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 |
|---|---|---|---|---|---|---|
What is the next term in the geometric sequence $$2, 6x, 18x^2, 54x^3, \ldots ?$$ Express your answer in terms of $x$. | 162x^4 | 1 | 1,486 | 1,486 | -1 | |
A plane flies from city A to city B against a wind in 120 minutes. On the return trip with the wind, it takes 10 minutes less than it would in still air. Determine the time in minutes for the return trip. | 110 | 0 | 8,116.3125 | -1 | 8,116.3125 | |
Squares $ABCD$ and $EFGH$ are congruent, $AB=10$, and $G$ is the center of square $ABCD$. The area of the region in the plane covered by these squares is | 175 | 1. **Identify the given information**: We are given two congruent squares $ABCD$ and $EFGH$ with side length $AB = 10$. The point $G$ is the center of square $ABCD$.
2. **Calculate the area of each square**:
- The area of square $ABCD$ is $AB^2 = 10^2 = 100$.
- Since $EFGH$ is congruent to $ABCD$, its area is a... | 0.0625 | 8,160.5 | 8,091 | 8,165.133333 |
There are three identical red balls, three identical yellow balls, and three identical green balls. In how many different ways can they be split into three groups of three balls each? | 10 | 0 | 7,829.1875 | -1 | 7,829.1875 | |
Find $1^{234} + 4^6 \div 4^4$. | 17 | 1 | 1,529.4375 | 1,529.4375 | -1 | |
For all integers $n$ greater than 1, define $a_n =
\dfrac{1}{\log_n 2002}$. Let $b = a_2 + a_3 + a_4 + a_5$ and $c=
a_{10} + a_{11} + a_{12} + a_{13} + a_{14}$. Find $b - c.$ | -1 | 0.875 | 3,756.5625 | 3,276.5 | 7,117 | |
Find the area of a trapezoid, whose diagonals are 7 and 8, and whose bases are 3 and 6. | 4 : 3 | 0 | 4,841.75 | -1 | 4,841.75 | |
A subset of the integers $1,2,\cdots,100$ has the property that none of its members is 3 times another. What is the largest number of members such a subset can have?
$\text{(A) } 50\quad \text{(B) } 66\quad \text{(C) } 67\quad \text{(D) } 76\quad \text{(E) } 78$
| 76 | 0 | 8,192 | -1 | 8,192 | |
Three circles of radius 1 are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle? Express your answer as a common fraction in simplest radical form.
[asy]
draw(Circle((0,-0.58),2.15),linewidth(0.7));
draw(Circle((-1,0),1),linewidth(0.7));
draw(Circle((1,0... | \frac{3+2\sqrt{3}}{3} | 0 | 5,446.1875 | -1 | 5,446.1875 | |
If $A\ \clubsuit\ B$ is defined as $A\ \clubsuit\ B = 3A + 2B + 5$, what is the value of $A$ for which $A\ \clubsuit\ 4 = 58$? | 15 | 1 | 1,295 | 1,295 | -1 | |
The probability that B sits exactly in the middle in a row of three seats, given that three people A, B, and C are randomly seated. | \frac{1}{3} | 0.8125 | 4,087.25 | 3,140 | 8,192 | |
Eight strangers are preparing to play bridge. How many ways can they be grouped into two bridge games, meaning into unordered pairs of unordered pairs of people? | 315 | 0.25 | 6,695.875 | 5,755.75 | 7,009.25 | |
In the middle of a vast prairie, a firetruck is stationed at the intersection of two perpendicular straight highways. The truck travels at $50$ miles per hour along the highways and at $14$ miles per hour across the prairie. Consider the set of points that can be reached by the firetruck within six minutes. The area of... | 731 | 0 | 8,052.5 | -1 | 8,052.5 | |
Sean is a biologist, and is looking at a string of length 66 composed of the letters $A, T, C, G$. A substring of a string is a contiguous sequence of letters in the string. For example, the string $AGTC$ has 10 substrings: $A, G, T, C, AG, GT, TC, AGT, GTC, AGTC$. What is the maximum number of distinct substrings of t... | 2100 | Let's consider the number of distinct substrings of length $\ell$. On one hand, there are obviously at most $4^{\ell}$ distinct substrings. On the other hand, there are $67-\ell$ substrings of length $\ell$ in a length 66 string. Therefore, the number of distinct substrings is at most $\sum_{\ell=1}^{66} \min \left(4^{... | 0.4375 | 7,011.875 | 6,301.428571 | 7,564.444444 |
Given the quadratic function $f(x)=3x^{2}-2x$, the sum of the first $n$ terms of the sequence $\{a_{n}\}$ is $S_{n}$, and the point $(n,S_{n})$ (where $n \in \mathbb{N}^{*}$) is on the graph of the function $y=f(x)$.
