectangle $ABCD$ have lengths $10$ and $11$. An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD$. The maximum possible area of such a triangle can be written in the form $p\sqrt{q}-r$, where $p$, $q$, and $r$ are positive integers, and $q$ is not divisible by the square of... | 554 | 0 | 8,119.75 | -1 | 8,119.75 | |
How many ways can the integers from -7 to 7 be arranged in a sequence such that the absolute value of the numbers in the sequence is nondecreasing? | 128 | 0.0625 | 8,149.125 | 7,506 | 8,192 | |
Express $4.\overline{054}$ as a common fraction in lowest terms. | \frac{150}{37} | 1 | 2,304.6875 | 2,304.6875 | -1 | |
Given an isosceles triangle with a vertex angle of 36°, the ratio of the base to the leg is equal to . | \frac{\sqrt{5}-1}{2} | 0 | 6,018.8125 | -1 | 6,018.8125 | |
Given the function $f(x)=\sin (2x+ \frac {π}{3})$, for any $x\_1$, $x\_2$, $x\_3$, and $0\leqslant x\_1 < x\_2 < x\_3\leqslant π$, the equation $|f(x\_1)-f(x\_2)|+|f(x\_2)-f(x\_3)|\leqslant m$ holds true. Find the minimum value of the real number $m$. | 3+ \frac { \sqrt {3}}{2} | 0 | 7,948.875 | -1 | 7,948.875 | |
Let $M$ be the intersection of diagonals of the convex quadrilateral $ABCD$, where $m(\widehat{AMB})=60^\circ$. Let the points $O_1$, $O_2$, $O_3$, $O_4$ be the circumcenters of the triangles $ABM$, $BCM$, $CDM$, $DAM$, respectively. Calculate the ratio of the area of quadrilateral $ABCD$ to the area of quadrilateral $... | 3/2 | 0 | 8,192 | -1 | 8,192 | |
5 students stand in a row for a photo, where students A and B must stand next to each other, and A cannot stand at either end. Calculate the total number of possible arrangements. | 36 | 0.3125 | 7,559.375 | 6,257.2 | 8,151.272727 | |
There are 11 children sitting in a circle playing a game. They are numbered clockwise from 1 to 11. The game starts with child number 1, and each child has to say a two-digit number. The number they say cannot have a digit sum of 6 or 9, and no child can repeat a number that has already been said. The game continues u... | 10 | 0.1875 | 7,038.125 | 5,618 | 7,365.846154 | |
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 | 8,178.125 | -1 | 8,178.125 | |
A parallelogram $ABCD$ is inscribed in the ellipse $\frac{x^{2}}{4}+y^{2}=1$, where the slope of the line $AB$ is $k_{1}=1$. Determine the slope of the line $AD$. | -\frac{1}{4} | 0.0625 | 8,078.875 | 6,382 | 8,192 | |
A square is inscribed in a circle. A smaller square has one side coinciding with a side of the larger square and has two vertices on the circle, as shown. What percent of the area of the larger square is the area of the smaller square?
