problem stringlengths 10 5.15k | answer dict |
|---|---|
Given a set $A_n = \{1, 2, 3, \ldots, n\}$, define a mapping $f: A_n \rightarrow A_n$ that satisfies the following conditions:
① For any $i, j \in A_n$ with $i \neq j$, $f(i) \neq f(j)$;
② For any $x \in A_n$, if the equation $x + f(x) = 7$ has $K$ pairs of solutions, then the mapping $f: A_n \rightarrow A_n$ is said to contain $K$ pairs of "good numbers." Determine the number of such mappings for $f: A_6 \rightarrow A_6$ that contain 3 pairs of good numbers. | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ten digits 0, 1, 2, 3, …, 9 and the imaginary unit i, determine the total number of distinct imaginary numbers that can be formed. | {
"answer": "90",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ a,b,c,d>0$ for which the following conditions:: $a)$ $(a-c)(b-d)=-4$ $b)$ $\frac{a+c}{2}\geq\frac{a^{2}+b^{2}+c^{2}+d^{2}}{a+b+c+d}$ Find the minimum of expression $a+c$ | {
"answer": "4\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 2 dimes of Chinese currency, how many ways can they be exchanged into coins (1 cent, 2 cents, and 5 cents)? | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the area of the portion of the circle defined by the equation $x^2 + 6x + y^2 = 50$ that lies below the $x$-axis and to the left of the line $y = x - 3$? | {
"answer": "\\frac{59\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( a_1, a_2, \ldots, a_{2020} \) be the roots of the polynomial
\[ x^{2020} + x^{2019} + \cdots + x^2 + x - 2022 = 0. \]
Compute
\[ \sum_{n = 1}^{2020} \frac{1}{1 - a_n}. \] | {
"answer": "2041210",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Amelia and Blaine are playing a modified game where they toss their respective coins. Amelia's coin lands on heads with a probability of $\frac{3}{7}$, and Blaine's lands on heads with a probability of $\frac{1}{3}$. They begin their game only after observing at least one head in a simultaneous toss of both coins. Once the game starts, they toss coins alternately with Amelia starting first, and the player to first toss heads twice consecutively wins the game. What is the probability that Amelia wins the game?
A) $\frac{1}{2}$
B) $\frac{9}{49}$
C) $\frac{2401}{6875}$
D) $\frac{21609}{64328}$ | {
"answer": "\\frac{21609}{64328}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Seven cards numbered $1$ through $7$ are to be lined up in a row. Find the number of arrangements of these seven cards where one of the cards can be removed leaving the remaining six cards in either ascending or descending order. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=ax^{3}+2bx^{2}+3cx+4d$, where $a,b,c,d$ are real numbers, $a < 0$, and $c > 0$, is an odd function, and when $x\in[0,1]$, the range of $f(x)$ is $[0,1]$. Find the maximum value of $c$. | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a large bag of decorative ribbons, $\frac{1}{4}$ are yellow, $\frac{1}{3}$ are purple, $\frac{1}{8}$ are orange, and the remaining 45 ribbons are silver. How many of the ribbons are orange? | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The water tank in the diagram below is in the shape of an inverted right circular cone. The radius of its base is 8 feet, and its height is 64 feet. The water in the tank is $40\%$ of the tank's capacity. The height of the water in the tank can be written in the form $a\sqrt[3]{b}$, where $a$ and $b$ are positive integers, and $b$ is not divisible by a perfect cube greater than 1. What is $a+b$? | {
"answer": "66",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ten circles of diameter 1 are arranged in the first quadrant of a coordinate plane. Five circles are in the base row with centers at $(0.5, 0.5)$, $(1.5, 0.5)$, $(2.5, 0.5)$, $(3.5, 0.5)$, $(4.5, 0.5)$, and the remaining five directly above the first row with centers at $(0.5, 1.5)$, $(1.5, 1.5)$, $(2.5, 1.5)$, $(3.5, 1.5)$, $(4.5, 1.5)$. Let region $\mathcal{S}$ be the union of these ten circular regions. Line $m,$ with slope $-2$, divides $\mathcal{S}$ into two regions of equal area. Line $m$'s equation can be expressed in the form $px=qy+r$, where $p, q,$ and $r$ are positive integers whose greatest common divisor is 1. Find $p^2+q^2+r^2$. | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an odd function defined on $\mathbb{R}$, when $x > 0$, $f(x)=x^{2}+2x-1$.
