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 |
|---|---|---|---|---|---|---|
How many ordered triples $(x,y,z)$ of positive integers satisfy $\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600 \text{ and lcm}(y,z)=900$? | 15 | 1. **Understanding the LCM conditions**: We are given three conditions involving the least common multiples (LCMs) of three pairs of variables $(x, y)$, $(x, z)$, and $(y, z)$:
- $\text{lcm}(x,y) = 72$
- $\text{lcm}(x,z) = 600$
- $\text{lcm}(y,z) = 900$
2. **Prime factorization of the LCMs**:
- $72 = 2^3 \... | 0.1875 | 7,025.1875 | 4,952.333333 | 7,503.538462 |
Find $\sec 120^\circ.$ | -2 | 1 | 2,227.6875 | 2,227.6875 | -1 | |
Simplify first, then evaluate: $\frac{1}{{{x^2}+2x+1}}\cdot (1+\frac{3}{x-1})\div \frac{x+2}{{{x^2}-1}$, where $x=2\sqrt{5}-1$. | \frac{\sqrt{5}}{10} | 0 | 3,335.4375 | -1 | 3,335.4375 | |
Let \( x, y, z \) be complex numbers such that
\[
xy + 5y = -25, \\
yz + 5z = -25, \\
zx + 5x = -25.
\]
Find all possible values of \( xyz \). | 125 | 0.6875 | 6,504.875 | 5,738 | 8,192 | |
What is the least integer a greater than $14$ so that the triangle with side lengths $a - 1$ , $a$ , and $a + 1$ has integer area? | 52 | 0.5 | 7,228.0625 | 6,264.125 | 8,192 | |
Evaluate $\lfloor0.999\rfloor+\lceil2.001\rceil$. | 3 | 0.9375 | 2,156.1875 | 2,224.4 | 1,133 | |
Given the set \( S = \left\{ z \mid |z - 7 - 8i| = |z_1^4 + 1 - 2z_1^2| ; z, z_1 \in \mathbb{C}, |z_1| = 1 \right\} \), find the area of the region corresponding to \( S \) in the complex plane. | 16\pi | 1 | 5,685.1875 | 5,685.1875 | -1 | |
In Mathville, the streets are all $30$ feet wide and the blocks they enclose are all squares of side length $500$ feet. Matt runs around the block on the $500$-foot side of the street, while Mike runs on the opposite side of the street. How many more feet than Matt does Mike run for every lap around the block? | 240 | 0.75 | 4,283 | 4,091.583333 | 4,857.25 | |
$(1)$ Given $x \gt 0$, $y \gt 0$, and $2x+3y=6$, find the maximum value of $xy$;<br/>$(2)$ Given $x＜\frac{1}{2}$, find the maximum value of $y=2x+\frac{4}{2x-1}$. | -3 | 0.8125 | 5,708.8125 | 5,340.461538 | 7,305 | |
In square $ABCD$, points $P$ and $Q$ lie on $\overline{AD}$ and $\overline{AB}$ respectively. Segments $\overline{BP}$ and $\overline{CQ}$ intersect at point $R$, with $BR = 8$ and $PR = 9$. If $\triangle BRP$ is a right triangle with $\angle BRP = 90^\circ$, what is the area of the square $ABCD$?
A) 144
B) 169
C) 225
... | 225 | 0 | 8,192 | -1 | 8,192 | |
Some people like to write with larger pencils than others. Ed, for instance, likes to write with the longest pencils he can find. However, the halls of MIT are of limited height $L$ and width $L$. What is the longest pencil Ed can bring through the halls so that he can negotiate a square turn? | 3 L | $3 L$. | 0 | 6,855.4375 | -1 | 6,855.4375 |
A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display? | 23 | 1. **Understanding the Display Format**: The digital watch displays time in a 12-hour format with AM and PM, showing hours and minutes. The hours can range from 01 to 12, and the minutes from 00 to 59.
2. **Maximizing the Hour Digits**:
- The hours are displayed as either 01, 02, ..., 12.
