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 a square piece of paper with side length $s$ folded in half diagonally, then cut along a line perpendicular to the fold from the midpoint of the hypotenuse to the opposite side, forming a large rectangle and two smaller, identical triangles, find the ratio of the perimeter of one of the small triangles to the per... | \frac{2}{3} | 0 | 8,107.625 | -1 | 8,107.625 | |
Suppose $F$ is a polygon with lattice vertices and sides parralell to x-axis and y-axis.Suppose $S(F),P(F)$ are area and perimeter of $F$ .
Find the smallest k that:
$S(F) \leq k.P(F)^2$ | 1/16 | 0.125 | 8,005.75 | 7,861.5 | 8,026.357143 | |
A regular octagon $ABCDEFGH$ has sides of length two. Find the area of $\bigtriangleup ADG$. Express your answer in simplest radical form. | 4+3\sqrt{2} | 0.25 | 8,089.5 | 7,782 | 8,192 | |
At an international table tennis championship, 200 participants arrived. The tournament follows a single-elimination format, meaning that in each match two players compete, the loser is eliminated from the championship, and the winner remains. Find the maximum possible number of participants who have won at least three... | 66 | 0.0625 | 8,103.0625 | 6,769 | 8,192 | |
What is the largest number, all of whose digits are either 5, 3, or 1, and whose digits add up to $15$? | 555 | 0.0625 | 7,916.125 | 5,648 | 8,067.333333 | |
A typesetter scattered part of a set - a set of a five-digit number that is a perfect square, written with the digits $1, 2, 5, 5,$ and $6$. Find all such five-digit numbers. | 15625 | 0.0625 | 8,118.9375 | 7,023 | 8,192 | |
Three students solved the same problem. The first one said: "The answer is an irrational number. It represents the area of an equilateral triangle with a side length of 2 meters." The second one said: "The answer is divisible by 4 (without remainder). It represents the radius of a circle whose circumference is 2 meters... | \frac{1}{\pi} | 0.0625 | 7,869.75 | 4,248 | 8,111.2 | |
Given that there are 10 light bulbs in a box, with 8 of good quality and 2 defective, calculate P(ξ=4), where ξ is the number of draws to draw 2 good quality bulbs, drawing one at a time without replacement. | \frac{1}{15} | 0.1875 | 7,586.875 | 6,416.666667 | 7,856.923077 | |
Simplify $\frac{{x}^{2}-4x+4}{{x}^{2}-1}÷\frac{{x}^{2}-2x}{x+1}+\frac{1}{x-1}$ first, then choose a suitable integer from $-2\leqslant x\leqslant 2$ as the value of $x$ to evaluate. | -1 | 0.875 | 4,449.0625 | 3,950.714286 | 7,937.5 | |
What is the sum of the six positive integer factors of 30, each multiplied by 2? | 144 | 0.25 | 5,255.5625 | 4,675 | 5,449.083333 | |
Roger initially has 20 socks in a drawer, each of which is either white or black. He chooses a sock uniformly at random from the drawer and throws it away. He repeats this action until there are equal numbers of white and black socks remaining. Suppose that the probability he stops before all socks are gone is $p$. If ... | 20738 | Let $b_{i}$ and $w_{i}$ be the number of black and white socks left after $i$ socks have been thrown out. In particular, $b_{0}+w_{0}=20$. The key observation is that the ratio $r_{i}=\frac{b_{i}}{b_{i}+w_{i}}$ is a martingale (the expected value of $r_{i+1}$ given $r_{i}$ is just $r_{i}$). Suppose WLOG that $b_{0}<w_{... | 0 | 8,192 | -1 | 8,192 |
A positive integer $n$ is called*bad*if it cannot be expressed as the product of two distinct positive integers greater than $1$ . Find the number of bad positive integers less than $100. $ *Proposed by Michael Ren* | 30 | 0.3125 | 6,696.0625 | 5,839.2 | 7,085.545455 | |
The graph of $y = \frac{p(x)}{q(x)}$ is shown below, where $p(x)$ and $q(x)$ are quadratic. (Assume that the grid lines are at integers.)