(1) Find the value of $f(-3)$;
(2) Find the analytic expression of the function $f(x)$. | {
"answer": "-14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $|m|=3$, $|n|=2$, and $m<n$, find the value of $m^2+mn+n^2$. | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles A, B, and C are $a$, $b$, and $c$, respectively. Given the equation $$2b\cos A - \sqrt{3}c\cos A = \sqrt{3}a\cos C$$.
(1) Find the value of angle A;
(2) If $\angle B = \frac{\pi}{6}$, and the median $AM = \sqrt{7}$ on side $BC$, find the area of $\triangle ABC$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram below, the circle with center $A$ is congruent to and tangent to the circle with center $B$ . A third circle is tangent to the circle with center $A$ at point $C$ and passes through point $B$ . Points $C$ , $A$ , and $B$ are collinear. The line segment $\overline{CDEFG}$ intersects the circles at the indicated points. Suppose that $DE = 6$ and $FG = 9$ . Find $AG$ .
[asy]
unitsize(5);
pair A = (-9 sqrt(3), 0);
pair B = (9 sqrt(3), 0);
pair C = (-18 sqrt(3), 0);
pair D = (-4 sqrt(3) / 3, 10 sqrt(6) / 3);
pair E = (2 sqrt(3), 4 sqrt(6));
pair F = (7 sqrt(3), 5 sqrt(6));
pair G = (12 sqrt(3), 6 sqrt(6));
real r = 9sqrt(3);
draw(circle(A, r));
draw(circle(B, r));
draw(circle((B + C) / 2, 3r / 2));
draw(C -- D);
draw(" $6$ ", E -- D);
draw(E -- F);
draw(" $9$ ", F -- G);
dot(A);
dot(B);
label(" $A$ ", A, plain.E);
label(" $B$ ", B, plain.E);
label(" $C$ ", C, W);
label(" $D$ ", D, dir(160));
label(" $E$ ", E, S);
label(" $F$ ", F, SSW);
label(" $G$ ", G, N);
[/asy] | {
"answer": "9\\sqrt{19}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The maximum value of the function $f(x) = \frac{\frac{1}{6} \cdot (-1)^{1+ C_{2x}^{x}} \cdot A_{x+2}^{5}}{1+ C_{3}^{2} + C_{4}^{2} + \ldots + C_{x-1}^{2}}$ ($x \in \mathbb{N}$) is ______. | {
"answer": "-20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the product of $1101_2 \cdot 111_2$. Express your answer in base 2. | {
"answer": "1100111_2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an obtuse triangle \(ABC\) with obtuse angle \(C\). Points \(P\) and \(Q\) are marked on its sides \(AB\) and \(BC\) respectively, such that \(\angle ACP = CPQ = 90^\circ\). Find the length of segment \(PQ\) if it is known that \(AC = 25\), \(CP = 20\), and \(\angle APC = \angle A + \angle B\). | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given several numbers, one of them, $a$ , is chosen and replaced by the three numbers $\frac{a}{3}, \frac{a}{3}, \frac{a}{3}$ . This process is repeated with the new set of numbers, and so on. Originally, there are $1000$ ones, and we apply the process several times. A number $m$ is called *good* if there are $m$ or more numbers that are the same after each iteration, no matter how many or what operations are performed. Find the largest possible good number. | {
"answer": "667",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate: $6 - 5\left[7 - (\sqrt{16} + 2)^2\right] \cdot 3.$ | {
"answer": "-429",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve the equations:<br/>$(1)x^{2}-4x-1=0$;<br/>$(2)\left(x+3\right)^{2}=x+3$. | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCD$ be a rectangle with side lengths $AB = CD = 5$ and $BC = AD = 10$ . $W, X, Y, Z$ are points on $AB, BC, CD$ and $DA$ respectively chosen in such a way that $WXYZ$ is a kite, where $\angle ZWX$ is a right angle. Given that $WX = WZ = \sqrt{13}$ and $XY = ZY$ , determine the length of $XY$ . | {
"answer": "\\sqrt{65}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are real numbers $a, b, c, d$ such that for all $(x, y)$ satisfying $6y^2 = 2x^3 + 3x^2 + x$ , if $x_1 = ax + b$ and $y_1 = cy + d$ , then $y_1^2 = x_1^3 - 36x_1$ . What is $a + b + c + d$ ? | {
"answer": "90",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = \frac{x}{\ln x}$, and $g(x) = f(x) - mx (m \in \mathbb{R})$,
(I) Find the interval of monotonic decrease for function $f(x)$.