- To find the maximum... | 0.125 | 7,945.4375 | 6,219.5 | 8,192 |
A certain organization consists of five leaders and some number of regular members. Every year, the current leaders are kicked out of the organization. Next, each regular member must find two new people to join as regular members. Finally, five new people are elected from outside the organization to become leaders. In ... | 2435 | 0.5625 | 4,761.4375 | 5,295 | 4,075.428571 | |
Given $x \gt -1$, $y \gt 0$, and $x+2y=1$, find the minimum value of $\frac{1}{x+1}+\frac{1}{y}$. | \frac{3+2\sqrt{2}}{2} | 0 | 5,637.75 | -1 | 5,637.75 | |
Suppose \( x_{1}, x_{2}, \ldots, x_{49} \) are real numbers such that
\[ x_{1}^{2} + 2 x_{2}^{2} + \cdots + 49 x_{49}^{2} = 1. \]
Find the maximum value of \( x_{1} + 2 x_{2} + \cdots + 49 x_{49} \). | 35 | 0.1875 | 6,766.6875 | 3,949.333333 | 7,416.846154 | |
Determine the smallest constant $n$, such that for any positive real numbers $x$, $y$, and $z$,
\[\sqrt{\frac{x}{y + 2z}} + \sqrt{\frac{y}{2x + z}} + \sqrt{\frac{z}{x + 2y}} > n.\] | \sqrt{3} | 0 | 8,192 | -1 | 8,192 | |
A solid cube of side length 4 cm is cut into two pieces by a plane that passed through the midpoints of six edges. Find the surface area of each half cube created. | 69 | 0 | 8,170.375 | -1 | 8,170.375 | |
Compute $\dbinom{1293}{1}$. | 1293 | 1 | 1,412.875 | 1,412.875 | -1 | |
Find
\[
\sum_{n = 1}^\infty \frac{3^n}{1 + 3^n + 3^{n + 1} + 3^{2n + 1}}.
\] | \frac{1}{4} | 0 | 5,266.625 | -1 | 5,266.625 | |
Paula the painter and her two helpers each paint at constant, but different, rates. They always start at 8:00 AM, and all three always take the same amount of time to eat lunch. On Monday the three of them painted 50% of a house, quitting at 4:00 PM. On Tuesday, when Paula wasn't there, the two helpers painted only 24%... | 48 | 1. **Define Variables:**
Let $p$ be the rate at which Paula paints (in house/hours), $h$ be the combined rate of the two helpers (in house/hours), and $L$ be the lunch break duration (in hours).
2. **Set Up Equations:**
From the problem, we can set up the following equations based on the work done each day:
-... | 0.375 | 7,421.125 | 6,578.333333 | 7,926.8 |
Given that the base edge length of a right prism is $1$ and the side edge length is $2$, and all the vertices of the prism lie on a sphere, find the radius of the sphere. | \frac{\sqrt{6}}{2} | 0 | 4,977 | -1 | 4,977 | |
Let $a, b, c, d$ be real numbers such that $\min (20 x+19,19 x+20)=(a x+b)-|c x+d|$ for all real numbers $x$. Find $a b+c d$. | 380 | In general, $\min (p, q)=\frac{p+q}{2}-\left|\frac{p-q}{2}\right|$. Letting $p=20 x+19$ and $q=19 x+20$ gives $a=b=19.5$ and $c=d= \pm 0.5$. Then the answer is $19.5^{2}-0.5^{2}=19 \cdot 20=380$. | 0.75 | 5,768.125 | 4,960.166667 | 8,192 |
How many units are in the sum of the lengths of the two longest altitudes in a triangle with sides $8,$ $15,$ and $17$? | 23 | 0.75 | 2,496.75 | 2,279.25 | 3,149.25 | |
If one side of a triangle is $12$ inches and the opposite angle is $30^{\circ}$, then the diameter of the circumscribed circle is: | 24 | 1. **Identify the Known Values:**
- One side of the triangle (let's call it $a$) is given as $12$ inches.
- The angle opposite to this side ($\angle A$) is $30^\circ$.
2. **Apply the Extended Law of Sines:**
- The Extended Law of Sines states that for any triangle, the diameter $D$ of the circumscribed circle... | 1 | 1,862.25 | 1,862.25 | -1 |
A point $(x,y)$ in the plane is called a lattice point if both $x$ and $y$ are integers. The area of the largest square that contains exactly three lattice points in its interior is closest to | 5.0 | To solve this problem, we need to find the largest square that contains exactly three lattice points in its interior. We will consider the properties of lattice points and the geometry of squares.
1. **Understanding Lattice Points and Squares**:
A lattice point is a point in the plane where both coordinates are int... | 0 | 8,158.5625 | -1 | 8,158.5625 |
The positive integers $A, B$, and $C$ form an arithmetic sequence, while the integers $B, C$, and $D$ form a geometric sequence. If $\frac{C}{B} = \frac{7}{3},$ what is the smallest possible value of $A + B + C + D$? | 76 | 0 | 8,153.125 | -1 | 8,153.125 | |
A numerical sequence is defined by the conditions: \( a_{1} = 1 \), \( a_{n+1} = a_{n} + \left\lfloor \sqrt{a_{n}} \right\rfloor \).
