task_type stringclasses 4
values | problem stringlengths 14 5.23k | solution stringlengths 1 8.29k | problem_tokens int64 9 1.02k | solution_tokens int64 1 1.98k |
|---|---|---|---|---|
math | Given that the function $f(x-1)$ is an odd function, and the function $f(x+3)$ is an even function, with $f(0)=1$, then $f(8)=$ ? | -1 | 45 | 2 |
math | The function $g$ is defined for all integers and satisfies the following conditions:
\[ g(n) = \begin{cases}
n-3 & \mbox{if } n \geq 500 \\
g(g(n+4)) & \mbox{if } n < 500
\end{cases} \]
Find $g(56)$. | 497 | 81 | 3 |
math | In equilateral triangle $ABC$ , the midpoint of $\overline{BC}$ is $M$ . If the circumcircle of triangle $MAB$ has area $36\pi$ , then find the perimeter of the triangle.
*Proposed by Isabella Grabski* | 36 | 66 | 2 |
math | Let $f(x)$ be an odd function defined on $\mathbb{R}$, and when $x \geq 0$, $f(x) = x^2$. If for any $x \in [a, a+2]$, the inequality $f(x+a) \geq f(3x+1)$ always holds, then the range of the real number $a$ is | (-\infty, -5] | 83 | 8 |
math | Define the determinant $\begin{vmatrix} a_{1} & a_{2} \\ a_{3} & a_{4}\end{vmatrix} = a_{1}a_{4} - a_{2}a_{3}$, and let the function $g(\theta) = \begin{vmatrix} \sin\theta & 3-\cos\theta \\ m & \sin\theta\end{vmatrix}$ (where $0 \leq \theta \leq \frac{\pi}{2}$).
(1) Find the value of $g\left( \frac{\pi}{2} \right)$;... | m = -1 | 163 | 4 |
math | If the line $y=mx+1$ intersects the ellipse $x^2+4y^2=1$ exactly once, then the value of $m^2$ is | \frac{3}{4} | 38 | 7 |
math | Find all integers \( n \) such that \( n^5 + 3 \) is divisible by \( n^2 + 1 \). | -3, -1, 0, 1, 2 | 30 | 14 |
math | Consider the set \( A = \{1, 2, 3, \ldots, 2011\} \). How many subsets of \( A \) exist such that the sum of their elements is 2,023,060? | 4 | 57 | 1 |
math | On the sides \(AB\) and \(AD\) of rectangle \(ABCD\), points \(M\) and \(N\) are marked, respectively. It is known that \(AN=7\), \(NC=39\), \(AM=12\), and \(MB=3\).
(a) Find the area of the rectangle \(ABCD\).
(b) Find the area of triangle \(MNC\). | 286.5 | 86 | 5 |
math | In triangle ABC, the sides opposite to angles A, B, C are a, b, c respectively. Given that $B= \frac {π}{3}, b=2 \sqrt {3}$, find the range of the perimeter of triangle ABC. | (4 \sqrt {3}, 6 \sqrt {3}] | 53 | 14 |
math | In the triangle \(ABC\), \(AB = 14\), \(BC = 16\), \(AC = 26\), \(M\) is the midpoint of \(BC\) and \(D\) is the point on \(BC\) such that \(AD\) bisects \(\angle BAC\). Let \(P\) be the foot of the perpendicular from \(B\) onto \(AD\). Determine the length of \(PM\). | 6 | 94 | 1 |
math | Let z = 1 - i, simplify the expression 2/z + z^2. | 1-i | 19 | 2 |
math | The square with vertices $(-a, -a), (a, -a), (-a, a), (a, a)$ is cut by the line $y = x/2$ into congruent quadrilaterals. The perimeter of one of these congruent quadrilaterals divided by $a$ equals what? Express your answer in simplified radical form. | 4+\sqrt{5} | 75 | 6 |
math | A checker can move in one direction on a divided strip into cells, moving either to the adjacent cell or skipping one cell in one move. In how many ways can it move 10 cells? 11 cells? | 144 | 45 | 3 |
math | Given an arithmetic sequence $\{a_n\}$ with the first term $a_1=1$ and a common difference $d>0$, and a geometric sequence $\{b_n\}$, satisfying $b_2=a_2$, $b_3=a_5$, $b_4=a_{14}$.
(1) Find the general term of the sequences $\{a_n\}$ and $\{b_n\}$.
