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 |
|---|---|---|---|---|---|---|
Given 8 planes in space, for each pair of planes, the line of their intersection is noted. For each pair of these noted lines, the point of their intersection is noted (if the lines intersect). What is the maximum number of noted points that could be obtained? | 56 | 0.0625 | 7,946.625 | 5,508 | 8,109.2 | |
How many ways are there to win tic-tac-toe in $\mathbb{R}^{n}$? (That is, how many lines pass through three of the lattice points $(a_{1}, \ldots, a_{n})$ in $\mathbb{R}^{n}$ with each coordinate $a_{i}$ in $\{1,2,3\}$? Express your answer in terms of $n$. | \left(5^{n}-3^{n}\right) / 2 | A line consists of three points. Each coordinate can do one of three things passing from the first point to the last point: increase by 1 each time, stay the same, or decrease by 1 each time. There are three ways to stay the same (three coordinates), one way to increase by 1, and one way to decrease by 1, so there are ... | 0 | 8,192 | -1 | 8,192 |
What is the sum of the interior numbers of the eighth row of Pascal's Triangle? | 126 | 0.375 | 3,128.5 | 2,689.833333 | 3,391.7 | |
The diagram shows an \(n \times (n+1)\) rectangle tiled with \(k \times (k+1)\) rectangles, where \(n\) and \(k\) are integers and \(k\) takes each value from 1 to 8 inclusive. What is the value of \(n\)? | 15 | 0.3125 | 7,095.8125 | 4,699.8 | 8,184.909091 | |
Given the function $y=\cos(2x+\frac{\pi}{4})$, determine the $x$-coordinate of one of the symmetric centers of the translated graph after translating it to the left by $\frac{\pi}{6}$ units. | \frac{11\pi}{24} | 0 | 7,985.1875 | -1 | 7,985.1875 | |
A quarry wants to sell a large pile of gravel. At full price, the gravel would sell for $3200$ dollars. But during the first week the quarry only sells $60\%$ of the gravel at full price. The following week the quarry drops the price by $10\%$ , and, again, it sells $60\%$ of the remaining gravel. Each week, the... | 3000 | 0.5625 | 6,912.1875 | 6,123.444444 | 7,926.285714 | |
Dots are spaced one unit part, horizontally and vertically. What is the number of square units enclosed by the polygon?
[asy]
/* AMC8 1998 #6P */
size(1inch,1inch);
pair a=(0,0), b=(10,0), c=(20,0), d=(30, 0);
pair e=(0,10), f=(10,10), g=(20,10), h=(30,10);
pair i=(0,20), j=(10,20), k=(20,20), l=(30,20);
pair m=(0,30),... | 6 | 0 | 8,172.8125 | -1 | 8,172.8125 | |
Find the number of solutions to the equation
\[\tan (3 \pi \cos \theta) = \cot (3 \pi \sin \theta)\]
where $\theta \in (0, 3\pi)$. | 16 | 0 | 8,187 | -1 | 8,187 | |
For how many positive integers $n \le 1000$ is$\left\lfloor \dfrac{998}{n} \right\rfloor+\left\lfloor \dfrac{999}{n} \right\rfloor+\left\lfloor \dfrac{1000}{n}\right \rfloor$not divisible by $3$? | 22 | We are tasked with finding how many positive integers \( n \leq 1000 \) make the expression
\[
\left\lfloor \frac{998}{n} \right\rfloor + \left\lfloor \frac{999}{n} \right\rfloor + \left\lfloor \frac{1000}{n} \right\rfloor
\]
not divisible by 3. We start by analyzing the behavior of the floor function in this context.
... | 0 | 8,149.75 | -1 | 8,149.75 |
Square $BCFE$ is inscribed in right triangle $AGD$, as shown below. If $AB = 28$ units and $CD = 58$ units, what is the area of square $BCFE$?