How many perfect squares are there among the first terms of this sequence that do not exceed \( 1{,}000{,}000 \)? | 10 | 0.125 | 8,117.4375 | 8,135 | 8,114.928571 | |
Define a number to be an anti-palindrome if, when written in base 3 as $a_{n} a_{n-1} \ldots a_{0}$, then $a_{i}+a_{n-i}=2$ for any $0 \leq i \leq n$. Find the number of anti-palindromes less than $3^{12}$ such that no two consecutive digits in base 3 are equal. | 126 | Note once the middle digit/pair of digits is determined, it suffices to choose the digits in the left half of the number and ensure no pair of consecutive digits are equal. For a number with an even number of digits, the middle pair is 02 or 20 while for a number with an odd number of digits, the middle digit is 1. We ... | 0 | 8,192 | -1 | 8,192 |
Given that $a = \sin (2015\pi - \frac {\pi}{6})$ and the function $f(x) = \begin{cases} a^{x}, & x > 0 \\ f(-x), & x < 0 \end{cases}$, calculate the value of $f(\log_{2} \frac {1}{6})$. | \frac {1}{6} | 1 | 3,045 | 3,045 | -1 | |
Find all integer values of $a$ so that the polynomial
\[x^3 + 3x^2 + ax + 7 = 0\]has at least one integer root. Enter all possible values of $a,$ separated by commas. | -71, -27, -11, 9 | 0.0625 | 2,463 | 3,948 | 2,364 | |
Given the function $f(x)=|2x-1|$.
(1) Solve the inequality $f(x) < 2$;
(2) If the minimum value of the function $g(x)=f(x)+f(x-1)$ is $a$, and $m+n=a$ $(m > 0,n > 0)$, find the minimum value of $\frac{m^{2}+2}{m}+\frac{n^{2}+1}{n}$. | 2\sqrt{2}-2 | 0 | 6,374.5625 | -1 | 6,374.5625 | |
Given real numbers $x$ and $y$ satisfying $x^{2}+y^{2}-4x-2y-4=0$, find the maximum value of $x-y$. | 1+3\sqrt{2} | 1 | 4,255.8125 | 4,255.8125 | -1 | |
Let $f(x)$ be an even function defined on $\mathbb{R}$, which satisfies $f(x+1) = f(x-1)$ for any $x \in \mathbb{R}$. If $f(x) = 2^{x-1}$ for $x \in [0,1]$, then determine the correctness of the following statements:
(1) 2 is a period of the function $f(x)$.
(2) The function $f(x)$ is increasing on the interval (2, 3)... | \frac{1}{2} | 0 | 5,488.9375 | -1 | 5,488.9375 | |
Given that \( x - \frac{1}{x} = \sqrt{3} \), find \( x^{2048} - \frac{1}{x^{2048}} \). | 277526 | 0 | 8,192 | -1 | 8,192 | |
If I have a $5\times5$ chess board, in how many ways can I place five distinct pawns on the board such that each column and row of the board contains no more than one pawn? | 14400 | 0.9375 | 4,232.25 | 3,968.266667 | 8,192 | |
A team won $40$ of its first $50$ games. How many of the remaining $40$ games must this team win so it will have won exactly $70 \%$ of its games for the season? | 23 | 1. **Identify the total number of games and the desired win percentage**: The team plays a total of $50 + 40 = 90$ games in the season and aims to win 70% of these games.
2. **Convert the percentage to a fraction**: The fraction corresponding to 70% is $\frac{70}{100} = \frac{7}{10}$.
3. **Set up the equation for th... | 1 | 2,447.6875 | 2,447.6875 | -1 |
What is the sum of all integer values of $n$ such that $\frac{20}{2n - 1}$ is an integer? | 2 | 1 | 2,765 | 2,765 | -1 | |
What is the volume of the region in three-dimensional space defined by the inequalities $|x|+|y|+|z|\le1$ and $|x|+|y|+|z-1|\le1$? | \frac{1}{6} | 0.1875 | 8,131.625 | 7,870 | 8,192 | |
A rich emir was admiring a new jewel, a small golden plate in the shape of an equilateral triangle, decorated with diamonds. He noticed that the shadow of the plate forms a right triangle, with the hypotenuse being the true length of each side of the plate.
What is the angle between the plane of the plate and the fla... | \frac{\sqrt{3}}{3} | 0 | 7,212.125 | -1 | 7,212.125 | |
The second and fourth terms of a geometric sequence are 2 and 6. Which of the following is a possible first term? Type the letter of the correct option.
A. $-\sqrt{3}$
B. $-\frac{2\sqrt{3}}{3}$
C. $-\frac{\sqrt{3}}{3}$
D. $\sqrt{3}$
E. $3$ | B | 0.9375 | 912.5 | 929.533333 | 657 | |
Given that the line $2mx+ny-4=0$ passes through the point of intersection of the function $y=\log _{a}(x-1)+2$ where $a>0$ and $a\neq 1$, find the minimum value of $\frac{1}{m}+\frac{4}{n}$. | 3+2\sqrt{2} | 0.125 | 7,705.4375 | 6,467.5 | 7,882.285714 | |
The value of $\sqrt{73}$ is between two positive, consecutive integers. What is the product of these two integers? | 72 | 1 | 1,222.4375 | 1,222.4375 | -1 | |
Given the universal set $U=\{1, 2, 3, 4, 5, 6, 7, 8\}$, a set $A=\{a_1, a_2, a_3, a_4\}$ is formed by selecting any four elements from $U$, and the set of the remaining four elements is denoted as $\complement_U A=\{b_1, b_2, b_3, b_4\}$. If $a_1+a_2+a_3+a_4 < b_1+b_2+b_3+b_4$, then the number of ways to form set $A$ i... | 31 | 0.625 | 6,791.75 | 6,074.3 | 7,987.5 | |
In triangle \( \triangle ABC \), it is known that \( \overrightarrow{AB} \cdot \overrightarrow{AC} + 2 \overrightarrow{BA} \cdot \overrightarrow{BC} = 3 \overrightarrow{CA} \cdot \overrightarrow{CB} \). Find the maximum value of \( \sin C \). | \frac{\sqrt{7}}{3} | 0 | 7,842.3125 | -1 | 7,842.3125 | |
A man buys a house for $20,000 and wants to earn a $6\%$ annual return on his investment. He pays $650 a year in taxes and sets aside $15\%$ of each month's rent for repairs and upkeep. Determine the required monthly rent (in dollars) to meet his financial goals.