[asy]
draw(Circle((0,0),1.4142));
draw((1,1)--(1,-1)--(-1,-1)--(-1,1)--cycle);
dr... | 4\% | 0 | 8,192 | -1 | 8,192 | |
How many non-similar quadrilaterals have angles whose degree measures are distinct positive integers in an arithmetic progression? | 29 | 0.4375 | 7,507.9375 | 6,765.285714 | 8,085.555556 | |
In the diagram, what is the measure of $\angle ACB$ in degrees? [asy]
size(250);
draw((-60,0)--(0,0));
draw((0,0)--(64.3,76.6)--(166,0)--cycle);
label("$A$",(64.3,76.6),N);
label("$93^\circ$",(64.3,73),S);
label("$130^\circ$",(0,0),NW);
label("$B$",(0,0),S);
label("$D$",(-60,0),S);
label("$C$",(166,0),S);
[/asy] | 37^\circ | 0.3125 | 8,071.375 | 7,806 | 8,192 | |
Lucas, Emma, and Noah collected shells at the beach. Lucas found four times as many shells as Emma, and Emma found twice as many shells as Noah. Lucas decides to share some of his shells with Emma and Noah so that all three will have the same number of shells. What fraction of his shells should Lucas give to Emma? | \frac{5}{24} | 0.3125 | 1,046.0625 | 1,550 | 817 | |
Equilateral $\triangle ABC$ has side length $600$. Points $P$ and $Q$ lie outside the plane of $\triangle ABC$ and are on opposite sides of the plane. Furthermore, $PA=PB=PC$, and $QA=QB=QC$, and the planes of $\triangle PAB$ and $\triangle QAB$ form a $120^{\circ}$ dihedral angle (the angle between the two planes). Th... | 450 | Draw a good diagram. Draw $CH$ as an altitude of the triangle. Scale everything down by a factor of $100\sqrt{3}$, so that $AB=2\sqrt{3}$. Finally, call the center of the triangle U. Draw a cross-section of the triangle via line $CH$, which of course includes $P, Q$. From there, we can call $OU=h$. There are two crucia... | 0 | 8,048.1875 | -1 | 8,048.1875 |
Given $$∫_{ 0 }^{ 2 }(\cos \frac {π}{4}x+ \sqrt {4-x^{2}})dx$$, evaluate the definite integral. | \pi+\frac{4}{\pi} | 0.1875 | 3,231.0625 | 2,657.666667 | 3,363.384615 | |
Find the largest possible value of $k$ for which $3^{11}$ is expressible as the sum of $k$ consecutive positive integers. | 486 | Let us write down one such sum, with $m$ terms and first term $n + 1$:
$3^{11} = (n + 1) + (n + 2) + \ldots + (n + m) = \frac{1}{2} m(2n + m + 1)$.
Thus $m(2n + m + 1) = 2 \cdot 3^{11}$ so $m$ is a divisor of $2\cdot 3^{11}$. However, because $n \geq 0$ we have $m^2 < m(m + 1) \leq 2\cdot 3^{11}$ so $m < \sqrt{2\cdot ... | 0.25 | 7,758.375 | 6,457.5 | 8,192 |
A five-digit number is called a "pit" if its first three digits are in descending order and the last three digits are in ascending order. For example, 73016 and 98346 are pits, while 88012 and 56821 are not pits. How many pits are there that are less than the number 22222? | 36 | 0.0625 | 8,008 | 7,251 | 8,058.466667 | |
How many natural numbers greater than 10 but less than 100 are relatively prime to 21? | 51 | 0 | 5,887.1875 | -1 | 5,887.1875 | |
A kite is inscribed in a circle with center $O$ and radius $60$ . The diagonals of the kite meet at a point $P$ , and $OP$ is an integer. The minimum possible area of the kite can be expressed in the form $a\sqrt{b}$ , where $a$ and $b$ are positive integers and $b$ is squarefree. Find $a+b$ . | 239 | 0.0625 | 8,134.6875 | 7,275 | 8,192 | |
If $x=1$ is an extremum point of the function $f(x)=(x^{2}+ax-1)e^{x-1}$, determine the maximum value of $f(x)$. | 5e^{-3} | 0 | 4,368 | -1 | 4,368 | |
The set of points $(x,y,z)$ that are equidistant to $(1,2,-5)$ and point $P$ satisfy an equation of the form
