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 |
|---|---|---|---|---|---|---|
A student's final score on a 150-point test is directly proportional to the time spent studying multiplied by a difficulty factor for the test. The student scored 90 points on a test with a difficulty factor of 1.5 after studying for 2 hours. What score would the student receive on a second test of the same format if t... | 300 | 0.9375 | 564.125 | 560.2 | 623 | |
How many degrees are there in the measure of angle $P?$
[asy]
size (5cm,5cm);
pair A,B,C,D,E;
A=(0,1.1);
B=(4.5,0);
C=(6.4,1.7);
D=(4.2,5);
E=(0.5,4.2);
draw (A--B--C--D--E--A,linewidth(1));
label("$P$",A,SW);
label("$128^\circ$",shift(0,0.6)*B);
label("$92^\circ$",C,W);
label("$113^\circ$",shift(-0.3,-0.5)*D);
la... | 96^\circ | 0.75 | 4,175.375 | 2,836.5 | 8,192 | |
Given a circle of radius $2$, there are many line segments of length $2$ that are tangent to the circle at their midpoints. Find the area of the region consisting of all such line segments. | \pi | 1. **Identify the Geometry of the Problem:**
Let $AB$ be a line segment of length $2$, tangent to a circle $O$ at its midpoint $P$. The radius of circle $O$, $OP$, is $2$. Since $P$ is the midpoint of $AB$, we have $AP = PB = 1$.
2. **Use the Right Triangle Property:**
Since $AB$ is tangent to the circle at $P$,... | 0 | 8,192 | -1 | 8,192 |
Find the smallest sum of six consecutive prime numbers that is divisible by 5. | 90 | 0.8125 | 5,480.5 | 4,854.769231 | 8,192 | |
How many $x$-intercepts does the graph of the parabola $x = -2y^2 + y + 1$ have? | 1 | 0.6875 | 5,641.5625 | 4,883.727273 | 7,308.8 | |
Given that $abc$ represents a three-digit number, if it satisfies $a \lt b$ and $b \gt c$, then we call this three-digit number a "convex number". The number of three-digit "convex" numbers without repeated digits is ______. | 204 | 0.1875 | 7,846.9375 | 7,253.333333 | 7,983.923077 | |
In triangle \( \triangle ABC \), \(\angle A\) is the smallest angle, \(\angle B\) is the largest angle, and \(2 \angle B = 5 \angle A\). If the maximum value of \(\angle B\) is \(m^{\circ}\) and the minimum value of \(\angle B\) is \(n^{\circ}\), then find \(m + n\). | 175 | 0.6875 | 5,347.9375 | 4,505.909091 | 7,200.4 | |
Given an arithmetic-geometric sequence ${a_n}$ with the sum of its first $n$ terms denoted as $S_n$, and it is known that $\frac{S_6}{S_3} = -\frac{19}{8}$ and $a_4 - a_2 = -\frac{15}{8}$. Find the value of $a_3$. | \frac{9}{4} | 0 | 8,192 | -1 | 8,192 | |
Suppose
\[
3 + \frac{1}{1 + \frac{1}{3 + \frac{3}{4+y}}} = \frac{169}{53}.
\]
Solve for the value of $y$. | \frac{-605}{119} | 0 | 4,902.375 | -1 | 4,902.375 | |
Sunshine High School is planning to order a batch of basketballs and jump ropes from an online store. After checking on Tmall, they found that each basketball is priced at $120, and each jump rope is priced at $25. There are two online stores, Store A and Store B, both offering free shipping and their own discount sche... | 5700 | 0.0625 | 7,080.875 | 5,303 | 7,199.4 | |
If $x \cdot (x+y) = x^2 + 8$, what is the value of $xy$? | 8 | 1 | 1,268.75 | 1,268.75 | -1 | |
A box contains 5 white balls and 6 black balls. You draw them out of the box, one at a time. What is the probability that the first four draws alternate in colors, starting with a black ball? | \frac{2}{33} | 0.125 | 4,660.75 | 5,793 | 4,499 | |
Sarah's six assignment scores are 87, 90, 86, 93, 89, and 92. What is the arithmetic mean of these six scores? | 89.5 | 1 | 2,240.0625 | 2,240.0625 | -1 | |
The numerator of a fraction is $6x + 1$, then denominator is $7 - 4x$, and $x$ can have any value between $-2$ and $2$, both included. The values of $x$ for which the numerator is greater than the denominator are: | \frac{3}{5} < x \le 2 | We are given a fraction with the numerator $6x + 1$ and the denominator $7 - 4x$. We need to find the values of $x$ for which the numerator is greater than the denominator. This can be set up as an inequality:
\[ 6x + 1 > 7 - 4x. \]
1. **Isolate $x$:**
\[ 6x + 1 > 7 - 4x \]
\[ 6x + 4x > 7 - 1 \] (adding $4x$ to... | 0 | 4,385.125 | -1 | 4,385.125 |
Let $f(x)=3x+4$ and $g(x)=2x-3$. If $h(x)=f(g(x))$, then what is the inverse of $h(x)$? | \frac{x+5}{6} | 1 | 1,466.625 | 1,466.625 | -1 | |
Given the complex numbers \( z_{1} = -\sqrt{3} - i \), \( z_{2} = 3 + \sqrt{3} i \), and \( z = (2 + \cos \theta) + i \sin \theta \), find the minimum value of \( \left|z - z_{1}\right| + \left|z - z_{2}\right| \). | 2 + 2\sqrt{3} | 0.0625 | 8,119.1875 | 8,154 | 8,116.866667 | |
In triangle $PQR$, $\angle Q=90^\circ$, $PQ=9$ and $QR=12$. Points $S$ and $T$ are on $\overline{PR}$ and $\overline{QR}$, respectively, and $\angle PTS=90^\circ$. If $ST=6$, then what is the length of $PS$? | 10 | 0 | 7,895.375 | -1 | 7,895.375 | |
Let $c = \frac{2\pi}{11}.$ What is the value of
\[\frac{\sin 3c \cdot \sin 6c \cdot \sin 9c \cdot \sin 12c \cdot \sin 15c}{\sin c \cdot \sin 2c \cdot \sin 3c \cdot \sin 4c \cdot \sin 5c}?\] | 1 | 1. **Define the constant and simplify the expression:**
Let \( c = \frac{2\pi}{11} \). We need to evaluate:
\[
\frac{\sin 3c \cdot \sin 6c \cdot \sin 9c \cdot \sin 12c \cdot \sin 15c}{\sin c \cdot \sin 2c \cdot \sin 3c \cdot \sin 4c \cdot \sin 5c}
\]
2. **Substitute \( c \) into the expression:**
\[
... | 0.125 | 7,790.75 | 6,712.5 | 7,944.785714 |
Circle $\Gamma$ with radius $1$ is centered at point $A$ on the circumference of circle $\omega$ with radius $7$ . Suppose that point $P$ lies on $\omega$ with $AP=4$ . Determine the product of the distances from $P$ to the two intersections of $\omega$ and $\Gamma$ .
*2018 CCA Math Bonanza Team Rou... | 15 | 0.9375 | 5,213.25 | 5,014.666667 | 8,192 | |
Compute: \( 4.165 \times 4.8 + 4.165 \times 6.7 - 4.165 \div \frac{2}{3} = \) | 41.65 | 0 | 551.6875 | -1 | 551.6875 | |
If the average of a sample $m$, $4$, $6$, $7$ is $5$, then the variance of this sample is ______. | \frac{5}{2} | 0.0625 | 3,470.375 | 7,624 | 3,193.466667 | |
How many three-digit whole numbers have at least one 8 or at least one 9 as digits? | 452 | 0.3125 | 6,936.625 | 4,320.4 | 8,125.818182 | |
What is the value of $23^2 + 2(23)(2) + 2^2$? | 625 | 1 | 1,767.75 | 1,767.75 | -1 | |
Find the largest solution to \[
\lfloor x \rfloor = 7 + 150 \{ x \},
\] where $\{x\} = x - \lfloor x \rfloor$. | 156.9933 | 0 | 3,851.75 | -1 | 3,851.75 | |
When the square root of \( x \) is cubed, the result is 100. What is the value of \( x \)? | 10^{\frac{4}{3}} | 0 | 5,258.625 | -1 | 5,258.625 | |
The polynomial \( g(x) = x^4 + ax^3 + bx^2 + cx + d \) has real coefficients, with the roots \( 3i \) and \( 1+2i \). Calculate the sum of the coefficients \( a + b + c + d \). | 39 | 0.9375 | 4,025.3125 | 3,931.333333 | 5,435 | |
Determine the number of ordered pairs of positive integers \((a, b)\) satisfying the equation
\[ 100(a + b) = ab - 100. \] | 18 | 0.3125 | 6,205.375 | 4,156.6 | 7,136.636364 | |
Three of the following test scores are Cyprian's and the other three are Margaret's: 85, 87, 92, 93, 94, 98. Cyprian's mean score is 90. What is Margaret's mean score? | 93 | 0.9375 | 2,921.1875 | 2,569.8 | 8,192 | |
As shown in the diagram, three circles intersect to create seven regions. Fill the integers $0 \sim 6$ into the seven regions such that the sum of the four numbers within each circle is the same. What is the maximum possible value of this sum? | 15 | 0.0625 | 7,828.5625 | 3,659 | 8,106.533333 | |
Given that M is a point on the parabola $y^2 = 2px$ ($p > 0$), F is the focus of the parabola $C$, and $|MF| = p$. K is the intersection point of the directrix of the parabola $C$ and the x-axis. Calculate the measure of angle $\angle MKF$. | 45 | 1 | 4,201.125 | 4,201.125 | -1 | |
Matt's four cousins are coming to visit. There are four identical rooms that they can stay in. If any number of the cousins can stay in one room, how many different ways are there to put the cousins in the rooms? | 15 | 0.5625 | 6,064.6875 | 4,702.777778 | 7,815.714286 | |