A) $165.25$
B) $172.50$
C) $181.38$
D) $190.75$
E) $200.... | 181.38 | 0 | 5,832.6875 | -1 | 5,832.6875 | |
Let $p>2$ be a prime number. $\mathbb{F}_{p}[x]$ is defined as the set of all polynomials in $x$ with coefficients in $\mathbb{F}_{p}$ (the integers modulo $p$ with usual addition and subtraction), so that two polynomials are equal if and only if the coefficients of $x^{k}$ are equal in $\mathbb{F}_{p}$ for each nonneg... | 4 p(p-1) | Answer: $4 p(p-1)$ Solution 1. First, notice that $(\operatorname{deg} f)(\operatorname{deg} g)=p^{2}$ and both polynomials are clearly nonconstant. Therefore there are three possibilities for the ordered pair $(\operatorname{deg} f, \operatorname{deg} g)$, which are $\left(1, p^{2}\right),\left(p^{2}, 1\right)$, and $... | 0 | 8,192 | -1 | 8,192 |
A solid in the shape of a right circular cone is 4 inches tall and its base has a 3-inch radius. The entire surface of the cone, including its base, is painted. A plane parallel to the base of the cone divides the cone into two solids, a smaller cone-shaped solid $C$ and a frustum-shaped solid $F,$ in such a way that t... | 512 | 0.125 | 7,878.9375 | 5,687.5 | 8,192 | |
Find the smallest real constant $\alpha$ such that for all positive integers $n$ and real numbers $0=y_{0}<$ $y_{1}<\cdots<y_{n}$, the following inequality holds: $\alpha \sum_{k=1}^{n} \frac{(k+1)^{3 / 2}}{\sqrt{y_{k}^{2}-y_{k-1}^{2}}} \geq \sum_{k=1}^{n} \frac{k^{2}+3 k+3}{y_{k}}$. | \frac{16 \sqrt{2}}{9} | We first prove the following lemma: Lemma. For positive reals $a, b, c, d$, the inequality $\frac{a^{3 / 2}}{c^{1 / 2}}+\frac{b^{3 / 2}}{d^{1 / 2}} \geq \frac{(a+b)^{3 / 2}}{(c+d)^{1 / 2}}$ holds. Proof. Apply Hölder's inequality in the form $\left(\frac{a^{3 / 2}}{c^{1 / 2}}+\frac{b^{3 / 2}}{d^{1 / 2}}\right)^{2}(c+d)... | 0 | 8,192 | -1 | 8,192 |
The houses on the south side of Crazy Street are numbered in increasing order starting at 1 and using consecutive odd numbers, except that odd numbers that contain the digit 3 are missed out. What is the number of the 20th house on the south side of Crazy Street?
A) 41
B) 49
C) 51
D) 59
E) 61 | 59 | 0.0625 | 5,384.75 | 804 | 5,690.133333 | |
Given the expression \( \left(1-\frac{1}{2^{2}}\right)\left(1-\frac{1}{3^{2}}\right)\ldots\left(1-\frac{1}{12^{2}}\right) \), compute its value. | \frac{13}{24} | 0.8125 | 5,493.125 | 4,870.307692 | 8,192 | |
Let $Q(z)$ and $R(z)$ be the unique polynomials such that $z^{2021}+1=(z^2+z+1)Q(z)+R(z)$ and the degree of $R$ is less than $2.$ What is $R(z)?$ | -z | 1. **Understanding the Problem**: We are given the equation \( z^{2021} + 1 = (z^2 + z + 1)Q(z) + R(z) \), where \( R(z) \) is a polynomial of degree less than 2. We need to find \( R(z) \).