[asy]
draw((9,15.6)--(10.7,14.6)--(11.7,16.3));
draw((0,0)--(10,17.3)--(40,0)--cycle);
draw((7,0)--(19,0)--(19,12.1)--(7,12.1)--cycle);
label("$A$",(0,0),SW);
label("$B$",(7,0)... | 1624 | 0 | 8,192 | -1 | 8,192 | |
For two lines $ax+2y+1=0$ and $3x+(a-1)y+1=0$ to be parallel, determine the value of $a$ that satisfies this condition. | -2 | 0.375 | 5,153.625 | 4,099.333333 | 5,786.2 | |
Calculate the sum $E(1)+E(2)+E(3)+\cdots+E(200)$ where $E(n)$ denotes the sum of the even digits of $n$, and $5$ is added to the sum if $n$ is a multiple of $10$. | 902 | 0 | 8,146.4375 | -1 | 8,146.4375 | |
On Arbor Day, a class at a certain school divided into 10 small groups to participate in tree planting activities. The number of trees planted by the 10 groups is shown in the table below:
| Number of Trees Planted | 5 | 6 | 7 |
|--------------------------|-----|-----|-----|
| Number of Groups | 3 | 4 ... | 0.6 | 0.6875 | 2,650.0625 | 3,015.545455 | 1,846 | |
Let $S$ be a subset with four elements chosen from \{1,2, \ldots, 10\}$. Michael notes that there is a way to label the vertices of a square with elements from $S$ such that no two vertices have the same label, and the labels adjacent to any side of the square differ by at least 4 . How many possibilities are there for... | 36 | Let the four numbers be $a, b, c, d$ around the square. Assume without loss of generality that $a$ is the largest number, so that $a>b$ and $a>d$. Note that $c$ cannot be simultaneously smaller than one of $b, d$ and larger than the other because, e.g. if $b>c>d$, then $a>b>c>d$ and $a \geq d+12$. Hence $c$ is either s... | 0 | 8,192 | -1 | 8,192 |
Two spheres with radii $36$ and one sphere with radius $13$ are each externally tangent to the other two spheres and to two different planes $\mathcal{P}$ and $\mathcal{Q}$. The intersection of planes $\mathcal{P}$ and $\mathcal{Q}$ is the line $\ell$. The distance from line $\ell$ to the point where the sphere with ra... | 335 | The isosceles triangle of centers $O_1 O_2 O$ ($O$ is the center of sphere of radii $13$) has sides $O_1 O = O_2 O = 36 + 13 = 49,$ and $O_1 O_2 = 36 + 36 = 72.$
Let $N$ be the midpoint $O_1 O_2$.
The isosceles triangle of points of tangency $T_1 T_2 T$ has sides $T_1 T = T_2 T = 2 \sqrt{13 \cdot 36} = 12 \sqrt{13}$ ... | 0 | 8,192 | -1 | 8,192 |
There are 2008 congruent circles on a plane such that no two are tangent to each other and each circle intersects at least three other circles. Let $ N$ be the total number of intersection points of these circles. Determine the smallest possible values of $ N$ . | 3012 | 0.75 | 6,645.5625 | 6,130.083333 | 8,192 | |
Sarah subscribes to a virtual fitness class platform that charges a monthly membership fee plus a per-class fee. If Sarah paid a total of $30.72 in February for 4 classes, and $54.72 in March for 8 classes, with the monthly membership fee increasing by 10% from February to March, calculate the fixed monthly membership ... | 7.47 | 0.4375 | 876.1875 | 893 | 863.111111 | |
You are standing at the edge of a river which is $1$ km wide. You have to go to your camp on the opposite bank . The distance to the camp from the point on the opposite bank directly across you is $1$ km . You can swim at $2$ km/hr and walk at $3$ km-hr . What is the shortest time you will take to reach your ca... | \frac{2 + \sqrt{5}}{6} | 0 | 7,108 | -1 | 7,108 | |
In a store where all items cost an integer number of rubles, there are two special offers:
1) A customer who buys at least three items simultaneously can choose one item for free, whose cost does not exceed the minimum of the prices of the paid items.
2) A customer who buys exactly one item costing at least $N$ rubles ... | 504 | 0 | 8,187.375 | -1 | 8,187.375 | |
The Gauss family has three boys aged $7,$ a girl aged $14,$ and a boy aged $15.$ What is the mean (average) of the ages of the children? | 10 | 0.5 | 2,202.4375 | 1,590.375 | 2,814.5 | |
In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively. Given that $b=1$, $c= \sqrt {3}$, and $\angle C= \frac {2}{3}\pi$, find the area $S_{\triangle ABC}$. | \frac { \sqrt {3}}{4} | 0 | 4,030.6875 | -1 | 4,030.6875 | |
A novice economist-cryptographer received a cryptogram from a ruler which contained a secret decree about implementing an itemized tax on a certain market. The cryptogram specified the amount of tax revenue that needed to be collected, emphasizing that a greater amount could not be collected in that market. Unfortunate... | 6480 | 0.0625 | 8,039.5625 | 8,192 | 8,029.4 | |