Let $a_{0}, a_{1}, \ldots$ be a sequence such that $a_{0}=3, a_{1}=2$, and $a_{n+2}=a_{n+1}+a_{n}$ for all $n \geq 0$. Find $\sum_{n=0}^{8} \frac{a_{n}}{a_{n+1} a_{n+2}}$ | \frac{105}{212} | We can re-write $\frac{a_{n}}{a_{n+1} a_{n+2}}$ as $\frac{a_{n+2}-a_{n+1}}{a_{n+1} a_{n+2}}=\frac{1}{a_{n+1}}-\frac{1}{a_{n+2}}$. We can thus re-write the sum as $$\left(\frac{1}{a_{1}}-\frac{1}{a_{2}}\right)+\left(\frac{1}{a_{2}}-\frac{1}{a_{3}}\right)+\left(\frac{1}{a_{4}}-\frac{1}{a_{3}}\right)+\ldots+\left(\frac{1}... | 0.875 | 5,304.5 | 4,892 | 8,192 |
Solve the following system of equations in integer numbers:
$$\begin{cases} x^2 = yz + 1 \\ y^2 = zx + 1 \\ z^2 = xy + 1 \end{cases}$$ | (1, 0, -1) |
To solve the given system of equations in integer numbers:
\[
\begin{cases}
x^2 = yz + 1 \\
y^2 = zx + 1 \\
z^2 = xy + 1
\end{cases}
\]
we need to find integer solutions \((x, y, z)\).
### Analysis
First, consider the symmetry of the problem; each equation is structurally similar, suggesting potential symmetry i... | 0.125 | 5,885.75 | 5,511.5 | 5,939.214286 |
One angle of a parallelogram is 150 degrees, and two consecutive sides have lengths of 10 inches and 20 inches. What is the area of the parallelogram? Express your answer in simplest radical form. | 100\sqrt{3} | 0 | 3,664.5 | -1 | 3,664.5 | |
Let $T_n$ be the sum of the reciprocals of the non-zero digits of the integers from $1$ to $5^n$ inclusive. Find the smallest positive integer $n$ for which $T_n$ is an integer. | 63 | 0 | 8,192 | -1 | 8,192 | |
If $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = 4,$ then find
\[\begin{vmatrix} a & 7a + 3b \\ c & 7c +3d \end{vmatrix}.\] | 12 | 0.8125 | 4,444.6875 | 3,579.923077 | 8,192 | |
If $m>0$ and the points $(m,3)$ and $(1,m)$ lie on a line with slope $m$, then $m=$ | \sqrt{3} | 1. **Identify the slope formula**: Given two points $(x_1, y_1)$ and $(x_2, y_2)$, the slope $m$ of the line passing through these points is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
For the points $(m, 3)$ and $(1, m)$, substituting into the slope formula, we have:
\[
m = \frac{m - 3}{1 - m}
... | 1 | 1,945.6875 | 1,945.6875 | -1 |
For any two positive integers, define the operation (represented by the operator ⊕): when both $m$ and $n$ are positive even numbers or both are positive odd numbers, $m⊕n=m+n$; when one of $m$ and $n$ is a positive even number and the other is a positive odd number, $m⊕n=m×n$. For example, $4⊕6=4+6=10$, $3⊕7=3+7=10$, ... | 15 | 0.5 | 5,920 | 4,918.75 | 6,921.25 | |
If the vertices of a smaller square are midpoints of the sides of a larger square, and the larger square has an area of 144, what is the area of the smaller square? | 72 | 1 | 3,171.5625 | 3,171.5625 | -1 | |
Determine the largest square number that is not divisible by 100 and, when its last two digits are removed, is also a square number. | 1681 | 0.25 | 8,113.8125 | 7,879.25 | 8,192 | |
The cards in a stack are numbered consecutively from 1 to $2n$ from top to bottom. The top $n$ cards are removed to form pile $A$ and the remaining cards form pile $B$. The cards are restacked by alternating cards from pile $B$ and $A$, starting with a card from $B$. Given this process, find the total number of cards (... | 402 | 0 | 7,746.0625 | -1 | 7,746.0625 | |
The ruler of a certain country, for purely military reasons, wanted there to be more boys than girls among his subjects. Under the threat of severe punishment, he decreed that each family should have no more than one girl. As a result, in this country, each woman's last - and only last - child was a girl because no wom... | 2/3 | 0 | 6,566.1875 | -1 | 6,566.1875 | |
A pentagon is drawn by placing an isosceles right triangle on top of a square as pictured. What percent of the area of the pentagon is the area of the right triangle?