[asy]
unitsize(0.6 cm);
real func (real x) {
return (2*(x - 1)/(x + 2));
}
int i;
for (i = -8; i <= 8; ++i) {
draw((i,-8)--(i,8),gray(0.7));
draw((-8,i)--(8,i),gray(0.7));... | \frac{4}{5} | 0.3125 | 7,285.8125 | 5,424.6 | 8,131.818182 | |
Xiao Kang exercises every day by doing several sets of push-ups, 8 per set, and several sets of pull-ups, 5 per set. On the first day, he did a total of 41 reps (combining both exercises), and he increases the number by 1 each day until the 100th day. If the number of sets of push-ups and pull-ups he does each day are ... | 106 | 0.6875 | 8,192 | 8,192 | 8,192 | |
Find the gradient of the function \( z = x^2 - xy + y^3 \) at the point \( A(1, -1) \) and the derivative in the direction of the vector \( \bar{a} = 3\bar{i} - 4\bar{j} \). | \frac{1}{5} | 1 | 2,211.4375 | 2,211.4375 | -1 | |
In triangle \(ABC\), the angle bisector of \(\angle ABC\) intersects side \(AC\) at point \(K\). It is given that \(BC = 2\), \(KC = 1\), and \(BK = \frac{3\sqrt{2}}{2}\). Find the area of triangle \(ABC\). | \frac{15 \sqrt{7}}{16} | 0 | 6,722.5625 | -1 | 6,722.5625 | |
Find all ordered pairs $(a,b)$ of positive integers for which the numbers $\dfrac{a^3b-1}{a+1}$ and $\dfrac{b^3a+1}{b-1}$ are both positive integers | \[
\{(2,2), (1,3), (3,3)\}
\] | Adding $1$ to both the given numbers we get:
$\dfrac{a^3b-1}{a+1} + 1$ is also a positive integer so we have:
$\dfrac{a^3b+a}{a+1}$ = $\dfrac{a(a^2b+1)}{a+1}$ is a positive integer
$\implies (a+1) \mid (a^2b+1)$ $\implies (a+1) \mid (((a+1) - 1)^2b+1)$ $\implies (a+1) \mid (b+1)$
Similarly,
$\dfrac{b^3a+1}{b-1} + 1$... | 0 | 7,795.75 | -1 | 7,795.75 |
Let \( n \) be the positive integer such that
\[ \frac{1}{9 \sqrt{11} + 11 \sqrt{9}} + \frac{1}{11 \sqrt{13} + 13 \sqrt{11}} + \frac{1}{13 \sqrt{15} + 15 \sqrt{13}} + \cdots + \frac{1}{n \sqrt{n+2} + (n+2) \sqrt{n}} = \frac{1}{9} . \]
Find the value of \( n \). | 79 | 0.375 | 6,037.5625 | 4,641 | 6,875.5 | |
Let $a^2=\frac{16}{44}$ and $b^2=\frac{(2+\sqrt{5})^2}{11}$, where $a$ is a negative real number and $b$ is a positive real number. If $(a+b)^3$ can be expressed in the simplified form $\frac{x\sqrt{y}}{z}$ where $x$, $y$, and $z$ are positive integers, what is the value of the sum $x+y+z$? | 181 | 1 | 4,493.375 | 4,493.375 | -1 | |
The hyperbola \[-x^2+2y^2-10x-16y+1=0\]has two foci. Find the coordinates of either of them. (Enter your answer as an ordered pair. Enter only one of the foci, not both.) | (-5, 1) | 0 | 3,595.8125 | -1 | 3,595.8125 | |
The circle inscribed in a right trapezoid divides its larger lateral side into segments of lengths 1 and 4. Find the area of the trapezoid. | 18 | 0 | 8,114.75 | -1 | 8,114.75 | |
Using three rectangular pieces of paper (A, C, D) and one square piece of paper (B), an area of 480 square centimeters can be assembled into a large rectangle. It is known that the areas of B, C, and D are all 3 times the area of A. Find the total perimeter of the four pieces of paper A, B, C, and D in centimeters. | 184 | 0 | 8,113.9375 | -1 | 8,113.9375 | |
In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$, respectively.
(1) If $A$, $B$, $C$ form an arithmetic sequence, find the range of $\cos A + \cos C$.
(2) If $a$, $b$, $c$ form a geometric sequence, and $\cos B = \frac{4}{5}$, find the value of $\frac{1}{\tan A} + \frac{1}{\tan C}$. | \frac{5}{3} | 0.5625 | 6,563.0625 | 5,296.111111 | 8,192 | |
A printer prints text pages at a rate of 17 pages per minute and graphic pages at a rate of 10 pages per minute. If a document consists of 250 text pages and 90 graphic pages, how many minutes will it take to print the entire document? Express your answer to the nearest whole number. | 24 | 0.9375 | 2,401.875 | 2,015.866667 | 8,192 | |
A contractor estimated that one of his two bricklayers would take $9$ hours to build a certain wall and the other $10$ hours.
However, he knew from experience that when they worked together, their combined output fell by $10$ bricks per hour.
Being in a hurry, he put both men on the job and found that it took exactly... | 900 | 1. **Define the rates of the bricklayers**: Let the total number of bricks in the wall be $x$. The first bricklayer can complete the wall in $9$ hours, so his rate is $\frac{x}{9}$ bricks per hour. The second bricklayer can complete the wall in $10$ hours, so his rate is $\frac{x}{10}$ bricks per hour.