\[10x - 4y + 24z = 55.\]Find the point $P.$ | (6,0,7) | 0.625 | 4,832.6875 | 2,817.1 | 8,192 | |
Let $\mathbf{a} = \begin{pmatrix} 4 \\ -3 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 2 \\ 6 \end{pmatrix}.$ Calculate the area of the triangle with vertices $\mathbf{0},$ $\mathbf{a},$ and $\mathbf{b}$. Also, determine the vector $\mathbf{a} + \mathbf{b}$. | 15 | 0.0625 | 1,460.6875 | 2,065 | 1,420.4 | |
How many positive four-digit integers are divisible by both 13 and 7? | 99 | 0.9375 | 3,986.625 | 3,706.266667 | 8,192 | |
Given that point \( P \) lies on the hyperbola \( \frac{x^2}{16} - \frac{y^2}{9} = 1 \), and the distance from \( P \) to the right directrix of this hyperbola is the arithmetic mean of the distances from \( P \) to the two foci of the hyperbola, find the x-coordinate of \( P \). | -\frac{64}{5} | 0.125 | 8,164 | 7,968 | 8,192 | |
$10 \cdot 52 \quad 1990-1980+1970-1960+\cdots-20+10$ equals: | 1000 | 0 | 7,891.875 | -1 | 7,891.875 | |
Given the function $f(x) = \cos x \cdot \sin \left( x + \frac{\pi}{3} \right) - \sqrt{3}\cos^2 x + \frac{\sqrt{3}}{4}, x \in \mathbb{R}$,
$(1)$ Find the smallest positive period of $f(x)$;
$(2)$ Find the maximum and minimum values of $f(x)$ on the closed interval $\left[ -\frac{\pi}{4}, \frac{\pi}{4} \right]$. | -\frac{1}{2} | 0.5625 | 6,530.0625 | 6,316 | 6,805.285714 | |
Four horizontal lines and four vertical lines are drawn in a plane. In how many ways can four lines be chosen such that a rectangular region is enclosed? | 36 | 0.9375 | 3,113 | 2,774.4 | 8,192 | |
Let $A$ , $B$ , and $C$ be distinct points on a line with $AB=AC=1$ . Square $ABDE$ and equilateral triangle $ACF$ are drawn on the same side of line $BC$ . What is the degree measure of the acute angle formed by lines $EC$ and $BF$ ?
*Ray Li* | 75 | 0.875 | 4,412.5 | 3,872.571429 | 8,192 | |
The scores on a $110$-point test were organized in the stem-and-leaf plot shown. $9 | 6$ represents $96$ points. What is the mode of the scores? \begin{tabular}{c|lllllll}
\multicolumn{8}{c}{\underline{Points on the Test}}\\
5 &0 & 0 & & & & &\\
6 &3 & & & & & &\\
7 &7 & 8 & & & & &\\
8 &2 & 6 & 7 & 9 & 9 & 9 & 9\\
9 &... | 89 | 0.9375 | 2,117.5 | 1,751.533333 | 7,607 | |
Semicircles of diameter 4 inches are aligned in a linear pattern, with a second row staggered under the first such that the flat edges of the semicircles in the second row touch the midpoints of the arcs in the first row. What is the area, in square inches, of the shaded region in an 18-inch length of this pattern? Exp... | 16\pi | 0 | 7,637.0625 | -1 | 7,637.0625 | |
Given the equation $x^2 + y^2 = |x| + 2|y|$, calculate the area enclosed by the graph of this equation. | \frac{5\pi}{4} | 0 | 8,192 | -1 | 8,192 | |
An abundant number is a positive integer such that the sum of its proper divisors is greater than the number itself. Find the smallest abundant number that is not a multiple of 10. | 12 | 0.9375 | 3,859.375 | 3,570.533333 | 8,192 | |
Calculate $\left(\sqrt{(\sqrt{5})^4}\right)^6$. | 15625 | 1 | 3,610.1875 | 3,610.1875 | -1 | |
Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function satisfying the following conditions:
(1) $f(1)=1$;
(2) $\forall n\in \mathbb{N}$, $3f(n) f(2n+1) =f(2n) ( 1+3f(n) )$;
(3) $\forall n\in \mathbb{N}$, $f(2n) < 6 f(n)$.
Find all solutions of equation $f(k) +f(l)=293$, where $k<l$.
($\mathbb{N}$ denotes the set of a... | (5, 47), (7, 45), (13, 39), (15, 37) |
Let \( f: \mathbb{N} \rightarrow \mathbb{N} \) be a function satisfying the following conditions:
1. \( f(1) = 1 \);