[asy]
size(50);
draw((0,0)--(0,-1)--(1,-1)--(1,0)--(0,0)--(.5,.5)--(1,0));
[/asy] | 20\% | 0.5625 | 6,181.5 | 5,376 | 7,217.142857 | |
The route from $A$ to $B$ is traveled by a passenger train 3 hours and 12 minutes faster than a freight train. In the time it takes the freight train to travel from $A$ to $B$, the passenger train travels 288 km more. If the speed of each train is increased by $10 \mathrm{km} / h$, the passenger train would travel from... | 360 | 0.3125 | 7,054.0625 | 4,550.6 | 8,192 | |
Given that the function $f\left(x\right)$ is an even function on $R$, and $f\left(x+2\right)$ is an odd function. If $f\left(0\right)=1$, then $f\left(1\right)+f\left(2\right)+\ldots +f\left(2023\right)=\_\_\_\_\_\_$. | -1 | 0.5 | 6,953.625 | 5,973.5 | 7,933.75 | |
Given real numbers \(a, b, c\), the polynomial
$$
g(x) = x^{3} + a x^{2} + x + 10
$$
has three distinct roots, and these three roots are also roots of the polynomial
$$
f(x) = x^{4} + x^{3} + b x^{2} + 100 x + c.
$$
Find the value of \(f(1)\). | -7007 | 0.75 | 4,805.0625 | 3,970.916667 | 7,307.5 | |
What is $\sqrt[4]{16} \cdot \sqrt[3]{8} \cdot \sqrt{4}$ expressed as a positive integer? | 8 | 1 | 2,618 | 2,618 | -1 | |
Let $S$ be the set of points whose coordinates $x,$ $y,$ and $z$ are integers that satisfy $0\le x\le2,$ $0\le y\le3,$ and $0\le z\le4.$ Two distinct points are randomly chosen from $S.$ The probability that the midpoint of the segment they determine also belongs to $S$ is $m/n,$ where $m$ and $n$ are relatively prime ... | 200 | 0.125 | 7,887.625 | 6,723.5 | 8,053.928571 | |
If $x,$ $y,$ and $k$ are positive real numbers such that \[3=k^2\left(\dfrac{x^2}{y^2}+\dfrac{y^2}{x^2}\right)+k\left(\dfrac{x}{y}+\dfrac{y}{x}\right),\]find the maximum possible value of $k.$ | \frac{-1+\sqrt7}{2} | 0 | 5,960.4375 | -1 | 5,960.4375 | |
Find the units digit of $7 \cdot 17 \cdot 1977 - 7^3$ | 0 | 0.9375 | 3,870.1875 | 3,582.066667 | 8,192 | |
An assembly line produces, on average, 85% first grade products. How many products need to be sampled so that, with a probability of 0.997, the deviation of the proportion of first grade products from 0.85 in absolute value does not exceed 0.01? | 11475 | 0.6875 | 5,100 | 4,066.090909 | 7,374.6 | |
From a deck of cards, 5 spades, 4 clubs, and 6 hearts, totaling 15 cards, are drawn. If drawing $m$ cards such that all three suits are present is a certain event, then the minimum value of $m$ is \_\_\_\_\_\_\_\_\_. | 12 | 0.875 | 2,821.875 | 2,519.142857 | 4,941 | |
What is the distance between the center of the circle with equation $x^2+y^2=-4x+6y-12$ and the point $(1,7)$? | 5 | 0.9375 | 2,547.3125 | 2,171 | 8,192 | |
Given the "ratio arithmetic sequence" $\{a_{n}\}$ with $a_{1}=a_{2}=1$, $a_{3}=3$, determine the value of $\frac{{a_{2019}}}{{a_{2017}}}$. | 4\times 2017^{2}-1 | 0 | 6,682.5625 | -1 | 6,682.5625 | |