$(1)$ Find the general formula for the sequence $\{a_{n}\}$;
$(2)$ Let $b_{n}=\frac{3}{a_{n}a_{n+1}}$... | 10 | 0.25 | 5,838.4375 | 7,000 | 5,451.25 | |
A point is randomly thrown onto the segment $[11, 18]$ and let $k$ be the resulting value. Find the probability that the roots of the equation $\left(k^{2}+2k-99\right)x^{2}+(3k-7)x+2=0$ satisfy the condition $x_{1} \leq 2x_{2}$. | \frac{2}{3} | 0.1875 | 7,925.25 | 7,213.333333 | 8,089.538462 | |
Let ($a_1$, $a_2$, ... $a_{10}$) 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 the constraints on the sequence and how they affect the arrangement of the numbers. The key constraint is that for each $2 \leq i \leq 10$, either $a_i + 1$ or $a_i - 1$ (or both) must appear before $a_i$ in the list. This constraint significantly restricts the order in whic... | 0.1875 | 7,910.75 | 6,784.666667 | 8,170.615385 |
Kelvin the Frog and 10 of his relatives are at a party. Every pair of frogs is either friendly or unfriendly. When 3 pairwise friendly frogs meet up, they will gossip about one another and end up in a fight (but stay friendly anyway). When 3 pairwise unfriendly frogs meet up, they will also end up in a fight. In all ot... | 28 | Consider a graph $G$ with 11 vertices - one for each of the frogs at the party - where two vertices are connected by an edge if and only if they are friendly. Denote by $d(v)$ the number of edges emanating from $v$; i.e. the number of friends frog $v$ has. Note that $d(1)+d(2)+\ldots+d(11)=2e$, where $e$ is the number ... | 0.0625 | 7,721.4375 | 7,619 | 7,728.266667 |
Two circles of radius 5 are externally tangent to each other and are internally tangent to a circle of radius 13 at points $A$ and $B$, as shown in the diagram. The distance $AB$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$ ?
[asy]
draw(circle((0,0)... | 69 | 0.5625 | 6,367.875 | 5,103.555556 | 7,993.428571 | |
The analysis of bank accounts revealed that the balances on each of them are more than 10 rubles. Additionally, there was a group of clients, each having the same amount of money on their account. This amount is a number consisting solely of ones. If the total amount of money on the accounts of this group of clients i... | 101 | 0.1875 | 8,098.25 | 7,692 | 8,192 | |
(1) Use the Euclidean algorithm to find the greatest common divisor (GCD) of 2146 and 1813.
(2) Use the Horner's method to calculate the value of $v_4$ for the function $f(x) = 2x^5 + 3x^4 + 2x^3 - 4x + 5$ when $x = 2$. | 60 | 0.625 | 4,595.625 | 4,628.8 | 4,540.333333 | |
The first term of a given sequence is 2, and each successive term is the sum of all the previous terms of the sequence. What is the value of the first term which exceeds 10000? | 16384 | 1 | 5,002.4375 | 5,002.4375 | -1 | |
Given the function $f(x)=\sin x+\cos x$, where $x\in \mathbb{R}$.
- (I) Find the value of $f\left( \frac{\pi}{2}\right)$;
- (II) Find the smallest positive period of the function $f(x)$;
- (III) Find the minimum value of the function $g(x)=f\left(x+ \frac{\pi}{4}\right)+f\left(x+ \frac{3\pi}{4}\right)$. | -2 | 0.875 | 4,449.4375 | 4,612.642857 | 3,307 | |
In the diagram, \(p, q, r, s\), and \(t\) represent five consecutive integers, not necessarily in order. The two integers in the leftmost circle add to 63. The two integers in the rightmost circle add to 57. What is the value of \(r\)? | 30 | 0.375 | 6,496.9375 | 4,056 | 7,961.5 | |
The height of a rhombus, drawn from the vertex of its obtuse angle, divides the side of the rhombus in the ratio $1:3$, measured from the vertex of its acute angle. What fraction of the area of the rhombus is occupied by the area of a circle inscribed in it? | \frac{\pi \sqrt{15}}{16} | 0 | 6,572.9375 | -1 | 6,572.9375 | |
Suppose the edge length of a regular tetrahedron $ABC D$ is 1 meter. A bug starts at point $A$ and moves according to the following rule: at each vertex, it chooses one of the three edges connected to this vertex with equal probability and crawls along this edge to the next vertex. What is the probability that the bug ... | 7/27 | 0.3125 | 7,262.8125 | 5,905 | 7,880 | |
Given that the circumferences of the two bases of a cylinder lie on the surface of a sphere with an area of $20\pi$, the maximum value of the lateral surface area of the cylinder is ____. | 10\pi | 0.4375 | 6,295.125 | 4,517.142857 | 7,678 | |
In triangle $ABC$, we have that $E$ and $F$ are midpoints of sides $\overline{AC}$ and $\overline{AB}$, respectively. The area of $\triangle ABC$ is 24 square units. How many square units are in the area of $\triangle CEF$? | 6 | 0.9375 | 5,486.75 | 5,306.4 | 8,192 | |
On the complex plane, the parallelogram formed by the points 0, $z,$ $\frac{1}{z},$ and $z + \frac{1}{z}$ has an area of $\frac{12}{13}.$ If the real part of $z$ is positive, compute the smallest possible value of $\left| z + \frac{1}{z} \right|^2.$ | \frac{36}{13} | 0.0625 | 7,465.125 | 7,402 | 7,469.333333 | |
How many three-digit whole numbers contain at least one 6 or at least one 8? | 452 | 0.3125 | 7,604.5 | 6,469 | 8,120.636364 | |
A line passing through the focus of the parabola $y^2=4x$ intersects the parabola at points $A(x_1, y_1)$ and $B(x_2, y_2)$. If $|AB|=7$, find the distance from the midpoint $M$ of $AB$ to the directrix of the parabola. | \frac{7}{2} | 0.8125 | 5,900.9375 | 5,372.230769 | 8,192 | |
Consider a parallelogram where each vertex has integer coordinates and is located at $(0,0)$, $(4,5)$, $(11,5)$, and $(7,0)$. Calculate the sum of the perimeter and the area of this parallelogram. | 9\sqrt{41} | 0 | 5,734.3125 | -1 | 5,734.3125 | |
In this diagram the center of the circle is $O$, the radius is $a$ inches, chord $EF$ is parallel to chord $CD$. $O$,$G$,$H$,$J$ are collinear, and $G$ is the midpoint of $CD$. Let $K$ (sq. in.) represent the area of trapezoid $CDFE$ and let $R$ (sq. in.) represent the area of rectangle $ELMF.$ Then, as $CD$ and $EF$ a... | \frac{1}{\sqrt{2}}+\frac{1}{2} | 1. **Identify the given information and setup the problem:**
- The center of the circle is $O$, and the radius is $a$ inches.