2. For all \( n \in \mathbb{N} \), \( 3f(n) f(2n+1) = f(2n) (1 + 3f(n)) \);
3. For all \( n \in \mathbb{N} \), \( f(2n) < 6 f(n) \).
We need to find all solutions of the equation \( f(k) + f(l) = 293 \... | 0 | 8,192 | -1 | 8,192 |
A sequence is defined as follows $a_1=a_2=a_3=1,$ and, for all positive integers $n, a_{n+3}=a_{n+2}+a_{n+1}+a_n.$ Given that $a_{28}=6090307, a_{29}=11201821,$ and $a_{30}=20603361,$ find the remainder when $\sum^{28}_{k=1} a_k$ is divided by 1000. | 834 | Since the problem only asks for the first 28 terms and we only need to calculate mod 1000, we simply bash the first 28 terms:
$a_{1}\equiv 1 \pmod {1000} \newline a_{2}\equiv 1 \pmod {1000} \newline a_{3}\equiv 1 \pmod {1000} \newline a_{4}\equiv 3 \pmod {1000} \newline a_{5}\equiv 5 \pmod {1000} \newline \cdots \newli... | 0.125 | 7,971.4375 | 6,427.5 | 8,192 |
Sixty points, of which thirty are coloured red, twenty are coloured blue and ten are coloured green, are marked on a circle. These points divide the circle into sixty arcs. Each of these arcs is assigned a number according to the colours of its endpoints: an arc between a red and a green point is assigned a number $1$... | 180 | 0 | 8,192 | -1 | 8,192 | |
The hypotenuse of a right triangle, where the legs are consecutive whole numbers, is 53 units long. What is the sum of the lengths of the two legs? | 75 | 0 | 8,192 | -1 | 8,192 | |
Given complex numbers \( z_{1}, z_{2}, z_{3} \) such that \( \left|z_{1}\right| \leq 1 \), \( \left|z_{2}\right| \leq 1 \), and \( \left|2 z_{3}-\left(z_{1}+z_{2}\right)\right| \leq \left|z_{1}-z_{2}\right| \). What is the maximum value of \( \left|z_{3}\right| \)? | \sqrt{2} | 0.25 | 8,067.6875 | 7,694.75 | 8,192 | |
In a grade, Class 1, Class 2, and Class 3 each select two students (one male and one female) to form a group of high school students. Two students are randomly selected from this group to serve as the chairperson and vice-chairperson. Calculate the probability of the following events:
- The two selected students are no... | \dfrac{2}{5} | 0.1875 | 7,521.625 | 6,106 | 7,848.307692 | |
The volume of the solid formed by rotating an isosceles right triangle with legs of length 1 around its hypotenuse is __________. | \frac{\sqrt{2}}{6}\pi | 0 | 7,727.125 | -1 | 7,727.125 | |
A three-digit positive integer \( n \) has digits \( a, b, c \). (That is, \( a \) is the hundreds digit of \( n \), \( b \) is the tens digit of \( n \), and \( c \) is the ones (units) digit of \( n \).) Determine the largest possible value of \( n \) for which:
- \( a \) is divisible by 2,
- the two-digit integer \(... | 870 | 0 | 8,192 | -1 | 8,192 | |
Given that the domains of functions $f(x)$ and $g(x)$ are both $\mathbb{R}$, and $f(x) + g(2-x) = 5$, $g(x) - f(x-4) = 7$. If the graph of $y = g(x)$ is symmetric about the line $x = 2$, $g(2) = 4$, find the sum of the values of $f(k)$ from $k=1$ to $k=22$. | -24 | 0.25 | 7,733 | 6,356 | 8,192 | |
The numbers $1, 2, 3, \dots, 1024$ are written on a blackboard. They are divided into pairs. Then each pair is wiped off the board and non-negative difference of its numbers is written on the board instead. $512$ numbers obtained in this way are divided into pairs and so on. One number remains on the blackboard after t... | 0, 2, 4, 6, \ldots, 1022 |
We begin with the numbers \(1, 2, 3, \ldots, 1024\) written on a blackboard. The task involves repeatedly dividing these numbers into pairs, taking the non-negative difference of each pair, and then continuing this process until only one number remains after ten such operations. We aim to determine all possible final ... | 0 | 8,192 | -1 | 8,192 |
A line containing the points $(9, 1)$ and $(5, 5)$ intersects the $x$ -axis at what point? | (10, 0) | 1 | 2,126.875 | 2,126.875 | -1 | |
Find the largest five-digit positive integer such that it is not a multiple of 11, and any number obtained by deleting some of its digits is also not divisible by 11. | 98765 | 0 | 8,034.3125 | -1 | 8,034.3125 | |
Seven people stand in a row. (Write out the necessary process, and use numbers for the answers)
(1) How many ways can person A and person B stand next to each other?