In convex quadrilateral \(ABCD\) with \(AB=11\) and \(CD=13\), there is a point \(P\) for which \(\triangle ADP\) and \(\triangle BCP\) are congruent equilateral triangles. Compute the side length of these triangles. | 7 | Evidently \(ABCD\) is an isosceles trapezoid with \(P\) as its circumcenter. Now, construct isosceles trapezoid \(ABB'C\) (that is, \(BB'\) is parallel to \(AC\).) Then \(AB'PD\) is a rhombus, so \(\angle B'CD=\frac{1}{2} \angle B'PD=60^{\circ}\) by the inscribed angle theorem. Also, \(B'C=11\) because the quadrilatera... | 0 | 8,192 | -1 | 8,192 |
If $e^{i \theta} = \frac{2 + i \sqrt{5}}{3},$ then find $\sin 4 \theta.$ | -\frac{8 \sqrt{5}}{81} | 0 | 4,905.3125 | -1 | 4,905.3125 | |
Let $F_1$ and $F_2$ be the left and right foci of the ellipse $\frac{x^2}{4}+\frac{y^2}{b^2}=1 \ (b > 0)$, respectively. A line $l$ passing through $F_1$ intersects the ellipse at points $A$ and $B$. If the maximum value of $|AF_2|+|BF_2|$ is $5$, determine the eccentricity of the ellipse. | \frac{1}{2} | 0.3125 | 7,537.1875 | 6,105.8 | 8,187.818182 | |
Let $(a_1, a_2, \dots ,a_{10})$ be a list of the first 10 positive integers such that for each $2 \le i \le 10$ either $a_i+1$ or $a_i-1$ or both appear somewhere before $a_i$ in the list. How many such lists are there? | 512 | To solve this problem, we need to understand how the lists can be constructed under the given constraints. The key constraint is that for each $a_i$ where $i \geq 2$, either $a_i + 1$ or $a_i - 1$ (or both) must appear before $a_i$ in the list. This constraint guides the order in which numbers can be added to the list.... | 0.25 | 7,534.3125 | 6,301 | 7,945.416667 |
For a finite graph $G$, let $f(G)$ be the number of triangles and $g(G)$ the number of tetrahedra formed by edges of $G$. Find the least constant $c$ such that \[g(G)^3\le c\cdot f(G)^4\] for every graph $G$.
[i] | \frac{3}{32} |
Let \( G \) be a finite graph. We denote by \( f(G) \) the number of triangles and by \( g(G) \) the number of tetrahedra in \( G \). We seek to establish the smallest constant \( c \) such that
\[
g(G)^3 \le c \cdot f(G)^4
\]
for every graph \( G \).
### Step 1: Understanding the Problem
A triangle in a graph con... | 0.0625 | 8,175.9375 | 7,935 | 8,192 |
Hexadecimal (base-16) numbers are written using numeric digits $0$ through $9$ as well as the letters $A$ through $F$ to represent $10$ through $15$. Among the first $1000$ positive integers, there are $n$ whose hexadecimal representation contains only numeric digits. What is the sum of the digits of $n$? | 21 | 1. **Convert 1000 to Hexadecimal**:
The decimal number 1000 can be converted to hexadecimal. We find that $1000_{10} = 3E8_{16}$. This conversion is done by dividing 1000 by 16 repeatedly and considering the remainders:
- $1000 \div 16 = 62$ remainder $8$
- $62 \div 16 = 3$ remainder $14$ (which is 'E' in hex... | 0.1875 | 7,696.8125 | 6,426 | 7,990.076923 |
Given positive numbers $a$ and $b$ satisfying $a+b=1$, determine the minimum value of $\sqrt{ab}$. | \frac{1}{2} | 0 | 6,444.25 | -1 | 6,444.25 | |
At the end of 1994, Walter was half as old as his grandmother. The sum of the years in which they were born was 3838. How old will Walter be at the end of 1999? | 55 | 1. **Assign variables to ages**: Let Walter's age in 1994 be $x$. Then, his grandmother's age in 1994 is $2x$ because Walter is half as old as his grandmother.