Let $f(x) = 4x^4+12x^3-9x^2+x+3$ and $d(x) = x^2+3x-2$. If $f(x) = q(x)d(x) + r(x)$ for some polynomials $q(x)$ and $r(x)$ where $\deg r < \deg d$, calculate $q(1)+r(-1)$. | 0 | 1 | 3,034.5625 | 3,034.5625 | -1 | |
Let $a, b, c$ be positive real numbers such that $a+b+c=10$ and $a b+b c+c a=25$. Let $m=\min \{a b, b c, c a\}$. Find the largest possible value of $m$. | \frac{25}{9} | Without loss of generality, we assume that $c \geq b \geq a$. We see that $3 c \geq a+b+c=10$. Therefore, $c \geq \frac{10}{3}$. Since $0 \leq(a-b)^{2} =(a+b)^{2}-4 a b =(10-c)^{2}-4(25-c(a+b)) =(10-c)^{2}-4(25-c(10-c)) =c(20-3 c)$ we obtain $c \leq \frac{20}{3}$. Consider $m=\min \{a b, b c, c a\}=a b$, as $b c \geq c... | 0 | 8,192 | -1 | 8,192 |
Oil, as an important strategic reserve commodity, has always been of concern to countries. According to reports from relevant departments, it is estimated that the demand for oil in China in 2022 will be 735,000,000 tons. Express 735,000,000 in scientific notation as ____. | 7.35 \times 10^{8} | 0.1875 | 336.875 | 353.333333 | 333.076923 | |
Express as a fraction in lowest terms: $0.\overline{23} + 0.\overline{056} + 0.\overline{004}$. | \frac{28917}{98901} | 0 | 5,457.1875 | -1 | 5,457.1875 | |
In a Go championship participated by three players: A, B, and C, the matches are conducted according to the following rules: the first match is between A and B; the second match is between the winner of the first match and C; the third match is between the winner of the second match and the loser of the first match; th... | 0.162 | 0 | 6,703.5 | -1 | 6,703.5 | |
If $3p+4q=8$ and $4p+3q=13$, what is $q$ equal to? | -1 | 1 | 1,817.4375 | 1,817.4375 | -1 | |
The function $g$ is defined on the set of integers and satisfies \[g(n)= \begin{cases} n-5 & \mbox{if }n\ge 1200 \\ g(g(n+7)) & \mbox{if }n<1200. \end{cases}\] Find $g(70)$. | 1195 | 0.0625 | 8,192 | 8,192 | 8,192 | |
Suppose $a, b$ and $c$ are integers such that the greatest common divisor of $x^{2}+a x+b$ and $x^{2}+b x+c$ is $x+1$ (in the ring of polynomials in $x$ with integer coefficients), and the least common multiple of $x^{2}+a x+b$ and $x^{2}+b x+c$ is $x^{3}-4 x^{2}+x+6$. Find $a+b+c$. | -6 | Since $x+1$ divides $x^{2}+a x+b$ and the constant term is $b$, we have $x^{2}+a x+b=(x+1)(x+b)$, and similarly $x^{2}+b x+c=(x+1)(x+c)$. Therefore, $a=b+1=c+2$. Furthermore, the least common multiple of the two polynomials is $(x+1)(x+b)(x+b-1)=x^{3}-4 x^{2}+x+6$, so $b=-2$. Thus $a=-1$ and $c=-3$, and $a+b+c=-6$. | 0.9375 | 3,994.5625 | 3,955.533333 | 4,580 |
Given the function $f(x)=\sin x\cos x-\cos ^{2}x$.
$(1)$ Find the interval where $f(x)$ is decreasing.
$(2)$ Let the zeros of $f(x)$ on $(0,+\infty)$ be arranged in ascending order to form a sequence $\{a_{n}\}$. Find the sum of the first $10$ terms of $\{a_{n}\}$. | \frac{95\pi}{4} | 0.625 | 6,849.8125 | 6,476.7 | 7,471.666667 | |
Let the real number \( a < -1 \) and the variable \( x \) satisfy \( x^2 + ax \leq -x \). The minimum value of \( x^2 + ax \) is \( -\frac{1}{2} \). What is \( a \)? | -\frac{3}{2} | 0.375 | 6,253.75 | 6,784 | 5,935.6 | |
A certain shopping mall's sales volume (unit: ten thousand pieces) for five consecutive years is as shown in the table below:
| $x$(year) | 1 | 2 | 3 | 4 | 5 |
|-----------|---|---|---|---|---|
| $y$(sales volume) | 5 | 5 | 6 | 7 | 7 |
$(1)$ Find the linear regression equation $\hat{y}=\hat{b}x+\hat{a}$ for the sales... | \frac{2}{5} | 1 | 3,611.375 | 3,611.375 | -1 | |
What is the smallest number of rectangles, each measuring $2 \mathrm{~cm}$ by $3 \mathrm{~cm}$, which are needed to fit together without overlap to form a rectangle whose sides are in the ratio 5:4? | 30 | 0.375 | 7,692.875 | 6,861 | 8,192 | |
A river boat travels at a constant speed from point A to point B. Along the riverbank, there is a road. The boat captain observes that every 30 minutes, a bus overtakes the boat from behind, and every 10 minutes, a bus approaches from the opposite direction. Assuming that the buses depart from points A and B uniformly ... | 15 | 0.0625 | 7,386.9375 | 2,540 | 7,710.066667 | |
In a certain year the price of gasoline rose by $20\%$ during January, fell by $20\%$ during February, rose by $25\%$ during March, and fell by $x\%$ during April. The price of gasoline at the end of April was the same as it had been at the beginning of January. To the nearest integer, what is $x$ | 17 | 1. **Assume the initial price**: Let's assume the initial price of gasoline at the beginning of January is $P_0 = 100$ dollars.