(II) If function $g(x)$ is monotonically decreasing on the interval $(1, +\infty)$, find the range of the real number $m$.
(III) If there exist $x_1, x_2 \in [e, e^2]$ such that $m \geq g(x_1) - g'(x_2)$ holds true, find the minimum value of the real number $m$. | {
"answer": "\\frac{1}{2} - \\frac{1}{4e^2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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? | {
"answer": "500",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A shepherd uses 15 segments of fencing, each 2 meters long, to form a square or rectangular sheep pen with one side against a wall. What is the maximum area of the sheep pen in square meters? | {
"answer": "112",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
It is planned to establish an additional channel for exchanging stereo audio signals (messages) for daily reporting communication sessions between two working sites of a deposit. Determine the required bandwidth of this channel in kilobits, considering that the sessions will be conducted for no more than 51 minutes. The requirements for a mono signal per second are given below:
- Sampling rate: 63 Hz
- Sampling depth: 17 bits
- Metadata volume: 47 bytes for every 5 kilobits of audio | {
"answer": "2.25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $A_1A_2A_3A_4A_5$ be a regular pentagon with side length 1. The sides of the pentagon are extended to form the 10-sided polygon shown in bold at right. Find the ratio of the area of quadrilateral $A_2A_5B_2B_5$ (shaded in the picture to the right) to the area of the entire 10-sided polygon.
[asy]
size(8cm);
defaultpen(fontsize(10pt));
pair A_2=(-0.4382971011,5.15554989475), B_4=(-2.1182971011,-0.0149584477027), B_5=(-4.8365942022,8.3510997895), A_3=(0.6,8.3510997895), B_1=(2.28,13.521608132), A_4=(3.96,8.3510997895), B_2=(9.3965942022,8.3510997895), A_5=(4.9982971011,5.15554989475), B_3=(6.6782971011,-0.0149584477027), A_1=(2.28,3.18059144705);
filldraw(A_2--A_5--B_2--B_5--cycle,rgb(.8,.8,.8));
draw(B_1--A_4^^A_4--B_2^^B_2--A_5^^A_5--B_3^^B_3--A_1^^A_1--B_4^^B_4--A_2^^A_2--B_5^^B_5--A_3^^A_3--B_1,linewidth(1.2)); draw(A_1--A_2--A_3--A_4--A_5--cycle);
pair O = (A_1+A_2+A_3+A_4+A_5)/5;
label(" $A_1$ ",A_1, 2dir(A_1-O));
label(" $A_2$ ",A_2, 2dir(A_2-O));
label(" $A_3$ ",A_3, 2dir(A_3-O));
label(" $A_4$ ",A_4, 2dir(A_4-O));
label(" $A_5$ ",A_5, 2dir(A_5-O));
label(" $B_1$ ",B_1, 2dir(B_1-O));
label(" $B_2$ ",B_2, 2dir(B_2-O));
label(" $B_3$ ",B_3, 2dir(B_3-O));
label(" $B_4$ ",B_4, 2dir(B_4-O));
label(" $B_5$ ",B_5, 2dir(B_5-O));
[/asy] | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the polynomial $$Q(x) = \left(1 + x + x^2 + \ldots + x^{20}\right)^2 - x^{20},$$ find the sum $$\beta_1 + \beta_2 + \beta_6$$ where the complex zeros of $Q(x)$ are written in the form, $\beta_k=r_k[\cos(2\pi\beta_k)+i\sin(2\pi\beta_k)]$, with $0<\beta_1\le\beta_2\le\ldots\le\beta_{41}<1$ and $r_k>0$. | {
"answer": "\\frac{3}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with its left and right foci being $F_1$ and $F_2$ respectively, and passing through the point $P(0, \sqrt{5})$, with an eccentricity of $\frac{2}{3}$, and $A$ being a moving point on the line $x=4$.