- Chords $CD$ and $EF$ are parallel, and $O$, $G$, $H$, $J$ are collinear with $G$ as the midpoint of $CD$.
- $OG = a - 2h$, where $h = JH = HG$.
- We need to find the ratio $K:R$ as... | 0 | 8,192 | -1 | 8,192 |
Given the triangular pyramid \( P-ABC \) with edge lengths \( PA=1 \), \( PB=2 \), and \( PC=3 \), and \( PA \perp PB \), \( PB \perp PC \), \( PC \perp PA \), determine the maximum distance from a point \( Q \) on the circumsphere of this pyramid to the face \( ABC \). | \frac{3}{7} + \frac{\sqrt{14}}{2} | 0 | 7,258.0625 | -1 | 7,258.0625 | |
Jane can walk any distance in half the time it takes Hector to walk the same distance. They set off in opposite directions around the outside of the 18-block area as shown. When they meet for the first time, they will be closest to | D | 1. **Understanding the Problem**: Jane and Hector start at the same point and walk in opposite directions around an 18-block area. Jane walks twice as fast as Hector.
2. **Setting Up the Problem**: Let's denote the speed of Hector as $s$ blocks per unit time. Therefore, Jane's speed is $2s$ blocks per unit time becaus... | 0 | 6,748.5 | -1 | 6,748.5 |
In a math test, there are easy and hard questions. The easy questions worth 3 points and the hard questions worth D points.\\
If all the questions begin to worth 4 points, the total punctuation of the test increases 16 points.\\
Instead, if we exchange the questions scores, scoring D points for the easy questions and 3... | 20 |
Let \( x \) be the number of hard questions and \( 9x \) be the number of easy questions in the test. Let the total number of questions be \( n = x + 9x = 10x \).
Given:
- Easy questions are worth 3 points each.
- Hard questions are worth \( D \) points each.
**Initial Total Points**
The initial total score of the... | 0.9375 | 3,045.3125 | 2,702.2 | 8,192 |
What is the area, in square units, of a regular hexagon inscribed in a circle whose area is $324\pi$ square units? Express your answer in simplest radical form. | 486 \sqrt{3} | 1 | 2,092.375 | 2,092.375 | -1 | |
If the area of the triangle formed by the tangent line to the curve $y=x^{-\frac{1}{2}}$ at the point $(a,a^{-\frac{1}{2}})$ and the two coordinate axes is $18$, find the real number $a=\_\_\_\_\_\_\_\_.$ | 64 | 1 | 2,957.875 | 2,957.875 | -1 | |
Vasya has:
a) 2 different volumes from the collected works of A.S. Pushkin, each volume is 30 cm high;
b) a set of works by E.V. Tarle in 4 volumes, each volume is 25 cm high;
c) a book of lyrical poems with a height of 40 cm, published by Vasya himself.
Vasya wants to arrange these books on a shelf so that his own wo... | 144 | 0 | 7,660 | -1 | 7,660 | |
How many different three-letter sets of initials are possible using the letters $A$ through $G$? | 343 | 0.1875 | 4,708.8125 | 4,236.333333 | 4,817.846154 | |
A cuckoo clock is on the wall. At the beginning of every hour, the cuckoo makes a number of "cuckoo" sounds equal to the hour displayed by the hour hand (for example, at 19:00 the cuckoo makes 7 sounds). One morning, Maxim approached the clock when it showed 9:05. He started turning the minute hand until he moved the t... | 43 | 0 | 4,949.875 | -1 | 4,949.875 | |
In $\triangle ABC$ shown in the figure, $AB=7$, $BC=8$, $CA=9$, and $\overline{AH}$ is an altitude. Points $D$ and $E$ lie on sides $\overline{AC}$ and $\overline{AB}$, respectively, so that $\overline{BD}$ and $\overline{CE}$ are angle bisectors, intersecting $\overline{AH}$ at $Q$ and $P$, respectively. What is $PQ$? | \frac{8}{15}\sqrt{5} | 1. **Calculate the semi-perimeter and area of $\triangle ABC$ using Heron's Formula:**
- Semi-perimeter, $s = \frac{AB + BC + CA}{2} = \frac{7 + 8 + 9}{2} = 12$.