(2) How many ways can person A and person B stand not next to each other?
(3) How many ways can person A, person B, and person C stand so that no tw... | 4320 | 0.1875 | 6,880.1875 | 4,656.333333 | 7,393.384615 | |
A regular hexagon is inscribed in another regular hexagon such that each vertex of the inscribed hexagon divides a side of the original hexagon into two parts in the ratio 2:1. Find the ratio of the area of the inscribed hexagon to the area of the larger hexagon. | 7/9 | 0.25 | 7,845.4375 | 7,448 | 7,977.916667 | |
Let point P be a fixed point inside a circle ⊙O with a radius of 5, and OP=4. The sum of all possible integer values of the chord lengths passing through point P is. | 40 | 0.8125 | 4,858.8125 | 4,666.307692 | 5,693 | |
James has 7 apples. 4 of them are red, and 3 of them are green. If he chooses 2 apples at random, what is the probability that both the apples he chooses are green? | \frac{1}{7} | 1 | 1,220.375 | 1,220.375 | -1 | |
Given two vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ in a plane, satisfying $|\overrightarrow {a}|=1$, $|\overrightarrow {b}|=2$, and the dot product $(\overrightarrow {a}+ \overrightarrow {b})\cdot (\overrightarrow {a}-2\overrightarrow {b})=-7$, find the angle between vectors $\overrightarrow {a}$ and $\o... | \frac{\pi}{2} | 0.6875 | 3,520.4375 | 3,549.727273 | 3,456 | |
Find the sum $$\frac{3^1}{9^1 - 1} + \frac{3^2}{9^2 - 1} + \frac{3^3}{9^3 - 1} + \frac{3^4}{9^4 - 1} + \cdots.$$ | \frac{1}{2} | 0 | 8,192 | -1 | 8,192 | |
There exists a unique positive integer $a$ for which the sum \[U=\sum_{n=1}^{2023}\left\lfloor\dfrac{n^{2}-na}{5}\right\rfloor\] is an integer strictly between $-1000$ and $1000$. For that unique $a$, find $a+U$.
(Note that $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x$.) | 944 | Consider the integral \[\int_{0}^{2023} \dfrac{n^2-na}{5} \, dn.\] We hope this will give a good enough appoximation of $U$ to find $a.$ However, this integral can be easily evaluated(if you know calculus) to be \[\dfrac{1}{15}2023^3-\dfrac{a}{10}2023^2=2023^2\left(\dfrac{2023}{15}-\dfrac{a}{10}\right).\] Because we wa... | 0 | 8,192 | -1 | 8,192 |
What is the sum of all two-digit positive integers whose squares end with the digits 25? | 495 | 0.875 | 3,351.0625 | 3,011.214286 | 5,730 | |
The mean age of Amanda's $4$ cousins is $8$, and their median age is $5$. What is the sum of the ages of Amanda's youngest and oldest cousins? | 22 | 1. **Calculate the total sum of the ages**: Given that the mean age of Amanda's 4 cousins is 8, the total sum of their ages is calculated by multiplying the mean by the number of cousins:
\[
\text{Total sum} = 4 \times 8 = 32
\]
2. **Understand the median**: The median age of the cousins is 5. Since there are... | 1 | 1,817.8125 | 1,817.8125 | -1 |
Simplify $\left(\dfrac{-1+i\sqrt{3}}{2}\right)^6+\left(\dfrac{-1-i\sqrt{3}}{2}\right)^6.$ | 2 | 0.875 | 4,520.0625 | 3,995.5 | 8,192 | |
Let the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ be $S_n$, and it satisfies $S_{2014} > 0$, $S_{2015} < 0$. For any positive integer $n$, it holds that $|a_n| \geqslant |a_k|$, determine the value of $k$. | 1008 | 0 | 7,900.6875 | -1 | 7,900.6875 | |
What is the shortest distance between the circles defined by $x^2-10x +y^2-4y-7=0$ and $x^2+14x +y^2+6y+49=0$? | 4 | 1 | 2,836.375 | 2,836.375 | -1 | |
How many perfect squares less than 20,000 can be represented as the difference of squares of two integers that differ by 2? | 70 | 0.125 | 7,208.125 | 2,752 | 7,844.714286 | |
In triangle $\triangle ABC$, the opposite sides of angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\sqrt{3}b\cos A - a\sin B = 0$. $D$ is the midpoint of $AB$, $AC = 2$, and $CD = 2\sqrt{3}$. Find:
$(Ⅰ)$ The measure of angle $A$;
$(Ⅱ)$ The value of $a$. | 2\sqrt{13} | 0.6875 | 6,010.25 | 5,018.545455 | 8,192 | |
Suppose the function \( g(x) \) is defined for \( x \in \{a, b, c\} \) such that the points \( (a, g(a)), (b, g(b)), (c, g(c)) \) form a triangle with an area of \( 24 \). The graph of \( y = 3g(3x) \) likewise contains just three points. Determine the area of this triangle. | 24 | 0.5625 | 5,941.625 | 5,108.222222 | 7,013.142857 | |
A quadratic equation $ax^2 - 2ax + b = 0$ has two real solutions. What is the average of these two solutions? | 1 | 1 | 2,290.6875 | 2,290.6875 | -1 | |
Find the product of all $x$ such that the expression $\frac{x^2+2x+1}{x^2+2x-3}$ is undefined. | -3 | 1 | 1,085.375 | 1,085.375 | -1 | |
A triangle ABC has vertices at points $A = (0,2)$, $B = (0,0)$, and $C = (10,0)$. A vertical line $x = a$ divides the triangle into two regions. Find the value of $a$ such that the area to the left of the line is one-third of the total area of triangle ABC.
A) $\frac{10}{3}$
B) $5$
C) $\frac{15}{4}$
D) $2$ | \frac{10}{3} | 0 | 8,192 | -1 | 8,192 | |
Let S be the statement "If the sum of the digits of the whole number $n$ is divisible by $6$, then $n$ is divisible by $6$." A value of $n$ which shows $S$ to be false is | 33 | To find a counterexample to the statement "If the sum of the digits of the whole number $n$ is divisible by $6$, then $n$ is divisible by $6$," we need to check each option to see if the sum of its digits is divisible by $6$, and then verify if the number itself is not divisible by $6$.
1. **Option A: $n = 30$**
- ... | 0 | 4,829.9375 | -1 | 4,829.9375 |
Given that $f(x) = \frac{1}{3^x + \sqrt{3}}$, find the value of $f(-12) + f(-11) + f(-10) + ... + f(0) + ... + f(11) + f(12) + f(13)$. | \frac{13\sqrt{3}}{3} | 0 | 8,031.1875 | -1 | 8,031.1875 | |
Let $\mathbf{A} = \begin{pmatrix} a & 1 \\ -2 & d \end{pmatrix}$ for some real numbers $a$ and $d.$ If
\[\mathbf{A} + \mathbf{A}^{-1} = \mathbf{0},\]then find $\det \mathbf{A}.$ | 1 | 0.9375 | 3,666.1875 | 3,735.466667 | 2,627 | |
In what ratio does the angle bisector of an acute angle of an isosceles right triangle divide the area of the triangle? | 1 : \sqrt{2} | 0 | 6,018.375 | -1 | 6,018.375 | |
What is the $100$th digit after the decimal point when $\frac{3}{26}$ is expressed as a decimal? | 3 | 0.3125 | 4,257.0625 | 5,167.2 | 3,843.363636 | |
12 points are placed around the circumference of a circle. How many ways are there to draw 6 non-intersecting chords joining these points in pairs? | 132 | $C$ (number of chords) $=C(6)=132$. | 0.75 | 4,836.4375 | 3,717.916667 | 8,192 |
Let QR = x, PR = y, and PQ = z. Given that the area of the square on side QR is 144 = x^2 and the area of the square on side PR is 169 = y^2, find the area of the square on side PQ. | 25 | 0.125 | 2,561.5 | 3,872 | 2,374.285714 | |
During the 2013 National Day, a city organized a large-scale group calisthenics performance involving 2013 participants, all of whom were students from the third, fourth, and fifth grades. The students wore entirely red, white, or blue sports uniforms. It was known that the fourth grade had 600 students, the fifth grad... | 213 | 0 | 8,192 | -1 | 8,192 | |
Two lines have the same non-zero $y$-intercept. The first line has a slope of 10 and an $x$-intercept of $(s, 0)$. The second line has a slope of 6 and an $x$-intercept of $(t, 0)$. What is the ratio of $s $ to $t$? Express your answer as a common fraction. | \frac 35 | 1 | 1,591.0625 | 1,591.0625 | -1 | |
Let $x$ be chosen randomly from the interval $(0,1)$. What is the probability that $\lfloor\log_{10}5x\rfloor - \lfloor\log_{10}x\rfloor = 0$? Here $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x$.