- (I) Find the equation of the ellipse $C$;
- (II) Point $B$ is on the ellipse $C$, satisfying $OA \perpendicular OB$, find the minimum length of segment $AB$. | {
"answer": "\\sqrt{21}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A mathematical demonstration showed that there were distinct positive integers such that $97^4 + 84^4 + 27^4 + 3^4 = m^4$. Calculate the value of $m$. | {
"answer": "108",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We denote by gcd (...) the greatest common divisor of the numbers in (...). (For example, gcd $(4, 6, 8)=2$ and gcd $(12, 15)=3$ .) Suppose that positive integers $a, b, c$ satisfy the following four conditions: $\bullet$ gcd $(a, b, c)=1$ , $\bullet$ gcd $(a, b + c)>1$ , $\bullet$ gcd $(b, c + a)>1$ , $\bullet$ gcd $(c, a + b)>1$ .
a) Is it possible that $a + b + c = 2015$ ?
b) Determine the minimum possible value that the sum $a+ b+ c$ can take. | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular piece of paper $A B C D$ is folded and flattened such that triangle $D C F$ falls onto triangle $D E F$, with vertex $E$ landing on side $A B$. Given that $\angle 1 = 22^{\circ}$, find $\angle 2$. | {
"answer": "44",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a triangle $\triangle ABC$ with its three interior angles $A$, $B$, and $C$ satisfying: $$A+C=2B, \frac {1}{\cos A}+ \frac {1}{\cos C}=- \frac { \sqrt {2}}{\cos B}$$, find the value of $$\cos \frac {A-C}{2}$$. | {
"answer": "\\frac { \\sqrt {2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest positive integer with exactly 16 positive divisors? | {
"answer": "384",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular prism with dimensions 1 cm by 1 cm by 2 cm has a dot marked in the center of the top face (1 cm by 2 cm face). It is sitting on a table, which is 1 cm by 2 cm face. The prism is rolled over its shorter edge (1 cm edge) on the table, without slipping, and stops once the dot returns to the top. Find the length of the path followed by the dot in terms of $\pi$. | {
"answer": "2\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The height $BD$ of the acute-angled triangle $ABC$ intersects with its other heights at point $H$. Point $K$ lies on segment $AC$ such that the angle $BKH$ is maximized. Find $DK$ if $AD = 2$ and $DC = 3$. | {
"answer": "\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a luxurious ocean liner, 3000 adults consisting of men and women embark on a voyage. If 55% of the adults are men and 12% of the women as well as 15% of the men are wearing sunglasses, determine the total number of adults wearing sunglasses. | {
"answer": "409",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$2.46\times 8.163\times (5.17+4.829)$ is approximately equal to what value? | {
"answer": "200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many distinct four-digit positive integers have a digit product equal to 18? | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the xy-plane, consider a right triangle $ABC$ with the right angle at $C$. The hypotenuse $AB$ is of length $50$. The medians through vertices $A$ and $B$ are described by the lines $y = x + 5$ and $y = 2x + 2$, respectively. Determine the area of triangle $ABC$. | {
"answer": "500",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Initially, the numbers 1 and 2 are written at opposite positions on a circle. Each operation consists of writing the sum of two adjacent numbers between them. For example, the first operation writes two 3's, and the second operation writes two 4's and two 5's. After each operation, the sum of all the numbers becomes three times the previous total. After sufficient operations, find the sum of the counts of the numbers 2015 and 2016 that are written. | {
"answer": "2016",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The Wolf and the three little pigs wrote a detective story "The Three Little Pigs-2", and then, together with Little Red Riding Hood and her grandmother, a cookbook "Little Red Riding Hood-2". The publisher gave the fee for both books to the pig Naf-Naf. He took his share and handed the remaining 2100 gold coins to the Wolf. The fee for each book is divided equally among its authors. How much money should the Wolf take for himself? | {