- Area, $K = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{12(12-7)(12-8)(12-9)} = \sqrt{12 \cdot 5 \cdot 4 \cdot 3} = \sqrt{720} = 12\sqrt{5}$.
2. **Find the height... | 0 | 4,638.3125 | -1 | 4,638.3125 |
Let $ABCD$ be a convex quadrilateral such that $AB + BC = 2021$ and $AD = CD$ . We are also given that $\angle ABC = \angle CDA = 90^o$ . Determine the length of the diagonal $BD$ . | \frac{2021 \sqrt{2}}{2} | 0 | 8,058.875 | -1 | 8,058.875 | |
Two groups have an equal number of students. Each student studies at least one language: English or French. It is known that 5 people in the first group and 5 in the second group study both languages. The number of students studying French in the first group is three times less than in the second group. The number of s... | 28 | 0.25 | 6,482.0625 | 5,352.25 | 6,858.666667 | |
What value should the real number $m$ take for the complex number $z = m + 1 + (m - 1)i$ to be:
(1) a real number; (2) a complex number; (3) a pure imaginary number? | -1 | 0.8125 | 4,575.25 | 4,115.230769 | 6,568.666667 | |
Players A and B have a Go game match, agreeing that the first to win 3 games wins the match. After the match ends, assuming in a single game, the probability of A winning is 0.6, and the probability of B winning is 0.4, with the results of each game being independent. It is known that in the first 2 games, A and B each... | 2.48 | 0 | 8,091.3125 | -1 | 8,091.3125 | |
Let \[p(x, y) =
\begin{cases}
xy &\quad \text{if } x \ge 0 \text{ and } y \ge 0, \\
x - 2y &\quad \text{if } x < 0 \text{ and } y < 0, \\
2x + 3y &\quad \text{if } x \ge 0 \text{ and } y < 0, \\
x + 3y &\quad \text{if } x < 0 \text{ and } y \ge 0 \\
3x + y &\quad \text{otherwise}.
\end{cases}
\]What is $p(p(2,-3),p(-1... | 28 | 1 | 2,037 | 2,037 | -1 | |
Given that $S = 6 \times 10,000 + 5 \times 1000 + 4 \times 10 + 3 \times 1$, calculate the value of $S$. | 65043 | 0.5 | 403.25 | 396.125 | 410.375 | |
Given the parametric equation of line $l$ as $$\begin{cases} x= \sqrt {3}+t \\ y=7+ \sqrt {3}t\end{cases}$$ ($t$ is the parameter), a coordinate system is established with the origin as the pole and the positive half of the $x$-axis as the polar axis. The polar equation of curve $C$ is $\rho \sqrt {a^{2}\sin^{2}\theta+... | \frac{2\sqrt{21}}{3} | 0 | 6,641.3125 | -1 | 6,641.3125 | |
A group of 8 boys and 8 girls was paired up randomly. Find the probability that there is at least one pair with two girls. Round your answer to the nearest hundredth. | 0.98 | 0 | 8,192 | -1 | 8,192 | |
Given that Isabella's fort has dimensions $15$ feet in length, $12$ feet in width, and $6$ feet in height, with one-foot thick floor and walls, determine the number of one-foot cubical blocks required to construct this fort. | 430 | 0.0625 | 8,087.5 | 6,520 | 8,192 | |
Let $A_1,A_2,A_3,\cdots,A_{12}$ be the vertices of a regular dodecagon. How many distinct squares in the plane of the dodecagon have at least two vertices in the set $\{A_1,A_2,A_3,\cdots,A_{12}\} ?$ | 183 | Proceed as above to initially get 198 squares (with overcounting). Then note that any square with all four vertices on the dodecagon has to have three sides "between" each vertex, giving us a total of three squares. However, we counted these squares with all four of their sides plus both of their diagonals, meaning we ... | 0 | 8,078.25 | -1 | 8,078.25 |
Given real numbers $x$ and $y$ satisfying $x^{2}+y^{2}-4x-2y-4=0$, calculate the maximum value of $x-y$. | 1+3\sqrt{2} | 1 | 4,169.3125 | 4,169.3125 | -1 | |
The numbers $1447$, $1005$ and $1231$ have something in common: each is a $4$-digit number beginning with $1$ that has exactly two identical digits. How many such numbers are there? | 432 | Suppose that the two identical digits are both $1$. Since the thousands digit must be $1$, only one of the other three digits can be $1$. This means the possible forms for the number are
$11xy,\qquad 1x1y,\qquad1xy1$
Because the number must have exactly two identical digits, $x\neq y$, $x\neq1$, and $y\neq1$. Hence, t... | 0 | 8,117.75 | -1 | 8,117.75 |
Santa Claus has a sack containing both chocolate and gummy candies, totaling 2023 pieces. The chocolate candies make up 75% of the gummy candies. How many chocolate candies does Santa Claus have in his sack? | 867 | 0.3125 | 546.9375 | 578.6 | 532.545455 | |
Find the number of square units in the area of the shaded region. [asy]
size(200);
draw((0,0)--(30,0)--(30,30)--(0, 30)--cycle);
dot((30,0));dot((0,0));dot((10,0));dot((30,20));dot((30,30)); dot((20,30));dot((0,30));dot((0,10));
filldraw((0,0)--(10,0)--(30,20)--(30,30)--(20,30)--(0,10)--(0,0)--cycle, lightgray);
draw((... | 500 | 0.5 | 7,472.3125 | 6,752.625 | 8,192 | |
A sequence of numbers $t_{1}, t_{2}, t_{3}, \ldots$ has its terms defined by $t_{n}=\frac{1}{n}-\frac{1}{n+2}$ for every integer $n \geq 1$. What is the largest positive integer $k$ for which the sum of the first $k$ terms is less than 1.499? | 1998 | We note that $t_{1}=\frac{1}{1}-\frac{1}{3}=\frac{2}{3} \approx 0.67$, $t_{1}+t_{2}=\left(\frac{1}{1}-\frac{1}{3}\right)+\left(\frac{1}{2}-\frac{1}{4}\right)=\frac{2}{3}+\frac{1}{4}=\frac{11}{12} \approx 0.92$, $t_{1}+t_{2}+t_{3}=\left(\frac{1}{1}-\frac{1}{3}\right)+\left(\frac{1}{2}-\frac{1}{4}\right)+\left(\frac{1}{3... | 0.625 | 7,389 | 6,907.2 | 8,192 |
Sasha records the numbers 1, 2, 3, 4, and 5 in some order, places arithmetic operation signs "+", "-", "x" and parentheses, and looks at the result of the expression obtained. For example, he can get the number 8 using the expression \((4-3) \times (2+5) + 1\). Can he get the number 123?