2. **Set up the equation for their birth years**: Walter was born in $1994 - x$ and his grandmother was born in $1994 - 2x$. The sum of their birth years is gi... | 0.9375 | 2,894.75 | 2,541.6 | 8,192 |
Given that Ben constructs a $4$-step staircase using $26$ toothpicks, determine the number of additional toothpicks needed to extend the staircase to a $6$-step staircase, | 22 | 0 | 8,025.6875 | -1 | 8,025.6875 | |
The graph of the equation $y = |x| - 3$ is translated two units to the left and three units down. What are the coordinates of the minimum point of the new graph? | (-2,-6) | 0.9375 | 2,033.875 | 2,078.6 | 1,363 | |
In the rectangular coordinate system on a plane, the parametric equations of curve $C$ are given by $\begin{cases} x = 2\cos θ \\ y = \sqrt{3}\sin θ \end{cases}$ ($θ$ is the parameter). A polar coordinate system is established with the coordinate origin as the pole and the positive half of the $x$-axis as the polar axi... | \frac{120}{19} | 0.625 | 6,315.5 | 5,906.6 | 6,997 | |
Given that the domain of the function $f(x)$ is $\mathbb{R}$, if $f(x+1)$ and $f(x-1)$ are both odd functions, then the function $y=f(x)$ has at least \_\_\_\_\_ zeros in the interval $[0,100]$. | 50 | 0.1875 | 7,981.875 | 7,071.333333 | 8,192 | |
Solve the equation $a^3 + b^3 + c^3 = 2001$ in positive integers. | \[
\boxed{(10,10,1), (10,1,10), (1,10,10)}
\] | Note that for all positive integers $n,$ the value $n^3$ is congruent to $-1,0,1$ modulo $9.$ Since $2001 \equiv 3 \pmod{9},$ we find that $a^3,b^3,c^3 \equiv 1 \pmod{9}.$ Thus, $a,b,c \equiv 1 \pmod{3},$ and the only numbers congruent to $1$ modulo $3$ are $1,4,7,10.$
WLOG , let $a \ge b \ge c.$ That means $a^3 \... | 0 | 8,192 | -1 | 8,192 |
Some unit squares in an infinite sheet of squared paper are colored red so that every 2 x 3 and 3 x 2 rectangle contains exactly two red squares. How many red squares are there in a 9 x 11 rectangle? | 33 | 0.0625 | 8,157.0625 | 7,633 | 8,192 | |
Let $A B C D$ be a square of side length 5. A circle passing through $A$ is tangent to segment $C D$ at $T$ and meets $A B$ and $A D$ again at $X \neq A$ and $Y \neq A$, respectively. Given that $X Y=6$, compute $A T$. | \sqrt{30} | Let $O$ be the center of the circle, and let $Z$ be the foot from $O$ to $A D$. Since $X Y$ is a diameter, $O T=Z D=3$, so $A Z=2$. Then $O Z=\sqrt{5}$ and $A T=\sqrt{O Z^{2}+25}=\sqrt{30}$. | 0.6875 | 6,123.1875 | 5,182.818182 | 8,192 |
How many positive perfect cubes are divisors of the product \(1! \cdot 2! \cdot 3! \cdots 10!\)? | 468 | 0.75 | 5,979.125 | 5,965.5 | 6,020 | |
In writing the integers from 10 through 99 inclusive, how many times is the digit 6 written? | 19 | 0.5625 | 5,676.6875 | 4,429.555556 | 7,280.142857 | |
In the geometric sequence $\{a_n\}$, $a_2a_3=5$ and $a_5a_6=10$. Calculate the value of $a_8a_9$. | 20 | 1 | 3,128.9375 | 3,128.9375 | -1 | |
Let \( s(n) \) denote the sum of the digits of the natural number \( n \). Solve the equation \( n + s(n) = 2018 \). | 2008 | 0.5625 | 6,840.625 | 5,789.555556 | 8,192 | |
Points $B$, $D$, and $J$ are midpoints of the sides of right triangle $ACG$. Points $K$, $E$, $I$ are midpoints of the sides of triangle $JDG$, etc. If the dividing and shading process is done 100 times (the first three are shown) and $AC=CG=6$, then the total area of the shaded triangles is nearest | 6 |
We are given a right triangle $ACG$ with $AC = CG = 6$ and a process of dividing and shading triangles that continues 100 times. We need to find the total area of the shaded triangles.
#### Step 1: Calculate the area of the initial triangle $ACG$.
Since $ACG$ is a right triangle with legs of length 6, its area is:
\[... | 1 | 4,805.875 | 4,805.875 | -1 |
If the inequality $((x+y)^2+4)((x+y)^2-2)\geq A\cdot (x-y)^2$ holds for every real numbers $x,y$ such that $xy=1$, determine the largest value of $A$. | 18 | 0.4375 | 6,558.25 | 4,775.571429 | 7,944.777778 | |
Given a geometric sequence \(\{a_n\}\) with the sum of the first \(n\) terms \(S_n\) such that \(S_n = 2^n + r\) (where \(r\) is a constant), let \(b_n = 2(1 + \log_2 a_n)\) for \(n \in \mathbb{N}^*\).