"answer": "700",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2|\overrightarrow{b}|$, and $(\overrightarrow{a}-\overrightarrow{b})\bot \overrightarrow{b}$, calculate the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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? | {
"answer": "59",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many two-digit numbers have digits whose sum is a prime number? | {
"answer": "31",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Seven cards numbered $1$ through $7$ are to be lined up in a row. Find the number of arrangements of these seven cards where one of the cards can be removed leaving the remaining six cards in either ascending or descending order. | {
"answer": "26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $(2x)_((-1)^{5}=a_0+a_1x+a_2x^2+...+a_5x^5$, find:
(1) $a_0+a_1+...+a_5$;
(2) $|a_0|+|a_1|+...+|a_5|$;
(3) $a_1+a_3+a_5$;
(4) $(a_0+a_2+a_4)^2-(a_1+a_3+a_5)^2$. | {
"answer": "-243",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a,$ $b,$ $c$ be real numbers such that $9a^2 + 4b^2 + 25c^2 = 1.$ Find the maximum value of
\[8a + 3b + 5c.\] | {
"answer": "\\frac{\\sqrt{373}}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest natural number \( n \) such that both \( n^2 \) and \( (n+1)^2 \) contain the digit 7. | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence $\{a\_n\}(n=1,2,3,...,2016)$, circle $C\_1$: $x^{2}+y^{2}-4x-4y=0$, circle $C\_2$: $x^{2}+y^{2}-2a_{n}x-2a_{2017-n}y=0$. If circle $C\_2$ bisects the circumference of circle $C\_1$, then the sum of all terms in the sequence $\{a\_n\}$ is $\_\_\_\_\_\_$. | {
"answer": "4032",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\triangle ABC$ be a right triangle with $\angle ABC = 90^\circ$, and let $AB = 10\sqrt{21}$ be the hypotenuse. Point $E$ lies on $AB$ such that $AE = 10\sqrt{7}$ and $EB = 20\sqrt{7}$. Let $F$ be the foot of the altitude from $C$ to $AB$. Find the distance $EF$. Express $EF$ in the form $m\sqrt{n}$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m + n$. | {
"answer": "31",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two adjacent faces of a tetrahedron, which are equilateral triangles with side length 3, form a dihedral angle of 30 degrees. The tetrahedron rotates around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plane containing this edge. | {
"answer": "\\frac{9 \\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a nonnegative integer $n$, let $r_7(3n)$ represent the remainder when $3n$ is divided by $7$. Determine the $22^{\text{nd}}$ entry in an ordered list of all nonnegative integers $n$ that satisfy $$r_7(3n)\le 4~.$$ | {
"answer": "29",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A district in a city is laid out in an $11 \times 11$ grid. Every day, a sprinkler truck departs from the bottom-left corner $A(0,0)$ and travels along the streets to reach the top-right corner $B(10,10)$. At each intersection, the driver randomly chooses a direction, as long as it does not deviate from the shortest path. One day, the street from $(9,9)$ to $(10,9)$ is blocked due to an accident, but the driver is not aware of this at the time of departure. What is the probability that the sprinkler truck can still reach $B$ without any issues? | {
"answer": "1 - \\frac{\\binom{18}{9}}{\\binom{20}{10}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate $|\omega^2 + 4\omega + 34|$ if $\omega = 5 + 3i$. | {
"answer": "\\sqrt{6664}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From the numbers 2, 3, 4, 5, 6, 7, 8, 9, two different numbers are selected to be the base and the exponent of a logarithm, respectively. How many different logarithmic values can be formed? | {
"answer": "52",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABC$ be a triangle with side lengths $AB=6, AC=7,$ and $BC=8.$ Let $H$ be the orthocenter of $\triangle ABC$ and $H'$ be the reflection of $H$ across the midpoint $M$ of $BC.$ $\tfrac{[ABH']}{[ACH']}$ can be expressed as $\frac{p}{q}$ . Find $p+q$ .