2. **Price after January's increase**: The price increased by 20% in January. Therefore, the new price at the end of January is:
\[
P_1 = P_0 + 0.20 \times P_0 = 100 + 20 = 120 \text{ d... | 0.9375 | 4,202.625 | 3,936.666667 | 8,192 |
Find the remainder when $2 \times 12 \times 22 \times 32 \times \ldots \times 72 \times 82 \times 92$ is divided by $5$. | 4 | 1 | 3,064.375 | 3,064.375 | -1 | |
When $1000^{100}$ is expanded out, the result is $1$ followed by how many zeros? | 300 | 1 | 1,928.3125 | 1,928.3125 | -1 | |
A nonempty set $S$ is called well-filled if for every $m \in S$, there are fewer than $\frac{1}{2}m$ elements of $S$ which are less than $m$. Determine the number of well-filled subsets of $\{1,2, \ldots, 42\}$. | \binom{43}{21}-1 | Let $a_{n}$ be the number of well-filled subsets whose maximum element is $n$ (setting $a_{0}=1$). Then it's easy to see that $a_{2k+1}=a_{2k}+a_{2k-1}+\cdots+a_{0}$ and $a_{2k+2}=(a_{2k+1}-C_{k})+a_{2k}+\cdots+a_{0}$ where $C_{k}$ is the number of well-filled subsets of size $k+1$ with maximal element $2k+1$. We proce... | 0 | 8,192 | -1 | 8,192 |
Two circles touch in $M$ , and lie inside a rectangle $ABCD$ . One of them touches the sides $AB$ and $AD$ , and the other one touches $AD,BC,CD$ . The radius of the second circle is four times that of the first circle. Find the ratio in which the common tangent of the circles in $M$ divides $AB$ and $CD$ . | 1:1 | 0 | 8,192 | -1 | 8,192 | |
Given the sequence 1, 2, 1, 2, 2, 1, 2, 2, 2, 1..., where each pair of 1s is separated by 2s, and there are n 2s between the nth pair of 1s, the sum of the first 1234 terms of this sequence is ______. | 2419 | 0.4375 | 6,416.875 | 5,570.428571 | 7,075.222222 | |
Two individuals, A and B, start traveling towards each other from points A and B, respectively, at the same time. They meet at point C, after which A continues to point B and B rests for 14 minutes before continuing to point A. Both A and B, upon reaching points B and A, immediately return and meet again at point C. Gi... | 1680 | 0 | 8,192 | -1 | 8,192 | |
A self-employed person plans to distribute two types of goods, A and B. According to a survey, when the investment amount is $x(x\geqslant 0)$ ten thousand yuan, the profits obtained from distributing goods A and B are $f(x)$ ten thousand yuan and $g(x)$ ten thousand yuan, respectively, where $f(x)=a(x-1)+2(a > 0)$; $g... | 12.6 | 0.75 | 4,140.75 | 3,333.916667 | 6,561.25 | |
What is the maximum length of a closed self-avoiding polygon that can travel along the grid lines of an $8 \times 8$ square grid? | 80 | 0 | 6,414.25 | -1 | 6,414.25 | |
If the digit represented by $\triangle$ satisfies the following addition problem $$ \begin{array}{c@{}c@{\;}c@{}c@{}c@{}c} & & 3 & 2 & 1 & \triangle_6\\ & & & \triangle & 4 & 0_6\\ &+ & & & \triangle & 2_6\\ \cline{2-6} & & 4 & 2 & \triangle & 1_6\\ \end{array} $$what is the value of $\triangle$? | 5 | 0.5625 | 5,923.375 | 4,158.888889 | 8,192 | |
Given that M and N are the intersection points of circle A: $x^2 + y^2 - 2x = 0$ and circle B: $x^2 + y^2 + 2x - 4y = 0$, find the area of triangle BMN. | \frac{3}{2} | 0.3125 | 6,518.1875 | 4,007.6 | 7,659.363636 | |
Let $P$ be a point inside regular pentagon $A B C D E$ such that $\angle P A B=48^{\circ}$ and $\angle P D C=42^{\circ}$. Find $\angle B P C$, in degrees. | 84^{\circ} | Since a regular pentagon has interior angles $108^{\circ}$, we can compute $\angle P D E=66^{\circ}, \angle P A E=60^{\circ}$, and $\angle A P D=360^{\circ}-\angle A E D-\angle P D E-\angle P A E=126^{\circ}$. Now observe that drawing $P E$ divides quadrilateral $P A E D$ into equilateral triangle $P A E$ and isosceles... | 0 | 8,192 | -1 | 8,192 |
Two points $A(x_1, y_1)$ and $B(x_2, y_2)$ are chosen on the graph of $f(x) = \ln x$, with $0 < x_1 < x_2$. The points $C$ and $D$ trisect $\overline{AB}$, with $AC < CB$. Through $C$ a horizontal line is drawn to cut the curve at $E(x_3, y_3)$. Find $x_3$ if $x_1 = 1$ and $x_2 = 1000$.