Forming numbers from multiple ... | 123 | 0.0625 | 7,965 | 4,560 | 8,192 | |
Sophie has written three tests. Her marks were $73\%$, $82\%$, and $85\%$. She still has two tests to write. All tests are equally weighted. Her goal is an average of $80\%$ or higher. With which of the following pairs of marks on the remaining tests will Sophie not reach her goal: $79\%$ and $82\%$, $70\%$ and $91\%$,... | 73\% and 83\% | For Sophie's average over 5 tests to be $80\%$, the sum of her marks on the 5 tests must be $5 \times 80\% = 400\%$. After the first 3 tests, the sum of her marks is $73\% + 82\% + 85\% = 240\%$. Therefore, she will reach her goal as long as the sum of her marks on the two remaining tests is at least $400\% - 240\% = 1... | 0 | 2,622.4375 | -1 | 2,622.4375 |
A regular triangle is inscribed in a circle with a diameter of $\sqrt{12}$. Another regular triangle is constructed on its height as a side, and a new circle is inscribed in this triangle. Find the radius of this circle. | \frac{3}{4} | 0.875 | 4,458.6875 | 3,925.357143 | 8,192 | |
Given that $(a + 1)x^2 + (a^2 + 1) + 8x = 9$ is a quadratic equation in terms of $x$, find the value of $a$. | 2\sqrt{2} | 0 | 7,888.9375 | -1 | 7,888.9375 | |
If $4^{n}=64^{2}$, what is the value of $n$? | 6 | We note that $64=4 \times 4 \times 4$. Thus, $64^{2}=64 \times 64=4 \times 4 \times 4 \times 4 \times 4 \times 4$. Since $4^{n}=64^{2}$, then $4^{n}=4 \times 4 \times 4 \times 4 \times 4 \times 4$ and so $n=6$. | 1 | 650.875 | 650.875 | -1 |
A sequence of functions $\, \{f_n(x) \} \,$ is defined recursively as follows: \begin{align*} f_1(x) &= \sqrt {x^2 + 48}, \quad \text{and} \\ f_{n + 1}(x) &= \sqrt {x^2 + 6f_n(x)} \quad \text{for } n \geq 1. \end{align*} (Recall that $\sqrt {\makebox[5mm]{}}$ is understood to represent the positive square root .) For ... | \[ x = 4 \] | We define $f_0(x) = 8$ . Then the recursive relation holds for $n=0$ , as well.
Since $f_n (x) \ge 0$ for all nonnegative integers $n$ , it suffices to consider nonnegative values of $x$ .