*2022 CCA Math Bonanza Individual Round #14* | {
"answer": "251",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest positive integer with exactly 12 positive integer divisors? | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( g_{1}(x) = \sqrt{4 - x} \), and for integers \( n \geq 2 \), define \[ g_{n}(x) = g_{n-1}\left(\sqrt{(n+1)^2 - x}\right). \] Find the largest \( n \) (denote this as \( M \)) for which the domain of \( g_n \) is nonempty. For this value of \( M \), if the domain of \( g_M \) consists of a single point \( \{d\} \), compute \( d \). | {
"answer": "-589",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular table $ 9$ rows $ \times$ $ 2008$ columns is fulfilled with numbers $ 1$ , $ 2$ , ..., $ 2008$ in a such way that each number appears exactly $ 9$ times in table and difference between any two numbers from same column is not greater than $ 3$ . What is maximum value of minimum sum in column (with minimal sum)? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $15^{-1} \equiv 31 \pmod{53}$, find $38^{-1} \pmod{53}$, as a residue modulo 53. | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that there are 20 cards numbered from 1 to 20 on a table, and Xiao Ming picks out 2 cards such that the number on one card is 2 more than twice the number on the other card, find the maximum number of cards Xiao Ming can pick. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the piecewise function $f(x)= \begin{cases} x+2 & (x\leq-1) \\ x^{2} & (-1<x<2) \\ 2x & (x\geq2)\end{cases}$, if $f(x)=3$, determine the value of $x$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The graph of the equation $10x + 270y = 2700$ is drawn on graph paper where each square represents one unit in each direction. A second line defined by $x + y = 10$ also passes through the graph. How many of the $1$ by $1$ graph paper squares have interiors lying entirely below both graphs and entirely in the first quadrant? | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular pasture is to be fenced off on three sides using part of a 100 meter rock wall as the fourth side. Fence posts are to be placed every 15 meters along the fence including at the points where the fence meets the rock wall. Given the dimensions of the pasture are 36 m by 75 m, find the minimum number of posts required. | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a Cartesian coordinate plane, call a rectangle $standard$ if all of its sides are parallel to the $x$ - and $y$ - axes, and call a set of points $nice$ if no two of them have the same $x$ - or $y$ - coordinate. First, Bert chooses a nice set $B$ of $2016$ points in the coordinate plane. To mess with Bert, Ernie then chooses a set $E$ of $n$ points in the coordinate plane such that $B\cup E$ is a nice set with $2016+n$ points. Bert returns and then miraculously notices that there does not exist a standard rectangle that contains at least two points in $B$ and no points in $E$ in its interior. For a given nice set $B$ that Bert chooses, define $f(B)$ as the smallest positive integer $n$ such that Ernie can find a nice set $E$ of size $n$ with the aforementioned properties. Help Bert determine the minimum and maximum possible values of $f(B)$ .
*Yannick Yao* | {
"answer": "2015",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two lines with slopes $\dfrac{1}{3}$ and $3$ intersect at $(3,3)$. Find the area of the triangle enclosed by these two lines and the line $x+y=12$. | {
"answer": "8.625",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A function is given by
$$
f(x)=\ln (a x+b)+x^{2} \quad (a \neq 0).
$$
(1) If the tangent line to the curve $y=f(x)$ at the point $(1, f(1))$ is $y=x$, find the values of $a$ and $b$.
(2) If $f(x) \leqslant x^{2}+x$ always holds, find the maximum value of $ab$. | {
"answer": "\\frac{e}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The inclination angle $\alpha$ of the line $l: \sqrt{3}x+3y+1=0$ is $\tan^{-1}\left( -\frac{\sqrt{3}}{3} \right)$. Calculate the value of the angle $\alpha$. | {
"answer": "\\frac{5\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify $\sqrt[3]{1+27} \cdot \sqrt[3]{1+\sqrt[3]{27}} \cdot \sqrt{4}$. | {
"answer": "2 \\cdot \\sqrt[3]{112}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Farmer Tim is lost in the densely-forested Cartesian plane. Starting from the origin he walks a sinusoidal path in search of home; that is, after $t$ minutes he is at position $(t,\sin t)$ .
Five minutes after he sets out, Alex enters the forest at the origin and sets out in search of Tim. He walks in such a way that after he has been in the forest for $m$ minutes, his position is $(m,\cos t)$ .
What is the greatest distance between Alex and Farmer Tim while they are walking in these paths? | {
"answer": "3\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the trapezoid \(ABCD\), the bases are given as \(AD = 4\) and \(BC = 1\), and the angles at \(A\) and \(D\) are \(\arctan 2\) and \(\arctan 3\) respectively.
Find the radius of the circle inscribed in triangle \(CBE\), where \(E\) is the intersection point of the diagonals of the trapezoid. | {
"answer": "\\frac{18}{25 + 2 \\sqrt{130} + \\sqrt{445}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the inequality about $x$, $2\log_2^2x - 5\log_2x + 2 \leq 0$, the solution set is $B$.