A) $\frac{1}{9}$
B) $\frac{1}{10}$
C) $\frac{1}{8}$
D) $\frac{1}{7}$
E) $\frac{1}{6}$ | \frac{1}{9} | 0 | 7,725.3125 | -1 | 7,725.3125 | |
We flip a fair coin 10 times. What is the probability that we get heads in exactly 8 of the 10 flips? | \dfrac{45}{1024} | 1 | 2,759.625 | 2,759.625 | -1 | |
Emma has $100$ fair coins. She flips all the coins once. Any coin that lands on tails is tossed again, and the process continues up to four times for coins that sequentially land on tails. Calculate the expected number of coins that finally show heads. | 93.75 | 0.6875 | 5,356.5625 | 5,196.454545 | 5,708.8 | |
The inclination angle of the line $\sqrt {3}x-y+1=0$ is \_\_\_\_\_\_. | \frac {\pi}{3} | 0.0625 | 1,279.5625 | 1,598 | 1,258.333333 | |
Find the sum of the $x$-coordinates of the distinct points of intersection of the plane curves given by $x^{2}=x+y+4$ and $y^{2}=y-15 x+36$. | 0 | Substituting $y=x^{2}-x-4$ into the second equation yields $$\begin{aligned} 0 & =\left(x^{2}-x-4\right)^{2}-\left(x^{2}-x-4\right)+15 x-36 \\ & =x^{4}-2 x^{3}-7 x^{2}+8 x+16-x^{2}+x+4+15 x-36 \\ & =x^{4}-2 x^{3}-8 x^{2}+24 x-16 \\ & =(x-2)\left(x^{3}-8 x+8\right)=(x-2)^{2}\left(x^{2}+2 x-4\right) \end{aligned}$$ This ... | 0.1875 | 7,718.9375 | 5,669 | 8,192 |
In $\triangle ABC$, $b^{2}=ac$, and $a+c=3$, $\cos B= \frac{3}{4}$, then $\overset{→}{AB} \cdot \overset{→}{BC} =$______. | -\frac{3}{2} | 0.5 | 6,463.1875 | 5,390.375 | 7,536 | |
How many triangles can be formed using the vertices of a regular dodecagon (a 12-sided polygon)? | 220 | 0.875 | 3,518.3125 | 3,057.785714 | 6,742 | |
The minimum number of digits to the right of the decimal point needed to express the fraction $\frac{987654321}{2^{30}\cdot 5^3}$ as a decimal. | 30 | 0.5625 | 6,059.75 | 4,401.333333 | 8,192 | |
On June 14, 2018, the 21st FIFA World Cup will kick off in Russia. A local sports channel organized fans to guess the outcomes of the matches for the top four popular teams: Germany, Spain, Argentina, and Brazil. Each fan can choose one team from the four, and currently, three people are participating in the guessing g... | \frac{17}{15} | 0.5625 | 6,663 | 5,989.333333 | 7,529.142857 | |
Alice, Bob, and Carol each start with $3. Every 20 seconds, a bell rings, at which time each of the players who has an odd amount of money must give $1 to one of the other two players randomly. What is the probability that after the bell has rung 100 times, each player has exactly $3?