We claim that the following set of relations hold true for all natural numbers $n$ and nonnegative reals $x$ : \begin{align*} f_n(... | 0 | 7,425.8125 | -1 | 7,425.8125 |
The sides of triangle $CAB$ are in the ratio of $2:3:4$. Segment $BD$ is the angle bisector drawn to the shortest side, dividing it into segments $AD$ and $DC$. What is the length, in inches, of the longer subsegment of side $AC$ if the length of side $AC$ is $10$ inches? Express your answer as a common fraction. | \frac {40}7 | 0 | 3,812.5 | -1 | 3,812.5 | |
The given arithmetic sequences $\{a_{n}\}$ and $\{b_{n}\}$ have respective sums of the first $n$ terms, denoted by $S_{n}$ and $T_{n}$. The ratio $\frac{S_{n}}{T_{n}} = \frac{3n + 1}{n + 3}$. Determine the value of $\frac{a_{2} + a_{20}}{b_{7} + b_{15}}$. | \frac{8}{3} | 0.6875 | 5,516.6875 | 5,092 | 6,451 | |
The skeletal structure of coronene, a hydrocarbon with the chemical formula $\mathrm{C}_{24} \mathrm{H}_{12}$, is shown below. Each line segment between two atoms is at least a single bond. However, since each carbon (C) requires exactly four bonds connected to it and each hydrogen $(\mathrm{H})$ requires exactly one b... | 20 | Note that each carbon needs exactly one double bond. Label the six carbons in the center $1,2,3,4,5,6$ clockwise. We consider how these six carbons are double-bonded. If a carbon in the center is not double-bonded to another carbon in the center, it must double-bond to the corresponding carbon on the outer ring. This w... | 0 | 8,192 | -1 | 8,192 |
A triple of positive integers $(a, b, c)$ is tasty if $\operatorname{lcm}(a, b, c) \mid a+b+c-1$ and $a<b<c$. Find the sum of $a+b+c$ across all tasty triples. | 44 | The condition implies $c \mid b+a-1$. WLOG assume $c>b>a$; since $b+a-1<2 c$ we must have $b+a-1=c$. Substituting into $b \mid a+c-1$ and $a \mid c+b-1$ gives $$\begin{aligned} & b \mid 2 a-2 \\ & a \mid 2 b-2 \end{aligned}$$ Since $2 a-2<2 b$ we must either have $a=1$ (implying $a=b$, bad) or $2 a-2=b \Longrightarrow ... | 0 | 8,192 | -1 | 8,192 |
Let $$\begin{aligned} & A=(1+2 \sqrt{2}+3 \sqrt{3}+6 \sqrt{6})(2+6 \sqrt{2}+\sqrt{3}+3 \sqrt{6})(3+\sqrt{2}+6 \sqrt{3}+2 \sqrt{6})(6+3 \sqrt{2}+2 \sqrt{3}+\sqrt{6}) \\ & B=(1+3 \sqrt{2}+2 \sqrt{3}+6 \sqrt{6})(2+\sqrt{2}+6 \sqrt{3}+3 \sqrt{6})(3+6 \sqrt{2}+\sqrt{3}+2 \sqrt{6})(6+2 \sqrt{2}+3 \sqrt{3}+\sqrt{6}) \end{alig... | 1 | Note that $$\begin{aligned} & A=((1+2 \sqrt{2})(1+3 \sqrt{3}))((2+\sqrt{3})(1+3 \sqrt{2}))((3+\sqrt{2})(1+2 \sqrt{3}))((3+\sqrt{3})(2+\sqrt{2})) \\ & B=((1+3 \sqrt{2})(1+2 \sqrt{3}))((2+\sqrt{2})(1+3 \sqrt{3}))((3+\sqrt{3})(1+2 \sqrt{2}))((2+\sqrt{3})(3+\sqrt{2})) \end{aligned}$$ It is not difficult to check that they ... | 0 | 8,192 | -1 | 8,192 |
The sequence $\{a_{n}\}$ satisfies $a_{1}+\frac{1}{2}{a_2}+\frac{1}{3}{a_3}+…+\frac{1}{n}{a_n}={a_{n+1}}-1$, $n\in N^{*}$, and $a_{1}=1$.<br/>$(1)$ Find the general formula for the sequence $\{a_{n}\}$;<br/>$(2)$ Let $S_{n}=a_{1}\cdot a_{n}+a_{2}\cdot a_{n-1}+a_{3}\cdot a_{n-2}+\ldots +a_{n}\cdot a_{1}$, $b_{n}=\frac{1... | 1012 | 0 | 7,547.3125 | -1 | 7,547.3125 | |
Find $x^2+y^2$ if $x$ and $y$ are positive integers such that \[\begin{aligned} xy+x+y&=71 \\ x^2y+xy^2 &= 880.\end{aligned}\] | 146 | 1 | 3,225.625 | 3,225.625 | -1 | |
There are 1000 lights and 1000 switches. Each switch simultaneously controls all lights whose numbers are multiples of the switch's number. Initially, all lights are on. Now, if switches numbered 2, 3, and 5 are pulled, how many lights will remain on? | 499 | 0 | 7,769.375 | -1 | 7,769.375 | |
Let $a$, $b$, $c$ be the three sides of a triangle, and let $\alpha$, $\beta$, $\gamma$ be the angles opposite them. If $a^2 + b^2 = 2020c^2$, determine the value of
\[\frac{\cot \gamma}{\cot \alpha + \cot \beta}.\] | 1009.5 | 0 | 4,233.3125 | -1 | 4,233.3125 | |
Rationalize the denominator: $\sqrt{\frac{3}{8}}.$ | \frac{\sqrt{6}}{4} | 0 | 1,882.1875 | -1 | 1,882.1875 | |
A certain quadratic polynomial is known to have the following properties: its leading coefficient is equal to one, it has integer roots, and its graph (parabola) intersects the line \( y = 2017 \) at two points with integer coordinates. Can the ordinate of the vertex of the parabola be uniquely determined based on this... | -1016064 | 0.125 | 7,790.8125 | 6,309 | 8,002.5 | |
Let $z_{1}, z_{2}, z_{3}, z_{4}$ be the solutions to the equation $x^{4}+3 x^{3}+3 x^{2}+3 x+1=0$. Then $\left|z_{1}\right|+\left|z_{2}\right|+\left|z_{3}\right|+\left|z_{4}\right|$ can be written as $\frac{a+b \sqrt{c}}{d}$, where $c$ is a square-free positive integer, and $a, b, d$ are positive integers with $\operat... | 7152 | Note that $x=0$ is clearly not a solution, so we can divide the equation by $x^{2}$ to get $\left(x^{2}+2+\frac{1}{x^{2}}\right)+3\left(x+\frac{1}{x}\right)+1=0$. Letting $y=x+\frac{1}{x}$, we get that $y^{2}+3 y+1=0$, so $y=x+\frac{1}{x}=\frac{-3 \pm \sqrt{5}}{2}$. Since $\frac{-3+\sqrt{5}}{2}$ has absolute value less... | 0.1875 | 7,686.9375 | 5,949.333333 | 8,087.923077 |