2. **Combined r... | 1 | 1,680.375 | 1,680.375 | -1 |
How many perfect squares are divisors of the product $1! \cdot 2! \cdot 3! \cdot \hdots \cdot 9!$? | 672 | 1. **Compute the Prime Factorization of the Product of Factorials:**
The product in question is $1! \cdot 2! \cdot 3! \cdot \ldots \cdot 9!$. To find the number of perfect square divisors, we first need the prime factorization of this product. We calculate the exponent of each prime in the product:
- **Prime ... | 0.8125 | 4,944.3125 | 4,829.615385 | 5,441.333333 |
How many solutions does the equation
\[ x^{2}+y^{2}+2xy-1988x-1988y=1989 \]
have in the set of positive integers? | 1988 | 0.9375 | 4,865.1875 | 4,643.4 | 8,192 | |
Let complex numbers $\omega_{1}=-\frac{1}{2}+\frac{\sqrt{3}}{2}i$ and $\omega_{2}=\cos\frac{\pi}{12}+\sin\frac{\pi}{12}i$. If $z=\omega_{1}\cdot\omega_{2}$, find the imaginary part of the complex number $z$. | \frac { \sqrt {2}}{2} | 0 | 5,194.3125 | -1 | 5,194.3125 | |
Seven cookies of radius 1 inch are cut from a circle of cookie dough, as shown. Neighboring cookies are tangent, and all except the center cookie are tangent to the edge of the dough. The leftover scrap is reshaped to form another cookie of the same thickness. What is the radius in inches of the scrap cookie?
[asy]
dra... | \sqrt{2} | 1. **Calculate the area of the large cookie dough**:
The large cookie dough is a circle with a radius of $3$ inches. The area of a circle is given by the formula $A = \pi r^2$. Therefore, the area of the large cookie dough is:
\[
A_{\text{large}} = \pi \times 3^2 = 9\pi \text{ square inches}.
\]
2. **Calc... | 0.9375 | 3,244.0625 | 2,914.2 | 8,192 |
A regular octagon is inscribed in a circle and another regular octagon is circumscribed about the same circle. What is the ratio of the area of the larger octagon to the area of the smaller octagon? Express your answer as a common fraction. | 12 + 8\sqrt{2} | 0 | 7,952.8125 | -1 | 7,952.8125 | |
Let $P$ be a regular $2006$-gon. A diagonal is called [i]good[/i] if its endpoints divide the boundary of $P$ into two parts, each composed of an odd number of sides of $P$. The sides of $P$ are also called [i]good[/i].
Suppose $P$ has been dissected into triangles by $2003$ diagonals, no two of which have a common poi... | 1003 |
Let \( P \) be a regular 2006-gon. We are tasked with finding the maximum number of isosceles triangles that can be formed by dissecting \( P \) using 2003 diagonals such that each triangle has two good sides, where a side is called good if it divides the boundary of \( P \) into two parts, each having an odd number o... | 0.0625 | 8,088.0625 | 6,529 | 8,192 |
The set containing three real numbers can be represented as $\{a, \frac{b}{a}, 1\}$, and also as $\{a^2, a+b, 0\}$. Find the value of $a^{2009} + b^{2009}$. | -1 | 0.75 | 4,878.1875 | 3,912.75 | 7,774.5 | |
Vasya has 9 different books by Arkady and Boris Strugatsky, each containing a single work by the authors. Vasya wants to arrange these books on a shelf in such a way that:
(a) The novels "Beetle in the Anthill" and "Waves Extinguish the Wind" are next to each other (in any order).