(2) Let the sequence $\{c_n\}$ satisfy $c_n=2a_n-18$, find the mini... | -40 | 157 | 3 |
math | Three different one-digit positive integers are placed in the bottom row of cells. Numbers in adjacent cells are added and the sum is placed in the cell above them. In the second row, continue the same process to obtain a number in the top cell. What is the difference between the largest and smallest numbers possible i... | 26 | 265 | 2 |
math | Determine all positive integers $n{}$ for which there exist pairwise distinct integers $a_1,\ldots,a_n{}$ and $b_1,\ldots, b_n$ such that \[\prod_{i=1}^n(a_k^2+a_ia_k+b_i)=\prod_{i=1}^n(b_k^2+a_ib_k+b_i)=0, \quad \forall k=1,\ldots,n.\] | 2 | 102 | 1 |
math | Right triangle $ABC$ has its right angle at $A$ . A semicircle with center $O$ is inscribed inside triangle $ABC$ with the diameter along $AB$ . Let $D$ be the point where the semicircle is tangent to $BC$ . If $AD=4$ and $CO=5$ , find $\cos\angle{ABC}$ .
[asy]
import olympiad;
pair A, B, C, D, O;
A = ... | $\frac{4}{5}$ | 258 | 7 |
math | The coordinates of the point $P(-2,-4)$ with respect to the origin are given by the rule $(x, y) = (-x, -y)$. | (-2, -4) | 35 | 6 |
math | In the diagram shown, $\overrightarrow{OA}\perp\overrightarrow{OB}$ and $\overrightarrow{OC}\perp\overrightarrow{OD}$. If $\angle{BOD}$ is 4 times $\angle{AOC}$, what is $\angle{BOD}$? [asy]
unitsize(1.5cm);
defaultpen(linewidth(.7pt)+fontsize(10pt));
dotfactor=4;
pair O=(0,0), A=dir(0), B=dir(90), C=dir(50), D=dir(14... | 144^\circ | 233 | 5 |
math | Given the function \( f(x) = x^3 + ax^2 + bx + c \) where \( a, b, \) and \( c \) are nonzero integers, if \( f(a) = a^3 \) and \( f(b) = b^3 \), determine the value of \( c \). | 16 | 68 | 2 |
math | $(xy-k)^2 + (x+y-1)^2$ is minimized for real numbers $x$ and $y$, where k is a real constant. | 1 | 33 | 1 |
math | There are \(100\) countries participating in an olympiad. Suppose \(n\) is a positive integers such that each of the \(100\) countries is willing to communicate in exactly \(n\) languages. If each set of \(20\) countries can communicate in exactly one common language, and no language is common to all \(100\) countries,... | 20 | 87 | 2 |
math | There are some identical square pieces of paper. If a part of them is paired up to form rectangles with a length twice their width, the total perimeter of all the newly formed rectangles is equal to the total perimeter of the remaining squares. Additionally, the total perimeter of all shapes after pairing is 40 centime... | 280 | 86 | 3 |
math | Given the function $f(x)=a\ln x-bx^{2}(x > 0)$.
(1) If the function $f(x)$ is tangent to the line $y=- \frac {1}{2}$ at $x=1$, find the maximum value of the function $f(x)$ on $\[ \frac {1}{e},e\]$.
(2) When $b=0$, if the inequality $f(x)\geqslant m+x$ holds for all $a\in\[1, \frac {3}{2}\]$, $x\in\[1,e^{2}\]$, find th... | (-\infty,2-e^{2}] | 144 | 10 |
math | If the foci of the hyperbola $\frac{{y}^{2}}{2}-\frac{{x}^{2}}{m}=1$ coincide with the endpoints of the major axis of the ellipse $\frac{{x}^{2}}{3}+\frac{{y}^{2}}{4}=1$, determine the value of $m$. | 2 | 75 | 1 |
math | Given the function $f(x)=\sin (\omega x-\varphi)$ $(\omega > 0,|\varphi| < \frac {\pi}{2})$ whose graph intersects with the x-axis at points that are a distance of $\frac {\pi}{2}$ apart, and it passes through the point $(0,- \frac {1}{2})$
$(1)$ Find the analytical expression of the function $f(x)$;
$(2)$ Let the ... | \frac {1}{2} | 144 | 7 |
math | The sum of the binomial coefficients of all terms in the quadratic expansion of $$(2- \sqrt {x})^{n}$$ is 256. The coefficient of the $x^4$ term in the expansion is \_\_\_\_\_. | 1 | 54 | 1 |
math | Given a set \( A = \{a_1, a_2, \cdots, a_n\} \) consisting of \( n \) positive integers such that the sum of the elements of any two different subsets of set \( A \) is not equal, find the minimum value of \( \sum_{i=1}^{n} \sqrt{a_i} \). | (\sqrt{2} + 1)(\sqrt{2^n} - 1) | 80 | 19 |
math | Jonah’s five cousins are visiting and there are four identical rooms for them to stay in. If any number of cousins can occupy any room, how many different ways can the cousins be arranged among the rooms? | 51 | 42 | 2 |
math | Given the function $f(x)$ with domain $R$, and $f(x+y)+f(x-y)=f(x)f(y)$, $f(1)=1$, find $\sum_{k=1}^{22}f(k)$. | -3 | 51 | 2 |
math | Denis has cards with numbers from 1 to 50. How many ways are there to choose two cards such that the difference of the numbers on the cards is 11, and their product is divisible by 5?