1. Find the sum of the first \(n\) terms of the sequence \(\{a_n b_n\}\), denoted as \(T_n\).
2. If for any positive ... | \frac{3}{4} \sqrt{2} | 0 | 8,192 | -1 | 8,192 | |
Let $P(x)$ be the monic polynomial with rational coefficients of minimal degree such that $\frac{1}{\sqrt{2}}$, $\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{4}}, \ldots, \frac{1}{\sqrt{1000}}$ are roots of $P$. What is the sum of the coefficients of $P$? | \frac{1}{16000} | For irrational $\frac{1}{\sqrt{r}},-\frac{1}{\sqrt{r}}$ must also be a root of $P$. Therefore $P(x)=\frac{\left(x^{2}-\frac{1}{2}\right)\left(x^{2}-\frac{1}{3}\right) \cdots\left(x^{2}-\frac{1}{1000}\right)}{\left(x+\frac{1}{2}\right)\left(x+\frac{1}{3}\right) \cdots\left(x+\frac{1}{31}\right)}$. We get the sum of the ... | 0 | 4,820.5 | -1 | 4,820.5 |
Compute
\[
\frac{(1 + 23) \left( 1 + \dfrac{23}{2} \right) \left( 1 + \dfrac{23}{3} \right) \dotsm \left( 1 + \dfrac{23}{25} \right)}{(1 + 27) \left( 1 + \dfrac{27}{2} \right) \left( 1 + \dfrac{27}{3} \right) \dotsm \left( 1 + \dfrac{27}{21} \right)}.
\] | 421200 | 0 | 7,380.8125 | -1 | 7,380.8125 | |
Given the sequence $$1, \frac{1}{2}, \frac{2}{1}, \frac{1}{3}, \frac{2}{2}, \frac{3}{1}, \frac{1}{4}, \frac{2}{3}, \frac{3}{2}, \frac{4}{1}, \ldots$$, find the position of $$\frac{8}{9}$$ in this sequence. | 128 | 0.3125 | 7,419.3125 | 6,422.2 | 7,872.545455 | |
Given an ellipse with the equation $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ and an eccentricity of $\frac{1}{2}$. $F\_1$ and $F\_2$ are the left and right foci of the ellipse, respectively. A line $l$ passing through $F\_2$ intersects the ellipse at points $A$ and $B$. The perimeter of $\triangle F\_1AB... | m = 4 | 0.0625 | 8,192 | 8,192 | 8,192 | |
Lucy begins a sequence with the first term at 4. Each subsequent term is generated as follows: If a fair coin flip results in heads, she triples the previous term and then adds 3. If it results in tails, she subtracts 3 and divides the result by 3. What is the probability that the fourth term in Lucy's sequence is an i... | \frac{3}{4} | 0 | 7,486 | -1 | 7,486 | |
The mean of the set of numbers $\{87,85,80,83,84,x\}$ is 83.5. What is the median of the set of six numbers? Express your answer as a decimal to the nearest tenth. | 83.5 | 1 | 1,872.8125 | 1,872.8125 | -1 | |
Find $\cot 45^\circ.$ | 1 | 1 | 1,836.4375 | 1,836.4375 | -1 | |
When any set of $k$ consecutive positive integers necessarily includes at least one positive integer whose digit sum is a multiple of 11, we call each of these sets of $k$ consecutive positive integers a "dragon" of length $k." Find the shortest dragon length. | 39 | 0 | 8,192 | -1 | 8,192 | |
The Evil League of Evil plans to set out from their headquarters at (5,1) to poison two pipes: one along the line \( y = x \) and the other along the line \( x = 7 \). They wish to determine the shortest distance they can travel to visit both pipes and then return to their headquarters. | 4\sqrt{5} | 0.0625 | 8,093.9375 | 7,283 | 8,148 | |
The numbers $a,$ $b,$ $c,$ $d$ are equal to 1, 2, 3, 4, in some order. Find the largest possible value of
\[ab + bc + cd + da.\] | 25 | 1 | 3,406.9375 | 3,406.9375 | -1 | |
Given that the four vertices of the quadrilateral $MNPQ$ are on the graph of the function $f(x)=\log_{\frac{1}{2}} \frac{ax+1}{x+b}$, and it satisfies $\overrightarrow{MN}= \overrightarrow{QP}$, where $M(3,-1)$, $N\left( \frac{5}{3},-2\right)$, then the area of the quadrilateral $MNPQ$ is \_\_\_\_\_\_. | \frac{26}{3} | 0.25 | 8,083.6875 | 7,758.75 | 8,192 | |