| 10 | 1 | 2,637.0625 | 2,637.0625 | -1 | |
In the polar coordinate system, find the length of the segment cut by the curve $\rho=1$ from the line $\rho\sin\theta-\rho\cos\theta=1$. | \sqrt{2} | 1 | 2,501.3125 | 2,501.3125 | -1 | |
The measures of a pair of supplementary angles are in the ratio of 7:2. How many degrees are in the measure of their positive difference? | 100^\circ | 1 | 1,167.5625 | 1,167.5625 | -1 | |
Given $x-y=1$ and $x^3-y^3=2$, find the values of $x^4+y^4$ and $x^5-y^5$. | \frac{29}{9} | 0.5 | 7,203.375 | 6,783.625 | 7,623.125 | |
One writes, initially, the numbers $1,2,3,\dots,10$ in a board. An operation is to delete the numbers $a, b$ and write the number $a+b+\frac{ab}{f(a,b)}$ , where $f(a, b)$ is the sum of all numbers in the board excluding $a$ and $b$ , one will make this until remain two numbers $x, y$ with $x\geq y$ . Find... | 1320 | 0 | 8,192 | -1 | 8,192 | |
If $z$ is a complex number such that
\[
z + z^{-1} = \sqrt{3},
\]what is the value of
\[
z^{2010} + z^{-2010} \, ?
\] | -2 | 0.9375 | 5,018.375 | 4,806.8 | 8,192 | |
How many integers $n$ satisfy $(n-2)(n+4)<0$? | 5 | 1 | 1,500.875 | 1,500.875 | -1 | |
Given the function $f(x) = x^3 - 3x^2 - 9x + 1$, find the intervals of monotonicity and the extrema of $f(x)$. | -26 | 0.375 | 3,004.8125 | 3,255.333333 | 2,854.5 | |
Define a T-grid to be a $3\times3$ matrix which satisfies the following two properties:
Exactly five of the entries are $1$'s, and the remaining four entries are $0$'s.
Among the eight rows, columns, and long diagonals (the long diagonals are $\{a_{13},a_{22},a_{31}\}$ and $\{a_{11},a_{22},a_{33}\}$, no more than one o... | 68 | The T-grid can be considered as a tic-tac-toe board: five $1$'s (or X's) and four $0$'s (or O's).
There are only $\dbinom{9}{5} = 126$ ways to fill the board with five $1$'s and four $0$'s. Now we just need to subtract the number of bad grids. Bad grids are ones with more than one person winning, or where someone has ... | 0 | 8,056.25 | -1 | 8,056.25 |
Given the function $f(x)=|x-2|+|x-a^{2}|$.
$(1)$ If the inequality $f(x)\leqslant a$ has solutions for $x$, find the range of the real number $a$;
$(2)$ If the positive real numbers $m$, $n$ satisfy $m+2n=a$, when $a$ takes the maximum value from $(1)$, find the minimum value of $\left( \dfrac {1}{m}+ \dfrac {1}{n}... | \dfrac {3}{2}+ \sqrt {2} | 0 | 5,416.3125 | -1 | 5,416.3125 | |
Assume that the probability of a certain athlete hitting the bullseye with a dart is $40\%$. Now, the probability that the athlete hits the bullseye exactly once in two dart throws is estimated using a random simulation method: first, a random integer value between $0$ and $9$ is generated by a calculator, where $1$, $... | 0.5 | 0.25 | 5,607.8125 | 5,491 | 5,646.75 | |
Suppose $x$, $y$, and $z$ are all positive real numbers, and $x^{2}+y^{2}+z^{2}=1$, then the minimum value of $\frac{(z+1)^{2}}{2xyz}$ is $\_\_\_\_\_\_$. | 3+2 \sqrt{2} | 0.125 | 8,060.9375 | 7,886 | 8,085.928571 | |
Given that $\tan \alpha = \frac{1}{3}$ and $\tan (\alpha + \beta) = \frac{1}{2}$, find the value of $\tan \beta$. | \frac{1}{7} | 1 | 2,378.0625 | 2,378.0625 | -1 | |
Triangle $ABC$ has sides of $6$ units, $8$ units, and $10$ units. The width of a rectangle, whose area is equal to the area of the triangle, is $4$ units. What is the perimeter of this rectangle, in units? | 20 | 1 | 1,203.125 | 1,203.125 | -1 | |