2. **Polynomial Division**: The equation represents a division of \( z^{2021} + 1 \) by \( z^2 + z + 1 \), where \( Q(z) \) is t... | 0.875 | 5,051.375 | 4,602.714286 | 8,192 |
The positive integers $A, B$ and $C$ form an arithmetic sequence while the integers $B, C$ and $D$ form a geometric sequence. If $\frac CB = \frac 53,$ what is the smallest possible value of $A + B + C + D$? | 52 | 1 | 2,600.8125 | 2,600.8125 | -1 | |
Determine the smallest positive integer $ n$ such that there exists positive integers $ a_1,a_2,\cdots,a_n$, that smaller than or equal to $ 15$ and are not necessarily distinct, such that the last four digits of the sum,
\[ a_1!\plus{}a_2!\plus{}\cdots\plus{}a_n!\]
Is $ 2001$. | 3 |
We are tasked with finding the smallest positive integer \( n \) such that there exist positive integers \( a_1, a_2, \ldots, a_n \) where each \( a_i \) is less than or equal to 15, and the last four digits of the sum \( a_1! + a_2! + \cdots + a_n! \) is 2001.
To solve this problem, we need to examine the behavior o... | 0 | 8,192 | -1 | 8,192 |
Points $A,B,C$ and $D$ lie on a line, in that order, with $AB = CD$ and $BC = 12$. Point $E$ is not on the line, and $BE = CE = 10$. The perimeter of $\triangle AED$ is twice the perimeter of $\triangle BEC$. Find $AB$. | 9 | 1. **Setup and Diagram**: Points $A, B, C, D$ are collinear with $AB = CD = x$ and $BC = 12$. Point $E$ is not on the line, and $BE = CE = 10$. We need to find $x$ given that the perimeter of $\triangle AED$ is twice the perimeter of $\triangle BEC$.
2. **Properties of $\triangle BEC$**: Since $BE = CE$, $\triangle BE... | 1 | 2,749.3125 | 2,749.3125 | -1 |
Our school's girls volleyball team has 14 players, including a set of 3 triplets: Alicia, Amanda, and Anna. In how many ways can we choose 6 starters if at most one of the triplets is in the starting lineup? | 1848 | 0.875 | 3,569.1875 | 2,908.785714 | 8,192 | |
In the Cartesian coordinate system $xOy$, the terminal side of angle $\theta$ with $Ox$ as the initial side passes through the point $\left( \frac{3}{5}, \frac{4}{5} \right)$, then $\sin \theta=$ ______, $\tan 2\theta=$ ______. | -\frac{24}{7} | 1 | 2,144.375 | 2,144.375 | -1 | |
Given that in triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\sqrt{3}a\cos C=c\sin A$.
$(1)$ Find the measure of angle $C$.
$(2)$ If $a > 2$ and $b-c=1$, find the minimum perimeter of triangle $\triangle ABC$. | 9 + 6\sqrt{2} | 0.0625 | 7,844.5625 | 8,192 | 7,821.4 | |
Given right triangle $ABC$, with $AB=4, BC=3$, and $CA=5$. Circle $\omega$ passes through $A$ and is tangent to $BC$ at $C$. What is the radius of $\omega$? | \frac{25}{8} | Let $O$ be the center of $\omega$, and let $M$ be the midpoint of $AC$. Since $OA=OC$, $OM \perp AC$. Also, $\angle OCM=\angle BAC$, and so triangles $ABC$ and $CMO$ are similar. Then, $CO/CM=AC/AB$, from which we obtain that the radius of $\omega$ is $CO=\frac{25}{8}$. | 1 | 3,497.6875 | 3,497.6875 | -1 |
What is the area, in square units, of a trapezoid bounded by the lines $y = x$, $y = 15$, $y = 5$ and the line $x = 5$? | 50 | 0.125 | 7,432.125 | 8,192 | 7,323.571429 | |
Determine the value of the expression $\sin 410^{\circ}\sin 550^{\circ}-\sin 680^{\circ}\cos 370^{\circ}$. | \frac{1}{2} | 0.75 | 4,409 | 3,973.5 | 5,715.5 | |
Let $\triangle ABC$ be an isosceles triangle such that $BC = 30$ and $AB = AC.$ We have that $I$ is the incenter of $\triangle ABC,$ and $IC = 18.$ What is the length of the inradius of the triangle? | 3\sqrt{11} | 0.75 | 5,593.1875 | 4,726.916667 | 8,192 | |
Among all proper fractions whose numerator and denominator are two-digit numbers, find the smallest fraction that is greater than \(\frac{4}{9}\). Provide the numerator of this fraction in your answer. | 41 | 0 | 8,043.5 | -1 | 8,043.5 | |
A certain item is always sold with a 30% discount, and the profit margin is 47%. During the shopping festival, the item is sold at the original price, and there is a "buy one get one free" offer. Calculate the profit margin at this time. (Note: Profit margin = (selling price - cost) ÷ cost) | 5\% | 0.25 | 5,629.5625 | 5,322.25 | 5,732 | |