Simplify the expression $\frac{{2x+4}}{{{x^2}-1}}÷\frac{{x+2}}{{{x^2}-2x+1}}-\frac{{2x}}{{x+1}}$, then substitute an appropriate number from $-2$, $-1$, $0$, $1$ to evaluate. | -2 | 0.875 | 4,346.625 | 4,042.928571 | 6,472.5 | |
Let $P$ be a point on the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$, $F_{1}$ and $F_{2}$ be the two foci of the ellipse, and $e$ be the eccentricity of the ellipse. Given $\angle P F_{1} F_{2}=\alpha$ and $\angle P F_{2} F_{1}=\beta$, express $\tan \frac{\alpha}{2} \cdot \tan \frac{\beta}{2}$ in terms of $e$. | \frac{1 - e}{1 + e} | 0.0625 | 7,944.125 | 4,226 | 8,192 | |
Let $N$ be the number of ordered pairs of nonempty sets $\mathcal{A}$ and $\mathcal{B}$ that have the following properties:
$\mathcal{A} \cup \mathcal{B} = \{1,2,3,4,5,6,7,8,9,10,11,12\}$,
$\mathcal{A} \cap \mathcal{B} = \emptyset$,
The number of elements of $\mathcal{A}$ is not an element of $\mathcal{A}$,
The number ... | 772 | Regardless of the size $n$ of $A$ (ignoring the case when $n = 6$), $n$ must not be in $A$ and $12 - n$ must be in $A$.
There are $10$ remaining elements whose placements have yet to be determined. Note that the actual value of $n$ does not matter; there is always $1$ necessary element, $1$ forbidden element, and $10$... | 0.0625 | 8,192 | 8,192 | 8,192 |
An octahedron consists of two square-based pyramids glued together along their square bases to form a polyhedron with eight faces. Imagine an ant that begins at the top vertex and walks to one of the four adjacent vertices that he randomly selects and calls vertex A. From vertex A, he will then walk to one of the four ... | \frac{1}{4} | 0.375 | 7,070.9375 | 5,541 | 7,988.9 | |
What is the probability that all 4 blue marbles are drawn before all 3 yellow marbles are drawn? | \frac{4}{7} | 0.4375 | 7,332.75 | 6,228 | 8,192 | |
Given the arithmetic sequence $\{a_{n}\}$ with $a_{1}=1$ and $a_{n} > 0$ for all $n \in \mathbb{N}^*$, let $S_{n}$ denote the sum of its first $n$ terms. If the sequence $\{\sqrt{S_{n}}\}$ is also an arithmetic sequence, find the maximum value of $\frac{S_{n+10}}{a_{n}^{2}}$. | 121 | 0.875 | 5,635.125 | 5,269.857143 | 8,192 | |
Six trees are equally spaced along one side of a straight road. The distance from the first tree to the fourth is 60 feet. What is the distance in feet between the first and last trees? | 100 | 1. **Identify the number of spaces between trees**:
- The problem states that there are six trees equally spaced along one side of a straight road.
- The trees are numbered from 1 to 6.
- The distance from the first tree to the fourth tree is given as 60 feet.
2. **Calculate the distance between consecutiv... | 0.9375 | 1,900.375 | 1,480.933333 | 8,192 |
Grady distributed $x$ pieces of candy evenly among nine Halloween bags such that every bag received the greatest possible number of whole pieces of candy, but some candy was left over. What is the greatest possible number of pieces that could have been left over? | 8 | 1 | 1,362.75 | 1,362.75 | -1 | |
The graph of the line $x+y=b$ is a perpendicular bisector of the line segment from $(0,3)$ to $(6,9)$. What is the value of b? | 9 | 1 | 1,760.8125 | 1,760.8125 | -1 | |
What is the greatest prime factor of $15! + 18!$? | 17 | 0 | 3,174 | -1 | 3,174 | |
Given a positive integer \( n \) such that \( n \leq 2016 \) and \(\left\{\frac{n}{2}\right\}+\left\{\frac{n}{4}\right\}+\left\{\frac{n}{6}\right\}+\left\{\frac{n}{12}\right\}=3\), where \(\{x\} = x - \lfloor x \rfloor\) and \(\lfloor x \rfloor\) denotes the greatest integer less than or equal to \( x \), find the numb... | 168 | 0.4375 | 7,455.1875 | 6,507.857143 | 8,192 | |
Find the area of a trapezoid with bases 11 and 4 and diagonals 9 and 12. | 54 | 0.75 | 4,847.9375 | 4,023.5 | 7,321.25 | |
Given the vector $A$ with components $(1, -1, -3, -4)$, | 3\sqrt{3} | 0.5625 | 4,842.5 | 2,834.333333 | 7,424.428571 | |
In triangle \(ABC\), a circle is constructed with diameter \(AC\), which intersects side \(AB\) at point \(M\) and side \(BC\) at point \(N\). Given that \(AC = 2\), \(AB = 3\), and \(\frac{AM}{MB} = \frac{2}{3}\), find \(AN\). | \frac{24}{\sqrt{145}} | 0 | 8,191.625 | -1 | 8,191.625 | |
If the graph of the function $y=\log_a(x+m)+n$ passes through the fixed point $(-1, -2)$, then find the value of $m \cdot n$. | -4 | 1 | 2,147.9375 | 2,147.9375 | -1 | |
Two jars each contain the same number of marbles, and every marble is either blue or green. In Jar $1$ the ratio of blue to green marbles is $9:1$, and the ratio of blue to green marbles in Jar $2$ is $8:1$. There are $95$ green marbles in all. How many more blue marbles are in Jar $1$ than in Jar $2$? | 5 | 1. **Set up the ratios for each jar:**
For Jar $1$, the ratio of blue to green marbles is $9:1$. This means for every $10$ marbles in Jar $1$, $9$ are blue and $1$ is green.