(b) The stories "Restlessness" and "A ... | 4 \cdot 7! | 0 | 5,570.5625 | -1 | 5,570.5625 | |
How many ordered pairs of positive integers $(M,N)$ satisfy the equation $\frac{M}{6}=\frac{6}{N}?$ | 9 | 1. **Start with the given equation and cross-multiply:**
\[
\frac{M}{6} = \frac{6}{N} \implies M \cdot N = 36
\]
2. **Factorize 36 into pairs of positive integers:**
\[
36 = 1 \cdot 36, \quad 2 \cdot 18, \quad 3 \cdot 12, \quad 4 \cdot 9, \quad 6 \cdot 6
\]
Each factorization represents a potentia... | 1 | 1,919.375 | 1,919.375 | -1 |
What is the greatest integer less than $-\frac{15}4$? | -4 | 0.75 | 1,828.75 | 1,593.583333 | 2,534.25 | |
Given a moving line $l$ that tangentially touches the circle $O: x^{2}+y^{2}=1$ and intersects the ellipse $\frac{x^{2}}{9}+y^{2}=1$ at two distinct points $A$ and $B$, find the maximum distance from the origin to the perpendicular bisector of line segment $AB$. | \frac{4}{3} | 0.375 | 7,654.1875 | 6,898.833333 | 8,107.4 | |
On the sides \(BC\) and \(CD\) of the square \(ABCD\), points \(M\) and \(K\) are marked respectively such that \(\angle BAM = 30^{\circ}\) and \(\angle CKM = 30^{\circ}\). Find the angle \(AKD\). | 75 | 0.75 | 6,787.3125 | 6,319.083333 | 8,192 | |
In a square with a side length of 12 cm, the midpoints of its adjacent sides are connected to each other and to the opposite side of the square. Find the radius of the circle inscribed in the resulting triangle. | 2\sqrt{5} - \sqrt{2} | 0 | 6,798.875 | -1 | 6,798.875 | |
A standard deck of 52 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 4 suits ($\spadesuit$, $\heartsuit$, $\diamondsuit$, and $\clubsuit$), such that there is exactly one card for any given rank and suit. Two of the suits ($\spadesuit$ and $\clubsuit$) are black and the other two suits ($\... | \dfrac{1}{52} | 1 | 1,092.4375 | 1,092.4375 | -1 | |
Two right circular cones with vertices facing down as shown in the figure below contain the same amount of liquid. The radii of the tops of the liquid surfaces are $3$ cm and $6$ cm. Into each cone is dropped a spherical marble of radius $1$ cm, which sinks to the bottom and is completely submerged without spilling any... | 4:1 |
#### Initial Scenario
Let the heights of the narrow cone and the wide cone be \( h_1 \) and \( h_2 \), respectively. The volumes of the cones before the marbles are dropped are given by:
- Narrow Cone: \( V_1 = \frac{1}{3} \pi r_1^2 h_1 = \frac{1}{3} \pi (3)^2 h_1 = 3\pi h_1 \)
- Wide Cone: \( V_2 = \frac{1}{3} \pi r_... | 0.0625 | 6,612.25 | 5,084 | 6,714.133333 |
Let \( p(x) \) be a monic quartic polynomial such that \( p(1) = 1, p(2) = 9, p(3) = 28, \) and \( p(4) = 65. \) Find \( p(5) \). | 126 | 0 | 7,166.625 | -1 | 7,166.625 | |
Six male middle school students from a school participated in a pull-up physical fitness test and scored respectively: $8$, $5$, $2$, $5$, $6$, $4$. Calculate the variance of this dataset. | \frac{10}{3} | 0.75 | 2,372.6875 | 2,972.25 | 574 | |
How many distinct triangles can be formed by selecting three vertices from a set of points arranged in a 2×4 grid (2 rows and 4 columns)? | 48 | 0.3125 | 7,679 | 6,934.4 | 8,017.454545 | |
Let $X=\{2^m3^n|0 \le m, \ n \le 9 \}$ . How many quadratics are there of the form $ax^2+2bx+c$ , with equal roots, and such that $a,b,c$ are distinct elements of $X$ ? | 9900 | 0.0625 | 8,153.5625 | 7,766 | 8,179.4 | |
A volunteer organizes a spring sports event and wants to form a vibrant and well-trained volunteer team. They plan to randomly select 3 people from 4 male volunteers and 3 female volunteers to serve as the team leader. The probability of having at least one female volunteer as the team leader is ____; given the conditi... | \frac{2}{17} | 0.6875 | 3,442.9375 | 2,311.545455 | 5,932 | |
The difference between a two-digit number and the number obtained by reversing its digits is $5$ times the sum of the digits of either number. What is the sum of the two digit number and its reverse? | 99 | 1. **Define the two-digit number**: Let the two-digit number be represented as $10a + b$ where $a$ is the tens digit and $b$ is the units digit. Here, $a$ and $b$ are integers such that $1 \leq a \leq 9$ and $0 \leq b \leq 9$.