1. Find set $B$.
2. If $x \in B$, find the maximum and minimum values of $f(x) = \log_2 \frac{x}{8} \cdot \log_2 (2x)$. | {
"answer": "-4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that both $α$ and $β$ are acute angles, and $\cos(α+β)= \frac{\sin α}{\sin β}$, find the maximum value of $\tan α$. | {
"answer": "\\frac{ \\sqrt {2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The highest power of 2 that is a factor of \(15.13^{4} - 11^{4}\) needs to be determined. | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $p = 101$ and let $S$ be the set of $p$ -tuples $(a_1, a_2, \dots, a_p) \in \mathbb{Z}^p$ of integers. Let $N$ denote the number of functions $f: S \to \{0, 1, \dots, p-1\}$ such that
- $f(a + b) + f(a - b) \equiv 2\big(f(a) + f(b)\big) \pmod{p}$ for all $a, b \in S$ , and
- $f(a) = f(b)$ whenever all components of $a-b$ are divisible by $p$ .
Compute the number of positive integer divisors of $N$ . (Here addition and subtraction in $\mathbb{Z}^p$ are done component-wise.)
*Proposed by Ankan Bhattacharya* | {
"answer": "5152",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(Geometry Proof Exercise) From a point A outside a circle ⊙O with radius 2, draw a line intersecting ⊙O at points C and D. A line segment AB is tangent to ⊙O at B. Given that AC=4 and AB=$4 \sqrt {2}$, find $tan∠DAB$. | {
"answer": "\\frac { \\sqrt {2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two teams, Team A and Team B, are playing in a basketball finals series that uses a "best of seven" format (the first team to win four games wins the series and the finals end). Based on previous game results, Team A's home and away schedule is arranged as "home, home, away, away, home, away, home". The probability of Team A winning at home is 0.6, and the probability of winning away is 0.5. The results of each game are independent of each other. Calculate the probability that Team A wins the series with a 4:1 score. | {
"answer": "0.18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a geometric sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, and $a_1=2$, if $\frac {S_{6}}{S_{2}}=21$, then the sum of the first five terms of the sequence $\{\frac {1}{a_n}\}$ is
A) $\frac {1}{2}$ or $\frac {11}{32}$
B) $\frac {1}{2}$ or $\frac {31}{32}$
C) $\frac {11}{32}$ or $\frac {31}{32}$
D) $\frac {11}{32}$ or $\frac {5}{2}$ | {
"answer": "\\frac {31}{32}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f\left( x \right)={x}^{2}+{\left( \ln 3x \right)}^{2}-2a(x+3\ln 3x)+10{{a}^{2}}(a\in \mathbf{R})$, determine the value of the real number $a$ for which there exists ${{x}_{0}}$ such that $f\left( {{x}_{0}} \right)\leqslant \dfrac{1}{10}$. | {
"answer": "\\frac{1}{30}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 3 different pairs of shoes in a shoe cabinet. If one shoe is picked at random from the left shoe set of 6 shoes, and then another shoe is picked at random from the right shoe set of 6 shoes, calculate the probability that the two shoes form a pair. | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a parameterized curve $ C: x\equal{}e^t\minus{}e^{\minus{}t},\ y\equal{}e^{3t}\plus{}e^{\minus{}3t}$ .
Find the area bounded by the curve $ C$ , the $ x$ axis and two lines $ x\equal{}\pm 1$ . | {
"answer": "\\frac{5\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A point $Q$ lies inside the triangle $\triangle DEF$ such that lines drawn through $Q$ parallel to the sides of $\triangle DEF$ divide it into three smaller triangles $u_1$, $u_2$, and $u_3$ with areas $16$, $25$, and $36$ respectively. Determine the area of $\triangle DEF$. | {
"answer": "77",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many 5-letter words with at least one consonant can be constructed from the letters $A$, $B$, $C$, $D$, $E$, $F$, $G$, and $I$? Each letter can be used more than once, and $B$, $C$, $D$, $F$, $G$ are consonants. | {
"answer": "32525",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the least positive integer $n$ such that $7350$ is a factor of $n!$? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $M: \frac{x^2}{a^2} + \frac{y^2}{3} = 1 (a > 0)$ with one of its foci at $F(-1, 0)$. Points $A$ and $B$ are the left and right vertices of the ellipse's major axis, respectively. A line $l$ passes through $F$ and intersects the ellipse at distinct points $C$ and $D$.