For Jar $2$, the ratio of blue to green marbles is $8:1$. This means for every $9$ marbles in Jar $2$, $8$ are blue and $1$ is green.
2... | 1 | 2,160.4375 | 2,160.4375 | -1 |
A snail crawls from one tree to another. In half a day, it covered \( l_{1}=5 \) meters. Then, it got tired of this and turned back, crawling \( l_{2}=4 \) meters. It got tired and fell asleep. The next day, the same process repeats. The distance between the trees is \( s=30 \) meters. On which day of its journey will ... | 26 | 0.125 | 6,913.625 | 5,361.5 | 7,135.357143 | |
Let $\mathcal{H}$ be the unit hypercube of dimension 4 with a vertex at $(x, y, z, w)$ for each choice of $x, y, z, w \in \{0,1\}$. A bug starts at the vertex $(0,0,0,0)$. In how many ways can the bug move to $(1,1,1,1)$ by taking exactly 4 steps along the edges of $\mathcal{H}$? | 24 | You may think of this as sequentially adding 1 to each coordinate of $(0,0,0,0)$. There are 4 ways to choose the first coordinate, 3 ways to choose the second, and 2 ways to choose the third. The product is 24. | 0.9375 | 2,613.9375 | 2,242.066667 | 8,192 |
Given vectors \(\boldsymbol{a}\), \(\boldsymbol{b}\), and \(\boldsymbol{c}\) such that
\[
|a|=|b|=3, |c|=4, \boldsymbol{a} \cdot \boldsymbol{b}=-\frac{7}{2},
\boldsymbol{a} \perp \boldsymbol{c}, \boldsymbol{b} \perp \boldsymbol{c}
\]
Find the minimum value of the expression
\[
|x \boldsymbol{a} + y \boldsymbol{b} + (... | \frac{4 \sqrt{33}}{15} | 0 | 7,298.375 | -1 | 7,298.375 | |
The hyperbola given by the equation \[\frac{y^2}{9}-\frac{x^2}{4} = 1\]has asymptotes $y = \pm mx,$ where $m$ is positive. Find $m.$ | \frac32 | 1 | 1,667.375 | 1,667.375 | -1 | |
Mat is digging a hole. Pat asks him how deep the hole will be. Mat responds with a riddle: "I am $90 \mathrm{~cm}$ tall and I have currently dug half the hole. When I finish digging the entire hole, the top of my head will be as far below the ground as it is above the ground now." How deep will the hole be when finishe... | 120 | 0 | 5,142 | -1 | 5,142 | |
Let $a,$ $b,$ $c$ be the roots of $x^3 + px + q = 0.$ Express
\[\begin{vmatrix} 1 + a & 1 & 1 \\ 1 & 1 + b & 1 \\ 1 & 1 & 1 + c \end{vmatrix}\]in terms of $p$ and $q.$ | p - q | 0.625 | 5,144.6875 | 3,668.7 | 7,604.666667 | |
What is the value of the sum
\[
\sum_z \frac{1}{{\left|1 - z\right|}^2} \, ,
\]where $z$ ranges over all 7 solutions (real and nonreal) of the equation $z^7 = -1$? | \frac{49}{4} | 0.0625 | 7,772.625 | 5,616 | 7,916.4 | |
Two distinct positive integers $a$ and $b$ are factors of 48. If $a\cdot b$ is not a factor of 48, what is the smallest possible value of $a\cdot b$? | 18 | 0.5625 | 7,482.125 | 7,117.555556 | 7,950.857143 | |
Given that $\sin \alpha = \frac{2\sqrt{2}}{3}$, $\cos(\alpha + \beta) = -\frac{1}{3}$, and both $\alpha$ and $\beta$ are within the interval $(0, \frac{\pi}{2})$, find the value of $\sin(\alpha - \beta)$. | \frac{10\sqrt{2}}{27} | 0 | 3,510.625 | -1 | 3,510.625 |
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