The order of the selected cards does not matter: for example, selecting cards with numbers 5 and 16, as well as selecting cards with nu... | 15 | 91 | 2 |
math | Let \( a \) and \( b \) be non-zero real numbers and \( x \in \mathbb{R} \). If
\[
\frac{\sin^{4} x}{a^{2}} + \frac{\cos^{4} x}{b^{2}} = \frac{1}{a^{2} + b^{2}},
\]
find the value of \(\frac{\sin^{2008} x}{a^{2006}} + \frac{\cos^{2008} x}{b^{2006}}\). | \frac{1}{(a^2 + b^2)^{1003}} | 122 | 20 |
math | Jia spent 1000 yuan to buy a stock and then sold it to Yi at a 10% profit. Yi then sold the stock back to Jia at a 10% loss, and Jia finally sold the stock back to Yi at 90% of the price Yi sold it to Jia. Determine the net result of these transactions for Jia. | 1 | 80 | 1 |
math | A square piece of wood is sawn from an isosceles triangle wooden board with a base of 2m and a height of 3m. The square has one side coinciding with the base of the triangle. The area of this square piece of wood is ____ square meters. | \frac{36}{25} | 59 | 9 |
math | Consider two lines: line $l$ parametrized as
\[
\begin{align*}
x &= 2 + 5t,\\
y &= 3 + 2t,
\end{align*}
\]
and the line $m$ parametrized as
\[
\begin{align*}
x &= -7 + 5s,\\
y &= 9 + 2s.
\end{align*}
\]
Let $A$ be a point on line $l$, $B$ be a point on line $m$, and let $P$ be the foot of the perpendicular from $A$... | \begin{pmatrix} -2 \\ 5 \end{pmatrix} | 217 | 17 |
math | Given the sequence $a_n = \frac{n(n+1)}{2}$, remove all the numbers in the sequence $\{a_n\}$ that are divisible by 2, and arrange the remaining numbers in ascending order to form the sequence $\{b_n\}$. Find the value of $b_{51}$. | 5151 | 68 | 4 |
math | Given that the sum of the first $n$ terms of a geometric series with positive terms is denoted as $S_n$, and it is known that $S_3 = 3$ and $S_9 - S_6 = 12$, find the value of $S_6$. | S_6 = 9 | 62 | 6 |
math | The sum of three different numbers is 75. The two larger numbers differ by 5 and the two smaller numbers differ by 4. What is the value of the largest number? | \frac{89}{3} | 38 | 8 |
math | If the absolute value of a number is equal to $4$, then the number is ____. | \pm 4 | 19 | 4 |
math | Find all positive integers $n$ such that there exist two complete residue systems modulo $n$, $a_{i}$ and $b_{i}$ $(1 \leqslant i \leqslant n)$, such that the products $a_{i} b_{i}$ $(1 \leqslant i \leqslant n)$ also form a complete residue system modulo $n$. | n = 1 \text{ and } 2 | 84 | 11 |
math | The opposite of the number $2023$ is $-2023$. | -2023 | 19 | 5 |
math | Given the following five propositions:
1. If the equation $x^{2}+(a-3)x+a=0$ has one positive real root and one negative real root, then $a < 0$;
2. The function $y= \sqrt {x^{2}-1}+ \sqrt {1-x^{2}}$ is an even function, but not an odd function;
3. If the range of the function $f(x)$ is $[-2,2]$, then the range of the... | 1, 5 | 271 | 4 |
math | Given vectors $a=(1,2)$ and $b=(3,1)$, then the coordinates of $a+b$ are __________, and $a\cdot b=$ __________. | 5 | 40 | 1 |
math | Find all real numbers \(x\) which satisfy \[\frac{x^2 - x - 6}{x-4} \ge 3.\] (Give your answer in interval notation.) | (-\infty, 4) \cup (4, \infty) | 39 | 17 |
math | Given vectors $a$ and $b$ are in the same plane, where $a=(1,2)$ and $|\mathbf{b}|=2\sqrt{5}$.