What is the area of the smallest square that can contain a circle of radius 5? | 100 | 0.5 | 4,459.375 | 3,082 | 5,836.75 | |
The number \( N \) is of the form \( p^{\alpha} q^{\beta} r^{\gamma} \), where \( p, q, r \) are prime numbers, and \( p q - r = 3, p r - q = 9 \). Additionally, the numbers \( \frac{N}{p}, \frac{N}{q}, \frac{N}{r} \) respectively have 20, 12, and 15 fewer divisors than the number \( N \). Find the number \( N \). | 857500 | 0.6875 | 6,450.1875 | 5,658.454545 | 8,192 | |
Given that $a$ and $b$ are positive numbers, and $a+b=1$, find the minimum value of $\frac{1}{2a} + \frac{1}{b}$. | \sqrt{2} + \frac{3}{2} | 0.0625 | 6,021 | 5,178 | 6,077.2 | |
For real numbers $a, b$ , and $c$ the polynomial $p(x) = 3x^7 - 291x^6 + ax^5 + bx^4 + cx^2 + 134x - 2$ has $7$ real roots whose sum is $97$ . Find the sum of the reciprocals of those $7$ roots. | 67 | 0.625 | 4,249.875 | 3,949.4 | 4,750.666667 | |
Find four consecutive odd numbers, none of which are divisible by 3, such that their sum is divisible by 5. What is the smallest possible value of this sum? | 40 | 0 | 8,192 | -1 | 8,192 | |
Alice places six ounces of coffee into a twelve-ounce cup and two ounces of coffee plus four ounces of cream into a second twelve-ounce cup. She then pours half the contents from the first cup into the second and, after stirring thoroughly, pours half the liquid in the second cup back into the first. What fraction of t... | \frac{4}{15} | 0 | 3,720.875 | -1 | 3,720.875 | |
The parabolas with equations \( y = x^2 - 2x - 3 \) and \( y = -x^2 + 4x + c \) intersect at points \( A \) and \( B \). Determine the value of \( c \) so that the sum of the \( x \)-coordinate and \( y \)-coordinate of the midpoint of \( AB \) is 2017. | 4031 | 0.75 | 4,814.5625 | 4,105.583333 | 6,941.5 | |
Our next object up for bid is an arithmetic progression of primes. For example, the primes 3,5, and 7 form an arithmetic progression of length 3. What is the largest possible length of an arithmetic progression formed of positive primes less than 1,000,000? Be prepared to justify your answer. | 12 | 12. We can get 12 with 110437 and difference 13860. | 0 | 6,341.125 | -1 | 6,341.125 |
Parallelogram $PQRS$ has vertices $P(4,4)$, $Q(-2,-2)$, $R(-8,-2)$, and $S(-2,4)$. If a point is selected at random from the region determined by the parallelogram, what is the probability that the point is not below the $x$-axis? Express your answer as a common fraction. | \frac{1}{2} | 0 | 7,893 | -1 | 7,893 | |
Find all three-digit numbers that are equal to the sum of all their digits plus twice the square of the sum of their digits. List all possible numbers in ascending order without spaces and enter the resulting concatenated multi-digit number. | 171465666 | 0.75 | 6,330.0625 | 5,805.083333 | 7,905 | |
An infinite arithmetic progression of positive integers contains the terms 7, 11, 15, 71, 75, and 79. The first term in the progression is 7. Kim writes down all the possible values of the one-hundredth term in the progression. What is the sum of the numbers Kim writes down? | 714 | 0.5 | 5,813.25 | 4,572.625 | 7,053.875 | |
Given a cube \(ABCD-A_1B_1C_1D_1\) with side length 1, and \(E\) as the midpoint of \(D_1C_1\), find the following:
1. The distance between skew lines \(D_1B\) and \(A_1E\).
2. The distance from \(B_1\) to plane \(A_1BE\).
3. The distance from \(D_1C\) to plane \(A_1BE\).
4. The distance between plane \(A_1DB\) and pla... | \frac{\sqrt{3}}{3} | 0 | 7,443.5625 | -1 | 7,443.5625 | |
A deck of 100 cards is numbered from 1 to 100. Each card has the same number printed on both sides. One side of each card is red and the other side is yellow. Barsby places all the cards, red side up, on a table. He first turns over every card that has a number divisible by 2. He then examines all the cards, and turns ... | 49 | 0.25 | 7,283.625 | 6,625.25 | 7,503.083333 | |
Suppose that we have an 8-sided die with 4 red faces, 3 yellow faces, and a blue face. What is the probability of rolling a yellow face? | \dfrac38 | 1 | 816.9375 | 816.9375 | -1 | |
How many positive integers, not exceeding 200, are multiples of 5 or 7 but not 10? | 43 | 0.25 | 7,795 | 7,556.5 | 7,874.5 | |
When the graph of $y = 2x^2 - x + 7$ is shifted four units to the right, we obtain the graph of $y = ax^2 + bx + c$. Find $a + b + c$. | 28 | 1 | 2,769.5 | 2,769.5 | -1 | |
From the set of integers $\{1,2,3,\dots,3009\}$, choose $k$ pairs $\{a_i,b_i\}$ with $a_i<b_i$ so that no two pairs share a common element. Each sum $a_i+b_i$ must be distinct and less than or equal to $3009$. Determine the maximum possible value of $k$. | 1504 | 0.125 | 7,895.375 | 5,819 | 8,192 | |
Let $r,$ $s,$ and $t$ be the roots of the equation $x^3 - 20x^2 + 18x - 7 = 0.$ Find the value of $\frac{r}{\frac{1}{r}+st} + \frac{s}{\frac{1}{s}+tr} + \frac{t}{\frac{1}{t}+rs}.$ | \frac{91}{2} | 0.9375 | 3,653.5 | 3,350.933333 | 8,192 | |
Given $ab+bc+cd+da = 30$ and $b+d = 5$, find $a+c$. | 6 | 1 | 1,579.875 | 1,579.875 | -1 | |
Points with coordinates $(1,1),(5,1)$ and $(1,7)$ are three vertices of a rectangle. What are the coordinates of the fourth vertex of the rectangle? | (5,7) | Since the given three points already form a right angle, then the fourth vertex of the rectangle must be vertically above the point $(5,1)$ and horizontally to the right of $(1,7)$. Therefore, the $x$-coordinate of the fourth vertex is 5 and the $y$-coordinate is 7. Thus, the coordinates of the fourth vertex are $(5,7)... | 0.4375 | 6,267.9375 | 3,794.142857 | 8,192 |
Let $m\in R$, the moving straight line passing through the fixed point $A$ with equation $x+my-2=0$ intersects the moving straight line passing through the fixed point $B$ with equation $mx-y+4=0$ at point $P\left(x,y\right)$. Find the maximum value of $|PA|\cdot |PB|$. | 10 | 0.125 | 8,192 | 8,192 | 8,192 | |
Given the vertices of a pyramid with a square base, and two vertices connected by an edge are called adjacent vertices, with the rule that adjacent vertices cannot be colored the same color, and there are 4 colors to choose from, calculate the total number of different coloring methods. | 72 | 0.1875 | 7,713 | 7,116.333333 | 7,850.692308 | |
Find the repetend in the decimal representation of $\frac{5}{17}$. | 294117647058823529 | 0 | 6,782.25 | -1 | 6,782.25 | |
Given a sequence $\{x_n\}$ that satisfies $x_{n+2}=|x_{n+2}-x_n|$ (where $n \in \mathbb{N}^*$), if $x_1=1$, $x_2=a$ (where $a \leqslant 1$ and $a \neq 0$), and $x_{n+3}=x_n$ for any positive integer $n$, then the sum of the first 2017 terms of the sequence $\{x_n\}$ is ______. | 1345 | 0.0625 | 7,615.3125 | 7,256 | 7,639.266667 | |
Eighty bricks, each measuring $3''\times9''\times18''$, are to be stacked one on top of another to form a tower 80 bricks tall. Each brick can be oriented so it contributes $3''$, $9''$, or $18''$ to the total height of the tower. How many different tower heights can be achieved using all eighty of the bricks? | 401 | 0 | 8,192 | -1 | 8,192 | |
Given that a 4-digit positive integer has four different digits, the leading digit is not zero, the integer is a multiple of 4, and 6 is the largest digit, determine the total count of such integers. | 56 | 0 | 8,012.1875 | -1 | 8,012.1875 | |
Solve the equations:<br/>$(1)x^{2}-4x-1=0$;<br/>$(2)\left(x+3\right)^{2}=x+3$. | -2 | 0.0625 | 1,115.1875 | 1,949 | 1,059.6 | |
In parallelogram $ABCD$, let $O$ be the intersection of diagonals $\overline{AC}$ and $\overline{BD}$. Angles $CAB$ and $DBC$ are each twice as large as angle $DBA$, and angle $ACB$ is $r$ times as large as angle $AOB$. Find $\lfloor 1000r \rfloor$. | 777 | [asy]size(180); pathpen = black+linewidth(0.7); pair B=(0,0), A=expi(pi/4), C=IP(A--A + 2*expi(17*pi/12), B--(3,0)), D=A+C, O=IP(A--C,B--D); D(MP("A",A,N)--MP("B",B)--MP("C",C)--MP("D",D,N)--cycle); D(B--D); D(A--C); D(MP("O",O,SE)); D(anglemark(D,B,A,4));D(anglemark(B,A,C,3.5));D(anglemark(B,A,C,4.5));D(anglemark(C,B,... | 0.125 | 6,984.25 | 6,875 | 6,999.857143 |