2. **Expression for the reversed number**: The number obtained by reversing the digits is $1... | 1 | 1,855.6875 | 1,855.6875 | -1 |
The image of the point with coordinates $(1,1)$ under the reflection across the line $y=mx+b$ is the point with coordinates $(9,5)$. Find $m+b$. | 11 | 0.9375 | 2,454.8125 | 2,072.333333 | 8,192 | |
Simplify
\[\cos \frac{2 \pi}{13} + \cos \frac{6 \pi}{13} + \cos \frac{8 \pi}{13}.\] | \frac{\sqrt{13} - 1}{4} | 0 | 8,192 | -1 | 8,192 | |
For a polynomial $p(x),$ define its munificence as the maximum value of $|p(x)|$ on the interval $-1 \le x \le 1.$ For example, the munificence of the polynomial $p(x) = -x^2 + 3x - 17$ is 21, since the maximum value of $|-x^2 + 3x - 17|$ for $-1 \le x \le 1$ is 21, occurring at $x = -1.$
Find the smallest possible m... | \frac{1}{2} | 0.625 | 7,600.8125 | 7,246.1 | 8,192 | |
The parabolas defined by the equations $y=2x^2-4x+4$ and $y=-x^2-2x+4$ intersect at points $(a,b)$ and $(c,d)$, where $c\ge a$. What is $c-a$? Express your answer as a common fraction. | \frac{2}{3} | 1 | 2,364 | 2,364 | -1 | |
In a number matrix as shown, the three numbers in each row are in arithmetic progression, and the three numbers in each column are also in arithmetic progression. Given that \( a_{22} = 2 \), find the sum of all 9 numbers in the matrix. | 18 | 0.4375 | 7,429.75 | 6,449.714286 | 8,192 | |
From the following infinite list of numbers, how many are integers? $$\sqrt{4096},\sqrt[3]{4096},\sqrt[4]{4096},\sqrt[5]{4096},\sqrt[6]{4096},\ldots$$ | 5 | 1 | 3,178.9375 | 3,178.9375 | -1 | |
A parallelogram is generated by the vectors $\begin{pmatrix} 2 \\ 1\\ 1 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ -1 \\ - 1 \end{pmatrix}.$
[asy]
unitsize(0.4 cm);
pair A, B, C, D;
A = (0,0);
B = (7,2);
C = (1,3);
D = B + C;
draw(A--B,Arrow(6));
draw(A--C,Arrow(6));
draw(B--D--C);
draw(B--C,dashed);
draw(A--D,dashed... | \frac{1}{3} | 0.9375 | 2,724.75 | 2,710.8 | 2,934 | |
Four prime numbers are randomly selected without replacement from the first ten prime numbers. What is the probability that the sum of the four selected numbers is odd? Express your answer as a common fraction. | \frac{2}{5} | 0.9375 | 4,083 | 3,809.066667 | 8,192 | |
Find the integer $n$, $0 \le n \le 5$, such that \[n \equiv -3736 \pmod{6}.\] | 2 | 1 | 3,081.25 | 3,081.25 | -1 | |
In the plane Cartesian coordinate system \( xO y \), the circle \( \Omega \) and the parabola \( \Gamma: y^{2} = 4x \) share exactly one common point, and the circle \( \Omega \) is tangent to the x-axis at the focus \( F \) of \( \Gamma \). Find the radius of the circle \( \Omega \). | \frac{4 \sqrt{3}}{9} | 0 | 7,285.0625 | -1 | 7,285.0625 | |
Let (b_1, b_2, ... b_7) be a list of the first 7 odd positive integers such that for each 2 ≤ i ≤ 7, either b_i + 2 or b_i - 2 (or both) must appear before b_i in the list. How many such lists are there? | 64 | 0 | 8,192 | -1 | 8,192 | |
Quadrilateral $ABCD$ is a rhombus with perimeter $52$ meters. The length of diagonal $\overline{AC}$ is $24$ meters. What is the area in square meters of rhombus $ABCD$? | 120 | 1. **Identify the length of each side of the rhombus**: Given that the perimeter of rhombus $ABCD$ is $52$ meters, and knowing that a rhombus has four sides of equal length, we calculate the length of each side as follows:
\[
\text{Side length} = \frac{\text{Perimeter}}{4} = \frac{52}{4} = 13 \text{ meters}
\]... | 1 | 1,620 | 1,620 | -1 |
A recipe calls for $4 \frac{1}{2}$ cups of flour. If you only make half of the recipe, then how many cups of flour do you need? Express your answer as a mixed number. | 2\frac{1}{4} | 1 | 1,362.9375 | 1,362.9375 | -1 | |
On side \(BC\) and on the extension of side \(AB\) through vertex \(B\) of triangle \(ABC\), points \(M\) and \(K\) are located, respectively, such that \(BM: MC = 4: 5\) and \(BK: AB = 1: 5\). Line \(KM\) intersects side \(AC\) at point \(N\). Find the ratio \(CN: AN\). | 5/24 | 0.3125 | 7,066.8125 | 6,039.8 | 7,533.636364 | |
What is the greatest prime factor of 99? | 11 | 1 | 861.875 | 861.875 | -1 | |
What describes the set of values of $a$ for which the curves $x^2+y^2=a^2$ and $y=x^2-a$ in the real $xy$-plane intersect at exactly 3 points? | a>\frac{1}{2} | To solve this problem, we need to analyze the intersection points of the circle $x^2 + y^2 = a^2$ and the parabola $y = x^2 - a$.