The Greater Eighteen Hockey League has three divisions, with six teams in each division. Each team plays each of the other teams in its own division three times and every team in the other divisions twice. How many league games are scheduled? | 351 | 0.4375 | 6,761.25 | 4,921.714286 | 8,192 | |
Suppose you have three children and 40 pieces of candy. How many ways are there to distribute the candy such that each child gets more than one but fewer than 20 pieces? | 171 | 0.4375 | 6,813.8125 | 5,041.857143 | 8,192 | |
Find the number of integers \( n \) that satisfy
\[ 15 < n^2 < 120. \] | 14 | 0.9375 | 3,577.125 | 3,565.133333 | 3,757 | |
Given that $\cos \left(\alpha+ \frac{\pi}{6}\right)= \frac{4}{5}$, find the value of $\sin \left(2\alpha+ \frac{\pi}{3}\right)$. | \frac{24}{25} | 0.3125 | 7,216.75 | 5,071.2 | 8,192 | |
In the Nanjing area, the weather in July and August is relatively hot. Xiaohua collected the highest temperature for ten consecutive days, obtaining the following set of data in sequence: 34, 35, 36, 34, 36, 37, 37, 36, 37, 37 (unit: ℃). The mode of this set of data is ▲, and the median is ▲. | 36 | 0.375 | 670 | 645.5 | 684.7 | |
Consider the polynomial \( p(x) = x^n + n x^{n-1} + a_2 x^{n-2} + \cdots + a_n \) having all real roots. If \( r_1^{16} + r_2^{16} + \cdots + r_n^{16} = n \), where the \( r_j \) are the roots of \( p(x) \), find all such roots. | -1 | 0.75 | 5,728 | 4,906.666667 | 8,192 | |
In triangle \( ABC \), angle \( B \) is \( 80^\circ \). On side \( BC \), point \( D \) is marked such that \( AB = AD = CD \). On side \( AB \), point \( F \) is marked such that \( AF = BD \). On segment \( AC \), point \( E \) is marked such that \( AB = AE \). Find angle \( AEF \). | 20 | 0.125 | 8,182.6875 | 8,192 | 8,181.357143 | |
Let \( a, b \) and \( c \) be positive integers such that \( a^{2} = 2b^{3} = 3c^{5} \). What is the minimum possible number of factors of \( abc \) (including 1 and \( abc \))? | 77 | 0.125 | 7,706.625 | 5,805.5 | 7,978.214286 | |
Given the function f(x) = 2sin(x - π/6)sin(x + π/3), x ∈ R.
(1) Find the smallest positive period of the function f(x) and the center of symmetry of its graph.
(2) In △ABC, if A = π/4, and acute angle C satisfies f(C/2 + π/6) = 1/2, find the value of BC/AB. | \sqrt{2} | 0.75 | 5,433.625 | 4,514.166667 | 8,192 | |
Compute $\sqrt[4]{5508^{3}+5625^{3}+5742^{3}}$, given that it is an integer. | 855 | Let $a=5625=75^{2}$ and $b=117$. Then we have $5508^{3}+5265^{3}+5742^{3}=(a-b)^{3}+a^{3}+(a+b)^{3}=3a^{3}+6ab^{2}=3a(a^{2}+2b^{2})$. We have $3a=3^{3} \cdot 5^{4}$, so $a^{2}+2b^{2}=3^{4} \cdot(625^{2}+2 \cdot 19^{2})$ should be 3 times a fourth power. This means $625^{2}+2 \cdot 19^{2}=3x^{4}$ for some integer $x$. B... | 0.1875 | 7,864.1875 | 6,443.666667 | 8,192 |
How many solutions in natural numbers does the equation $\left\lfloor \frac{x}{10} \right\rfloor = \left\lfloor \frac{x}{11} \right\rfloor + 1$ have? | 110 | 0 | 8,170.6875 | -1 | 8,170.6875 | |
Let \(O\) be the origin. There exists a scalar \(k'\) so that for any points \(A\), \(B\), \(C\), and \(D\) if
\[4 \overrightarrow{OA} - 3 \overrightarrow{OB} + 6 \overrightarrow{OC} + k' \overrightarrow{OD} = \mathbf{0},\]
then the four points \(A\), \(B\), \(C\), and \(D\) are coplanar. Find \(k'\). | -7 | 0.5625 | 5,917.75 | 4,587.333333 | 7,628.285714 | |
A sequence $\left\{a_{n}\right\}_{n \geq 1}$ of positive reals is defined by the rule $a_{n+1} a_{n-1}^{5}=a_{n}^{4} a_{n-2}^{2}$ for integers $n>2$ together with the initial values $a_{1}=8$ and $a_{2}=64$ and $a_{3}=1024$. Compute $$\sqrt{a_{1}+\sqrt{a_{2}+\sqrt{a_{3}+\cdots}}}$$ | 3\sqrt{2} | Taking the base-2 $\log$ of the sequence $\left\{a_{n}\right\}$ converts the multiplicative rule to a more familiar additive rule: $\log _{2}\left(a_{n+1}\right)-4 \log _{2}\left(a_{n}\right)+5 \log _{2}\left(a_{n-1}\right)-2 \log _{2}\left(a_{n-2}\right)=0$. The characteristic equation is $0=x^{3}-4 x^{2}+5 x-2=(x-1)^... | 0 | 8,192 | -1 | 8,192 |