Two numbers are independently selected from the set of positive integers less than or equal to 7. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction. | \frac{36}{49} | 0 | 6,709 | -1 | 6,709 | |
Suppose two equally strong tennis players play against each other until one player wins three games in a row. The results of each game are independent, and each player will win with probability $\frac{1}{2}$ . What is the expected value of the number of games they will play? | 14 | 0.0625 | 7,425.25 | 3,596 | 7,680.533333 | |
Points \( M \) and \( N \) are taken on the diagonals \( AB_1 \) and \( BC_1 \) of the faces of the parallelepiped \( ABCD A_1 B_1 C_1 D_1 \), and the segments \( MN \) and \( A_1 C \) are parallel. Find the ratio of these segments. | 1:3 | 0 | 4,381.625 | -1 | 4,381.625 | |
Given that spinner S has numbers 1, 4, 3, spinner T has numbers 2, 4, 6, and spinner U has numbers 2, 3, 5, determine the probability that the sum of the numbers obtained from spinning each of the three spinners is an even number. | \frac{5}{9} | 0.4375 | 6,652.8125 | 4,673.857143 | 8,192 | |
Triangles \(ABC\) and \(ABD\) are inscribed in a semicircle with diameter \(AB = 5\). A perpendicular from \(D\) to \(AB\) intersects segment \(AC\) at point \(Q\), ray \(BC\) at point \(R\), and segment \(AB\) at point \(P\). It is known that \(PR = \frac{27}{10}\), and \(PQ = \frac{5}{6}\). Find the length of segment... | 1.5 | 0.0625 | 8,138.375 | 8,192 | 8,134.8 | |
How many positive integers less than $800$ are either a perfect cube or a perfect square? | 35 | 0 | 4,456 | -1 | 4,456 | |
Given cos($$α+ \frac {π}{6}$$)= $$\frac {1}{3}$$, find the value of sin($$ \frac {5π}{6}+2α$$). | -$$\frac {7}{9}$$ | 0 | 5,948.3125 | -1 | 5,948.3125 | |
Evaluate $\sqrt[3]{1+27} \cdot \sqrt[3]{1+\sqrt[3]{27}}$. | \sqrt[3]{112} | 0.0625 | 2,919.1875 | 5,603 | 2,740.266667 | |
Consider the following pair of equations:
\[120x^4 + ax^3 + bx^2 + cx + 18 = 0\] and
\[18x^5 + dx^4 + ex^3 + fx^2 + gx + 120 = 0\]
These equations have a common rational root $k$ which is not an integer and is positive. Determine $k$. | \frac{1}{2} | 0 | 8,192 | -1 | 8,192 | |
Given a tetrahedron $P-ABC$, in the base $\triangle ABC$, $\angle BAC=60^{\circ}$, $BC=\sqrt{3}$, $PA\perp$ plane $ABC$, $PA=2$, then the surface area of the circumscribed sphere of this tetrahedron is ______. | 8\pi | 0.75 | 6,051.0625 | 5,337.416667 | 8,192 | |
A two-meter gas pipe has rusted in two places. Determine the probability that all three resulting pieces can be used as connections to gas stoves, given that according to regulations, a stove should not be located closer than 50 cm to the main gas pipe. | 1/16 | 0.25 | 7,945.5 | 7,467.25 | 8,104.916667 | |
The vertices of a cube have coordinates $(0,0,0),$ $(0,0,4),$ $(0,4,0),$ $(0,4,4),$ $(4,0,0),$ $(4,0,4),$ $(4,4,0),$ and $(4,4,4).$ A plane cuts the edges of this cube at the points $P = (0,2,0),$ $Q = (1,0,0),$ $R = (1,4,4),$ and two other points. Find the distance between these two points. | \sqrt{29} | 0.5 | 6,649.1875 | 5,106.375 | 8,192 | |
Let $a_1, a_2, \dots, a_{2018}$ be a strictly increasing sequence of positive integers such that $a_1 + a_2 + \cdots + a_{2018} = 2018^{2018}$. What is the remainder when $a_1^3 + a_2^3 + \cdots + a_{2018}^3$ is divided by $6$? | 2 | 1. **Sum of Cubes Relation**: We start by expanding the cube of the sum of the sequence:
\[
(a_1 + a_2 + \cdots + a_{2018})^3 = a_1^3 + a_2^3 + \cdots + a_{2018}^3 + 3\sum_{i=1}^{2018} a_i^2 \left(\sum_{j=1}^{2018} a_j - a_i\right) + 6\sum_{i\neq j\neq k} a_i a_j a_k
\]
Here, the term $3\sum_{i=1}^{2018} a_... | 0 | 3,027.1875 | -1 | 3,027.1875 |
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