1. **Substitute the expression for $y$ from the parabola into the circle equation:**
\[
x^2 + (x^2 - a)^2 = a^2
\]
Simplifying, we get:
\[
x^2 + x^4 - 2ax^2 + a^2 = a^2
... | 0.25 | 7,369.3125 | 7,317 | 7,386.75 |
Points $A=(8,15)$ and $B=(14,9)$ lie on circle $\omega$ in the plane. Suppose the tangent lines to $\omega$ at $A$ and $B$ intersect at a point on the x-axis. Find the area of $\omega$. | 306\pi | 0 | 6,851.0625 | -1 | 6,851.0625 | |
A pentagonal prism is used as the base of a new pyramid. One of the seven faces of this pentagonal prism will be chosen as the base of the pyramid. Calculate the maximum value of the sum of the exterior faces, vertices, and edges of the resulting structure after this pyramid is added. | 42 | 0 | 7,321.375 | -1 | 7,321.375 | |
For the power function $f(x) = (m^2 - m - 1)x^{m^2 + m - 3}$ to be a decreasing function on the interval $(0, +\infty)$, then $m = \boxed{\text{answer}}$. | -1 | 0.0625 | 8,156.0625 | 7,617 | 8,192 | |
Given vectors $\overrightarrow{a} = (\cos \frac{3x}{2}, \sin \frac{3x}{2})$ and $\overrightarrow{b} = (\cos \frac{x}{2}, -\sin \frac{x}{2})$, with $x \in \left[-\frac{\pi}{3}, \frac{\pi}{4}\right]$,
(Ⅰ) Find $\overrightarrow{a} \cdot \overrightarrow{b}$ and $|\overrightarrow{a} + \overrightarrow{b}|$.
(Ⅱ) Let $f(x) = \... | -\frac{3}{2} | 0.75 | 6,393.1875 | 6,038.166667 | 7,458.25 | |
A projectile is fired with an initial speed $v$ from the ground at an angle between $0^\circ$ and $90^\circ$ to the horizontal. The trajectory of the projectile can be described by the parametric equations
\[
x = vt \cos \theta, \quad y = vt \sin \theta - \frac{1}{2} gt^2,
\]
where $t$ is the time, $g$ is the accelerat... | \frac{\pi}{16} | 0.0625 | 7,983.3125 | 7,530 | 8,013.533333 | |
Find out how many positive integers $n$ not larger than $2009$ exist such that the last digit of $n^{20}$ is $1$ . | 804 | 0.625 | 5,453.9375 | 4,473.7 | 7,087.666667 | |
Find the minimum value of the constant \(\lambda\) such that for all positive integers \(n\) and any positive real numbers \(x_{k}\) \((1 \leq k \leq n)\) summing to 1, the following inequality holds:
$$
\lambda \prod_{k=1}^{n}\left(1-x_{k}\right) \geq 1-\sum_{k=1}^{n} x_{k}^{2}.
$$ | e | 0.125 | 7,954.4375 | 6,291.5 | 8,192 | |
A fat coin is one which, when tossed, has a $2 / 5$ probability of being heads, $2 / 5$ of being tails, and $1 / 5$ of landing on its edge. Mr. Fat starts at 0 on the real line. Every minute, he tosses a fat coin. If it's heads, he moves left, decreasing his coordinate by 1; if it's tails, he moves right, increasing hi... | \[
\frac{1}{3}
\] | For $n \in \mathbb{Z}$, let $a_{n}$ be the fraction of the time Mr. Fat spends at $n$. By symmetry, $a_{n}=a_{-n}$ for all $n$. For $n>0$, we have $a_{n}=\frac{2}{5} a_{n-1}+\frac{2}{5} a_{n+1}$, or $a_{n+1}=\frac{5}{2} a_{n}-a_{n-1}$. This Fibonacci-like recurrence can be solved explicitly to obtain $$a_{n}=\alpha \cd... | 0 | 8,192 | -1 | 8,192 |
Let $n$ be an integer, and let $\triangle ABC$ be a right-angles triangle with right angle at $C$ . It is given that $\sin A$ and $\sin B$ are the roots of the quadratic equation \[(5n+8)x^2-(7n-20)x+120=0.\] Find the value of $n$ | 66 | 0.8125 | 6,113.25 | 5,633.538462 | 8,192 | |
Consider a cube with side length 4 units. Determine the volume of the set of points that are inside or within 2 units outside of the cube. | 1059 | 0 | 7,808.0625 | -1 | 7,808.0625 | |
Six distinct integers are picked at random from $\{1,2,3,\ldots,10\}$ . What is the probability that, among those selected, the second smallest is $3$ ? | $\frac{1}{3}$ | 0 | 6,260.9375 | -1 | 6,260.9375 | |