The sequence \(a_{0}, a_{1}, a_{2}, \cdots, a_{n}\) satisfies \(a_{0}=\sqrt{3}\), \(a_{n+1}=\left\lfloor a_{n}\right\rfloor + \frac{1}{\left\{a_{n}\right\}}\), where \(\left\lfloor a_{n}\right\rfloor\) and \(\left\{a_{n}\right\}\) represent the integer part and fractional part of \(a_{n}\), respectively. Find \(a_{2016... | 3024 + \sqrt{3} | 0 | 8,036.4375 | -1 | 8,036.4375 | |
For any positive integer $x_{}$, let $S(x)$ be the sum of the digits of $x_{}$, and let $T(x)$ be $|S(x+2)-S(x)|.$ For example, $T(199)=|S(201)-S(199)|=|3-19|=16.$ How many values of $T(x)$ do not exceed 1999? | 223 | For most values of $x$, $T(x)$ will equal $2$. For those that don't, the difference must be bumping the number up a ten, a hundred, etc. If we take $T(a999)$ as an example, \[|(a + 1) + 0 + 0 + 1 - (a + 9 + 9 + 9)| = |2 - 9(3)|\] And in general, the values of $T(x)$ will then be in the form of $|2 - 9n| = 9n - 2$. From... | 0.0625 | 7,984.25 | 4,868 | 8,192 |
Two congruent right circular cones each with base radius $3$ and height $8$ have the axes of symmetry that intersect at right angles at a point in the interior of the cones a distance $3$ from the base of each cone. A sphere with radius $r$ lies withing both cones. The maximum possible value of $r^2$ is $\frac{m}{n}$, ... | 298 | 0 | 8,189.625 | -1 | 8,189.625 | |
What is the minimum number of equilateral triangles, of side length 1 unit, needed to cover an equilateral triangle of side length 15 units? | 225 | 1 | 5,363.9375 | 5,363.9375 | -1 | |
A ball was floating in a lake when the lake froze. The ball was removed (without breaking the ice), leaving a hole $24$ cm across as the top and $8$ cm deep. What was the radius of the ball (in centimeters)?
$\textbf{(A)}\ 8 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 8\sqrt{3} \qquad \textbf{(... | 13 | 0 | 1,964.75 | -1 | 1,964.75 | |
In Mr. Lee's classroom, there are six more boys than girls among a total of 36 students. What is the ratio of the number of boys to the number of girls? | \frac{7}{5} | 0.9375 | 559.75 | 565.866667 | 468 | |
How many perfect squares are two-digit and divisible by $3?$ | 2 | 1 | 2,469.3125 | 2,469.3125 | -1 | |
Both roots of the quadratic equation $x^2 - 63 x + k = 0$ are prime numbers. How many possible values of $k$ are there? | 1 | 1 | 3,107.1875 | 3,107.1875 | -1 | |
What is the largest number of squares with side length 2 that can be arranged, without overlapping, inside a square with side length 8? | 16 | By arranging 4 rows of 4 squares of side length 2, a square of side length 8 can be formed. Thus, $4 \cdot 4=16$ squares can be arranged in this way. Since these smaller squares completely cover the larger square, it is impossible to use more $2 \times 2$ squares, so 16 is the largest possible number. | 1 | 5,085.3125 | 5,085.3125 | -1 |
The teacher wrote a positive number $x$ on the board and asked Kolya, Petya, and Vasya to raise this number to the 3rd, 4th, and 12th powers, respectively. It turned out that Kolya's number had at least 9 digits before the decimal point, and Petya's number had no more than 11 digits before the decimal point. How many d... | 33 | 0.25 | 7,308 | 5,981 | 7,750.333333 | |
A high school math team received 5 college students for a teaching internship, who are about to graduate. They need to be assigned to three freshman classes: 1, 2, and 3, with at least one and at most two interns per class. Calculate the number of different allocation schemes. | 90 | 0.25 | 7,471.75 | 5,311 | 8,192 | |
There are many ways in which the list \(0,1,2,3,4,5,6,7,8,9\) can be separated into groups. For example, this list could be separated into the four groups \(\{0,3,4,8\}\), \(\{1,2,7\}\), \{6\}, and \{5,9\}. The sum of the numbers in each of these four groups is \(15\), \(10\), \(6\), and \(14\), respectively. In how ma... | 32 | 0 | 8,192 | -1 | 8,192 | |
Find
\[
\cos \left( 8 \arccos \frac{1}{5} \right).