1. Find the equation of the ellipse $M$;
2. When the line $l$ has an angle of $45^{\circ}$, find the length of the line segment $CD$;
3. Let $S_1$ and $S_2$ represent the areas of triangles $\Delta ABC$ and $\Delta ABD$, respectively. Find the maximum value of $|S_1 - S_2|$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The Greenhill Soccer Club has 25 players, including 4 goalies. During an upcoming practice, the team plans to have a competition in which each goalie will try to stop penalty kicks from every other player, including the other goalies. How many penalty kicks are required for every player to have a chance to kick against each goalie? | {
"answer": "96",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each two-digit is number is coloured in one of $k$ colours. What is the minimum value of $k$ such that, regardless of the colouring, there are three numbers $a$ , $b$ and $c$ with different colours with $a$ and $b$ having the same units digit (second digit) and $b$ and $c$ having the same tens digit (first digit)? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( M = \{1, 2, \cdots, 10\} \) and let \( T \) be a collection of certain two-element subsets of \( M \), such that for any two different elements \(\{a, b\} \) and \(\{x, y\} \) in \( T \), the condition \( 11 \nmid (ax + by)(ay + bx) \) is satisfied.
Find the maximum number of elements in \( T \). | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, a line segment of length $\sqrt{2}+1$ has its endpoints $C$ and $D$ sliding on the $x$-axis and $y$-axis, respectively. It is given that $\overrightarrow{CP} = \sqrt{2} \overrightarrow{PD}$. Let the trajectory of point $P$ be curve $E$.
(I) Find the equation of curve $E$;
(II) A line $l$ passing through point $(0,1)$ intersects curve $E$ at points $A$ and $B$, and $\overrightarrow{OM} = \overrightarrow{OA} + \overrightarrow{OB}$. When point $M$ is on curve $E$, find the area of quadrilateral $OAMB$. | {
"answer": "\\frac{\\sqrt{6}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given points $A(2, 0)$, $B(0, 2)$, $C(\cos\alpha, \sin\alpha)$ and $0 < \alpha < \pi$:
1. If $|\vec{OA} + \vec{OC}| = \sqrt{7}$, find the angle between $\vec{OB}$ and $\vec{OC}$.
2. If $\vec{AC} \perp \vec{BC}$, find the value of $\cos\alpha$. | {
"answer": "\\frac{1 + \\sqrt{7}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find an axis of symmetry for the function $f(x) = \cos(2x + \frac{\pi}{6})$. | {
"answer": "\\frac{5\\pi}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose 9 people are arranged in a line randomly. What is the probability that person A is in the middle, and persons B and C are adjacent? | {
"answer": "\\frac{1}{42}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the random variable $\xi$ follow the normal distribution $N(1, \sigma^2)$ ($\sigma > 0$). If $P(0 < \xi < 1) = 0.4$, then find the value of $P(\xi > 2)$. | {
"answer": "0.2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $min\{ a,b\}$ represents the smaller value between the real numbers $a$ and $b$, and the vectors $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ satisfy $(\vert\overrightarrow{a}\vert=1,\vert\overrightarrow{b}\vert=2,\overrightarrow{a}\cdot\overrightarrow{b}=0,\overrightarrow{c}=\lambda\overrightarrow{a}+\mu\overrightarrow{b}(\lambda+\mu=1))$, find the maximum value of $min\{\overrightarrow{c}\cdot\overrightarrow{a}, \overrightarrow{c}\cdot\overrightarrow{b}\}$ and the value of $\vert\overrightarrow{c}\vert$. | {
"answer": "\\frac{2\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two congruent 30-60-90 triangles are placed such that one triangle is translated 2 units vertically upwards, while their hypotenuses originally coincide when not translated. The hypotenuse of each triangle is 10. Calculate the area common to both triangles when one is translated. | {
"answer": "25\\sqrt{3} - 10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
All of the roots of $x^3+ax^2+bx+c$ are positive integers greater than $2$ , and the coefficients satisfy $a+b+c+1=-2009$ . Find $a$ | {
"answer": "-58",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the constants $\alpha$ and $\beta$ such that $\frac{x-\alpha}{x+\beta} = \frac{x^2 - 64x + 975}{x^2 + 99x - 2200}$. What is $\alpha+\beta$? | {
"answer": "138",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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