If \(A\ \clubsuit\ B\) is defined as \(A\ \clubsuit\ B = 3A^2 + 2B + 7\), what is the value of \(A\) for which \(A\ \clubsuit\ 7 = 61\)? | \frac{2\sqrt{30}}{3} | 0 | 3,346.3125 | -1 | 3,346.3125 | |
Let $a/b$ be the probability that a randomly chosen positive divisor of $12^{2007}$ is also a divisor of $12^{2000}$ , where $a$ and $b$ are relatively prime positive integers. Find the remainder when $a+b$ is divided by $2007$ . | 79 | 0.9375 | 6,180.0625 | 6,067.333333 | 7,871 | |
Isabella took 9 math tests and received 9 different scores, each an integer between 88 and 100, inclusive. After each test, she noticed that the average of her test scores was always an integer. Her score on the ninth test was 93. What was her score on the eighth test? | 96 | 0.0625 | 7,769.875 | 5,948 | 7,891.333333 | |
In the calculations shown, each letter stands for a digit. They are used to make some two-digit numbers. The two numbers on the left have a total of 79. What is the total of the four numbers on the right? | 158 | 0 | 2,249.125 | -1 | 2,249.125 | |
Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $2500$ are not factorial tails? | 500 | 0 | 8,192 | -1 | 8,192 | |
Let the set
\[ S=\{1, 2, \cdots, 12\}, \quad A=\{a_{1}, a_{2}, a_{3}\} \]
where \( a_{1} < a_{2} < a_{3}, \quad a_{3} - a_{2} \leq 5, \quad A \subseteq S \). Find the number of sets \( A \) that satisfy these conditions. | 185 | 0.125 | 7,848.75 | 6,867.5 | 7,988.928571 | |
Consider the geometric sequence $3$, $\dfrac{9}{2}$, $\dfrac{27}{4}$, $\dfrac{81}{8}$, $\ldots$. Find the eighth term of the sequence. Express your answer as a common fraction. | \frac{6561}{128} | 1 | 2,037.25 | 2,037.25 | -1 | |
Given the function $f(x) = |2x+a| + |2x-2b| + 3$
(Ⅰ) If $a=1$, $b=1$, find the solution set of the inequality $f(x) > 8$;
(Ⅱ) When $a>0$, $b>0$, if the minimum value of $f(x)$ is $5$, find the minimum value of $\frac{1}{a} + \frac{1}{b}$. | \frac{3+2\sqrt{2}}{2} | 0 | 6,352.375 | -1 | 6,352.375 | |
Walter has exactly one penny, one nickel, one dime and one quarter in his pocket. What percent of one dollar is in his pocket? | 41\% | 1. **Identify the coins and their values**: Walter has the following coins:
- One penny = $1$ cent
- One nickel = $5$ cents
- One dime = $10$ cents
- One quarter = $25$ cents
2. **Calculate the total value of the coins**:
\[
1 \text{ cent} + 5 \text{ cents} + 10 \text{ cents} + 25 \text{ cents} = 41 ... | 1 | 950 | 950 | -1 |
Find the number of ordered integer pairs \((a, b)\) such that the equation \(x^{2} + a x + b = 167 y\) has integer solutions \((x, y)\), where \(1 \leq a, b \leq 2004\). | 2020032 | 0.0625 | 8,152.3125 | 7,557 | 8,192 | |
In the set of positive integers from 1 to \( n \), the numbers that have the most divisors are called the "wang numbers" of these \( n \) positive integers. For example, in the set of positive integers from 1 to 20, the numbers with the most divisors are 12, 18, and 20. Therefore, 12, 18, and 20 are all wang numbers in... | 10080 | 0.625 | 6,486.25 | 5,462.8 | 8,192 | |
The value of the product \(\cos \frac{\pi}{15} \cos \frac{2 \pi}{15} \cos \frac{3 \pi}{15} \cdots \cos \frac{7 \pi}{15}\). | \frac{1}{128} | 0.1875 | 7,725.625 | 5,704.666667 | 8,192 | |
In $\triangle ABC$, $AB=6$, $AC=8$, $BC=10$, and $D$ is the midpoint of $\overline{BC}$. What is the sum of the radii of the circles inscribed in $\triangle ADB$ and $\triangle ADC$? | 3 | 1. **Identify the triangles and their properties**:
Given $\triangle ABC$ with $AB = 6$, $AC = 8$, and $BC = 10$. Point $D$ is the midpoint of $\overline{BC}$, so $BD = DC = \frac{10}{2} = 5$.