Let $m$ be a positive integer and let the lines $13x+11y=700$ and $y=mx-1$ intersect in a point whose coordinates are integers. Then m can be: | 6 | 1. **Substitute the equation of the line $y = mx - 1$ into the equation $13x + 11y = 700$:**
\[
13x + 11(mx - 1) = 700
\]
Simplifying this, we get:
\[
13x + 11mx - 11 = 700
\]
\[
(13 + 11m)x = 711
\]
2. **Factorize 711 to find possible values for $13 + 11m$:**
\[
711 = 3 \times 3 \t... | 0.75 | 6,068.875 | 5,361.166667 | 8,192 |
If the three points $$\begin{aligned} & (1, a, b) \\ & (a, 2, b) \\ & (a, b, 3) \end{aligned}$$ are collinear (in 3-space), what is the value of $a+b$ ? | 4 | The first two points are distinct (otherwise we would have $a=1$ and $a=2$ simultaneously), and they both lie on the plane $z=b$, so the whole line is in this plane and $b=3$. Reasoning similarly with the last two points gives $a=1$, so $a+b=4$. | 0.9375 | 4,295.25 | 4,035.466667 | 8,192 |
Given that \(A, B, C,\) and \(D\) are points on a circle with radius 1, \(\overrightarrow{AB} + 2 \overrightarrow{AC} = \overrightarrow{AD}\), and \(|AC| = 1\). Find the area of the quadrilateral \(ABDC\). | \frac{3 \sqrt{3}}{4} | 0 | 8,117.75 | -1 | 8,117.75 | |
The points $(0,0)\,$, $(a,11)\,$, and $(b,37)\,$ are the vertices of an equilateral triangle. Find the value of $ab\,$. | 315 | Consider the points on the complex plane. The point $b+37i$ is then a rotation of $60$ degrees of $a+11i$ about the origin, so:
\[(a+11i)\left(\mathrm{cis}\,60^{\circ}\right) = (a+11i)\left(\frac 12+\frac{\sqrt{3}i}2\right)=b+37i.\]
Equating the real and imaginary parts, we have:
\begin{align*}b&=\frac{a}{2}-\frac{11\... | 0.75 | 6,342.75 | 5,726.333333 | 8,192 |
Compute the number of ordered quintuples of nonnegative integers $(a_{1}, a_{2}, a_{3}, a_{4}, a_{5})$ such that $0 \leq a_{1}, a_{2}, a_{3}, a_{4}, a_{5} \leq 7$ and 5 divides $2^{a_{1}}+2^{a_{2}}+2^{a_{3}}+2^{a_{4}}+2^{a_{5}}$. | 6528 | Let $f(n)$ denote the number of $n$-tuples $(a_{1}, \ldots, a_{n})$ such that $0 \leq a_{1}, \ldots, a_{n} \leq 7$ and $5 \mid 2^{a_{1}}+\ldots+2^{a_{n}}$. To compute $f(n+1)$ from $f(n)$, we note that given any $n$-tuple $(a_{1}, \ldots, a_{n})$ such that $0 \leq a_{1}, \ldots, a_{n} \leq 7$ and $5 \nmid 2^{a_{1}}+\ld... | 0.25 | 7,624.5625 | 6,555.5 | 7,980.916667 |
The sum of 18 consecutive positive integers is a perfect square. What is the smallest possible value of this sum? | 225 | 0.9375 | 4,744 | 4,514.133333 | 8,192 | |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $(\sqrt{3}\cos10°-\sin10°)\cos(B+35°)=\sin80°$.
$(1)$ Find angle $B$.
$(2)$ If $2b\cos \angle BAC=c-b$, the angle bisector of $\angle BAC$ intersects $BC$ at point $D$, and $AD=2$, find $c$. | \sqrt{6}+\sqrt{2} | 0.125 | 8,072.3125 | 7,494 | 8,154.928571 | |
Point $D$ lies on side $\overline{BC}$ of $\triangle ABC$ so that $\overline{AD}$ bisects $\angle BAC.$ The perpendicular bisector of $\overline{AD}$ intersects the bisectors of $\angle ABC$ and $\angle ACB$ in points $E$ and $F,$ respectively. Given that $AB=4,BC=5,$ and $CA=6,$ the area of $\triangle AEF$ can be writ... | 36 | Points are defined as shown. It is pretty easy to show that $\triangle AFE \sim \triangle AGH$ by spiral similarity at $A$ by some short angle chasing. Now, note that $AD$ is the altitude of $\triangle AFE$, as the altitude of $AGH$. We need to compare these altitudes in order to compare their areas. Note that Stewart'... | 0.375 | 7,286.375 | 6,522 | 7,745 |
The segment \( PQ \) is divided into several smaller segments. On each of them, a square is constructed (see illustration).
What is the length of the path along the arrows if the length of the segment \( PQ \) is 73? If necessary, round your answer to 0.01 or write the answer as a common fraction. | 219 | 0.1875 | 4,723.1875 | 4,980 | 4,663.923077 | |
In some 16 cells of an $8 \times 8$ board, rooks are placed. What is the minimum number of pairs of rooks that can attack each other in this configuration? | 16 | 0.5 | 7,133.875 | 6,075.75 | 8,192 | |
If \(\log_{2} a + \log_{2} b \geq \gamma\), determine the smallest positive value \(\delta\) for \(a+b\). | 16 | 0 | 3,232.25 | -1 | 3,232.25 |
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