\] | \frac{-15647}{390625} | 0 | 7,863.625 | -1 | 7,863.625 | |
Miyuki texted a six-digit integer to Greer. Two of the digits of the six-digit integer were 3s. Unfortunately, the two 3s that Miyuki texted did not appear and Greer instead received the four-digit integer 2022. How many possible six-digit integers could Miyuki have texted? | 15 | The six-digit integer that Miyuki sent included the digits 2022 in that order along with two 3s. If the two 3s were consecutive digits, there are 5 possible integers: 332022, 233022, 203322, 202332, 202233. If the two 3s are not consecutive digits, there are 10 possible pairs of locations for the 3s: 1st/3rd, 1st/4th, ... | 0.125 | 7,557.0625 | 6,413.5 | 7,720.428571 |
Divide the product of the first six positive composite integers by the product of the next six composite integers. Express your answer as a common fraction. | \frac{1}{49} | 0 | 7,233.0625 | -1 | 7,233.0625 | |
Andy the Ant lives on a coordinate plane and is currently at $(-20, 20)$ facing east (that is, in the positive $x$-direction). Andy moves $1$ unit and then turns $90^{\circ}$ left. From there, Andy moves $2$ units (north) and then turns $90^{\circ}$ left. He then moves $3$ units (west) and again turns $90^{\circ}$ left... | $(-1030, -990)$ | To solve this problem, we need to understand the pattern of Andy's movements and how his position changes with each move. Andy starts at $(-20, 20)$ and moves in a spiral pattern, increasing the distance he moves by $1$ unit after each turn, and always turning left.
1. **Initial Moves and Pattern Recognition:**
- *... | 0 | 7,377.4375 | -1 | 7,377.4375 |
A sequence $(c_n)$ is defined as follows: $c_1 = 1$, $c_2 = \frac{1}{3}$, and
\[c_n = \frac{2 - c_{n-1}}{3c_{n-2}}\] for all $n \ge 3$. Find $c_{100}$. | \frac{1}{3} | 0 | 8,190.5 | -1 | 8,190.5 | |
How many multiples of 10 are between 11 and 103? | 9 | 0.875 | 1,508.5625 | 1,652.357143 | 502 | |
Consider a square ABCD with side length 4 units. Points P and R are the midpoints of sides AB and CD, respectively. Points Q is located at the midpoint of side BC, and point S is located at the midpoint of side AD. Calculate the fraction of the square's total area that is shaded when triangles APQ and CSR are shaded.
... | \frac{1}{4} | 0.5 | 5,561.0625 | 5,576 | 5,546.125 | |
From a deck of 32 cards which includes three colors (red, yellow, and blue) with each color having 10 cards numbered from $1$ to $10$, plus an additional two cards (a small joker and a big joker) both numbered $0$, a subset of cards is selected. The score for each card is calculated as $2^{k}$, where $k$ is the number ... | 1006009 | 0 | 8,134.5625 | -1 | 8,134.5625 | |
Find $\log _{n}\left(\frac{1}{2}\right) \log _{n-1}\left(\frac{1}{3}\right) \cdots \log _{2}\left(\frac{1}{n}\right)$ in terms of $n$. | (-1)^{n-1} | Using $\log \frac{1}{x}=-\log x$ and $\log _{b} a=\frac{\log a}{\log b}$, we get that the product equals $\frac{(-\log 2)(-\log 3) \cdots(-\log n)}{\log n \cdots \log 3 \log 2}=(-1)^{n-1}$. | 0.625 | 5,473.375 | 4,264 | 7,489 |
Let $f(x)=c x(x-1)$, where $c$ is a positive real number. We use $f^{n}(x)$ to denote the polynomial obtained by composing $f$ with itself $n$ times. For every positive integer $n$, all the roots of $f^{n}(x)$ are real. What is the smallest possible value of $c$? | 2 | We first prove that all roots of $f^{n}(x)$ are greater than or equal to $-\frac{c}{4}$ and less than or equal to $1+\frac{c}{4}$. Suppose that $r$ is a root of $f^{n}(x)$. If $r=-\frac{c}{4}, f^{-1}(r)=\left\{\frac{1}{2}\right\}$ and $-\frac{c}{4}<\frac{1}{2}<1+\frac{c}{4}$ since $c$ is positive. Suppose $r \neq-\frac... | 0 | 8,090.375 | -1 | 8,090.375 |
Given the ellipse $\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1\ (a > b > 0)$, with $F\_{1}$ as the left focus, $A$ as the right vertex, and $B\_{1}$, $B\_{2}$ as the upper and lower vertices respectively. If the four points $F\_{1}$, $A$, $B\_{1}$, and $B\_{2}$ lie on the same circle, find the eccentricity of this elli... | \dfrac{\sqrt{5}-1}{2} | 0 | 5,390.6875 | -1 | 5,390.6875 | |
Calculate the total area of a pentagon with sides of lengths 18, 25, 30, 28, and 25 units, assuming it can be divided into a right triangle and a trapezoid. | 995 | 0 | 8,161.125 | -1 | 8,161.125 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.