2. **Use the triangle side lengths to find the length of $AD$**:
Since $D$ is the midpoint of $BC$, we can use Apolloni... | 0 | 4,759.8125 | -1 | 4,759.8125 |
If $a$, $b$, $c$, $d$, $e$, and $f$ are integers such that $8x^3 + 64 = (ax^2 + bx + c)(dx^2 + ex + f)$ for all $x$, then what is $a^2 + b^2 + c^2 + d^2 + e^2 + f^2$? | 356 | 0.0625 | 8,035.0625 | 6,143 | 8,161.2 | |
Four cubes with edge lengths $2$, $3$, $4$, and $5$ are stacked with their bottom faces on the $xy$-plane, and one vertex at the origin $(0,0,0)$. The stack sequence follows the increasing order of cube sizes from the bottom. If point $X$ is at $(0,0,0)$ and point $Y$ is at the top vertex of the uppermost cube, determi... | 4\sqrt{3} | 0.0625 | 7,132.1875 | 4,480 | 7,309 | |
If $10x+y=75$ and $10y+x=57$ for some positive integers $x$ and $y$, what is the value of $x+y$? | 12 | Since $10x+y=75$ and $10y+x=57$, then $(10x+y)+(10y+x)=75+57$ and so $11x+11y=132$. Dividing by 11, we get $x+y=12$. (We could have noticed initially that $(x, y)=(7,5)$ is a pair that satisfies the two equations, thence concluding that $x+y=12$.) | 1 | 2,183.25 | 2,183.25 | -1 |
Find the arithmetic mean of the reciprocals of the first four prime numbers, including the number 7 instead of 5. | \frac{493}{1848} | 0.125 | 6,304.75 | 4,080 | 6,622.571429 | |
Let \( R \) be a semicircle with diameter \( XY \). A trapezoid \( ABCD \) in which \( AB \) is parallel to \( CD \) is circumscribed about \( R \) such that \( AB \) contains \( XY \). If \( AD = 4 \), \( CD = 5 \), and \( BC = 6 \), determine \( AB \). | 10 | 0 | 8,192 | -1 | 8,192 | |
Compute the smallest positive integer $n$ such that $(n + i), (n + i)^3,$ and $(n + i)^4$ are the vertices of a triangle in the complex plane whose area is greater than 3000. | 10 | 0.0625 | 7,618.0625 | 7,140 | 7,649.933333 | |
Given a right rectangular prism $B$ with edge lengths $1,$ $3,$ and $4$, and the set $S(r)$ of points within a distance $r$ of some point in $B$, express the volume of $S(r)$ as $ar^{3} + br^{2} + cr + d$ and determine the ratio $\frac{bc}{ad}$. | 19 | 0.6875 | 4,693.875 | 4,365.181818 | 5,417 | |
If there exists a real number $x$ such that the inequality $\left(e^{x}-a\right)^{2}+x^{2}-2ax+a^{2}\leqslant \dfrac{1}{2}$ holds with respect to $x$, determine the range of real number $a$. | \left\{\dfrac{1}{2}\right\} | 0 | 8,192 | -1 | 8,192 | |
Let $O$ be the origin, and $F$ be the right focus of the ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ where $a > b > 0$. The line $l$ passing through $F$ intersects the ellipse $C$ at points $A$ and $B$. Two points $P$ and $Q$ on the ellipse satisfy
$$
\overrightarrow{O P}+\overrightarrow{O A}+\overrightarrow... | \frac{\sqrt{2}}{2} | 0 | 8,192 | -1 | 8,192 | |
The minimum value of the polynomial $x^2 + y^2 - 6x + 8y + 7$ is ______. | -18 | 1 | 1,884.625 | 1,884.625 | -1 | |
Let $\mathcal{P}$ be a convex polygon with $50$ vertices. A set $\mathcal{F}$ of diagonals of $\mathcal{P}$ is said to be *$minimally friendly$* if any diagonal $d \in \mathcal{F}$ intersects at most one other diagonal in $\mathcal{F}$ at a point interior to $\mathcal{P}.$ Find the largest possible ... | 72 | 0 | 8,192 | -1 | 8,192 | |
Car A and Car B are traveling from point A to point B. Car A departs 6 hours later than Car B. The speed ratio of Car A to Car B is 4:3. 6 hours after Car A departs, its speed doubles, and both cars arrive at point B simultaneously. How many hours in total did Car A take to travel from A to B? | 8.4 | 0.0625 | 6,268.1875 | 8,192 | 6,139.933333 | |
Choose one of the following three conditions:①$a_{2}=60$, ②the sum of binomial coefficients is $64$, ③the maximum term of the binomial coefficients is the $4$th term. Fill in the blank below. Given ${(1-2x)}^{n}={a}_{0}+{a}_{1}x+{a}_{2}{x}^{2}+…+{a}_{n}{x}^{n}(n∈{N}_{+})$,_____, find:<br/>$(1)$ the value of $n$;<br/>$(... | 63 | 0.3125 | 7,236.6875 | 6,080.2 | 7,762.363636 | |
If $\tan (\alpha+ \frac{\pi}{3})=2 \sqrt {3}$, find the value of $\tan (\alpha- \frac{2\pi}{3})$ and $2\sin^{2}\alpha-\cos^{2}\alpha$. | -\frac{43}{52} | 0.8125 | 4,174 | 3,838.153846 | 5,629.333333 |
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