A) $\frac{1}{3}\quad$ B) $\frac{1}... | \frac{8}{13} | 0 | 8,192 | -1 | 8,192 | |
The set \( M = \left\{(x, y) \mid \log_{4} x + \log_{4} y \leq 1, x, y \in \mathbf{N}^{*}\right\} \) has how many subsets? | 256 | 1 | 2,754.6875 | 2,754.6875 | -1 | |
A painting $18$" X $24$" is to be placed into a wooden frame with the longer dimension vertical. The wood at the top and bottom is twice as wide as the wood on the sides. If the frame area equals that of the painting itself, the ratio of the smaller to the larger dimension of the framed painting is: | 2:3 | 1. **Define the variables:**
Let the width of the frame on the sides be $x$ inches. Then, the width of the frame on the top and bottom is $2x$ inches, since it is twice as wide.
2. **Determine the overall dimensions of the framed painting:**
- The width of the framed painting (horizontal dimension) is $18 + 2x$ ... | 0.0625 | 5,342.9375 | 8,192 | 5,153 |
If \( p \) and \( q \) are prime numbers, the number of divisors \( d(a) \) of a natural number \( a = p^{\alpha} q^{\beta} \) is given by the formula
$$
d(a) = (\alpha+1)(\beta+1).
$$
For example, \( 12 = 2^2 \times 3^1 \), the number of divisors of 12 is
$$
d(12) = (2+1)(1+1) = 6,
$$
and the divisors are \( 1, 2, 3... | 450 | 0.1875 | 7,274.6875 | 4,744.666667 | 7,858.538462 | |
A magic square is an array of numbers in which the sum of the numbers in each row, in each column, and along the two main diagonals are equal. The numbers in the magic square shown are not written in base 10. For what base will this be a magic square?
[asy]
unitsize(0.75cm);
for (int i=0; i<4; ++i) {
draw((0,i)--(3,i)... | 5 | 0 | 8,192 | -1 | 8,192 | |
Given the function $f(x)=2\sin ωx (ω > 0)$, if the minimum value of $f(x)$ in the interval $\left[ -\frac{\pi}{3}, \frac{\pi}{4} \right]$ is $-2$, find the minimum value of $ω$. | \frac{3}{2} | 0.3125 | 7,378.75 | 6,320.2 | 7,859.909091 | |
Only nine out of the original thirteen colonies had to ratify the U.S. Constitution in order for it to take effect. What is this ratio, nine to thirteen, rounded to the nearest tenth? | 0.7 | 1 | 2,505.0625 | 2,505.0625 | -1 | |
How many ways can we put 3 math books and 5 English books on a shelf if all the math books must stay together and all the English books must also stay together? (The math books are all different and so are the English books.) | 1440 | 0.875 | 4,282.125 | 3,723.571429 | 8,192 | |
Given the sets $A={1,4,x}$ and $B={1,2x,x^{2}}$, if $A \cap B={4,1}$, find the value of $x$. | -2 | 0.5625 | 6,764.75 | 6,989.333333 | 6,476 | |
In triangle \(ABC\), the angle at vertex \(B\) is \(\frac{\pi}{3}\), and the line segments connecting the incenter to vertices \(A\) and \(C\) are 4 and 6, respectively. Find the radius of the circle inscribed in triangle \(ABC\). | \frac{6 \sqrt{3}}{\sqrt{19}} | 0 | 7,757.625 | -1 | 7,757.625 | |
Let \( a \), \( b \), and \( c \) be the roots of the polynomial equation \( x^3 - 2x^2 + x - 1 = 0 \). Calculate \( \frac{1}{a-2} + \frac{1}{b-2} + \frac{1}{c-2} \). | -5 | 0.75 | 5,766.5625 | 4,958.083333 | 8,192 | |
Given a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a>0, b>0$) with a point C on it, a line passing through the center of the hyperbola intersects the hyperbola at points A and B. Let the slopes of the lines AC and BC be $k_1$ and $k_2$ respectively. Find the eccentricity of the hyperbola when $\frac{2}{k_1 k_2... | \sqrt{3} | 0.25 | 7,613.3125 | 5,877.25 | 8,192 | |
The sum of three consecutive one-digit, positive, odd integers is one-seventh of the product of the same three integers. What is the middle integer when the three integers are listed in ascending order? | 5 | 1 | 2,110.625 | 2,110.625 | -1 |
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