(Ⅰ) If $a\parallel b$, find the coordinates of vector $b$;
(Ⅱ) If $(2a-3b)\cdot(2a+b)=-20$, find the value of the angle $\theta$ between $a$ and $b$. | \theta=\frac{2\pi}{3} | 98 | 11 |
math | Compute the number of real solutions $(x,y,z,w)$ to the system of equations:
\begin{align*}
x &= \sin(a+b+c), \\
y &= \sin(b+c+d), \\
z &= \sin(c+d+a), \\
w &= \sin(d+a+b),
\end{align*}
where \(a,b,c,d\) are angles measured in degrees. | 1 | 78 | 1 |
math | When a fair six-sided die is tossed on a table top, the bottom face cannot be seen. What is the probability that the product of the numbers on the five faces that can be seen is divisible by 6? | 1 | 46 | 1 |
math | Given the equation of an ellipse $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1(a>b>0)$$ where A and B are the left and right vertices, respectively, and F is the right focus. If the maximum distance from a point on the ellipse to the focus F is 3, and the eccentricity is a root of the equation $2x^2-5x+2=0$,
(1) Fin... | 6 | 158 | 1 |
math | Given that Jessie moves from 0 to 24 in six steps, and travels four steps to reach point x, then one more step to reach point z, and finally one last step to point y, calculate the value of y. | 24 | 48 | 2 |
math | Given 18 parking spaces in a row, 14 cars arrive and occupy spaces at random, followed by Auntie Em, who requires 2 adjacent spaces, determine the probability that the remaining spaces are sufficient for her to park. | \frac{113}{204} | 48 | 11 |
math | From the ten digits $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$, select nine digits to form a two-digit number, a three-digit number, and a four-digit number such that the sum of these three numbers equals 2010. What is the digit that is not selected? | 6 | 78 | 1 |
math | Given an integer \( n \geq 2 \). There are \( n \) sets written on the blackboard. The following operation is performed: select two sets \( A \) and \( B \) on the blackboard that do not contain each other, erase them, and then write \( A \cap B \) and \( A \cup B \). This is referred to as one operation. Continue perf... | \frac{n(n-1)}{2} | 115 | 10 |
math | Find the sum of the squares of the natural divisors of the number 1800. (For example, the sum of the squares of the natural divisors of the number 4 is \(1^{2} + 2^{2} + 4^{2} = 21\)). | 5035485 | 63 | 7 |
math | Find the value of
$$
\frac{x^{4}-6 x^{3}-2 x^{2}+18 x+23}{x^{2}-8 x+15}
$$
when \( x = \sqrt{19-8 \sqrt{3}} \). | 5 | 61 | 1 |
math | Triangle $ABC$ has $\angle BAC = 50^{\circ}$, $\angle CBA \leq 100^{\circ}$, $BC=2$, and $AC \geq AB$. Let $H$, $I$, and $O$ be the orthocenter, incenter, and circumcenter of $\triangle ABC$, respectively. Assume the area of pentagon $BCOIH$ is maximized. Find the value of $\angle CBA$. | 90^\circ | 101 | 4 |
math | Given that the number $121_b$, written in the integral base $b$, is the square of an integer, determine the possible values of $b$. | b > 2 | 33 | 4 |
math | Given circle $C$: $(x-2)^2 + y^2 = 4$, and line $l$: $x - \sqrt{3}y = 0$, the probability that the distance from point $A$ on circle $C$ to line $l$ is not greater than $1$ is $\_\_\_\_\_\_\_.$ | \frac{1}{2} | 74 | 7 |
math | Given a function $f(x)$ defined on $\mathbb{R}$ and satisfying $f(\frac{3}{2}+x)=f(\frac{3}{2}-x)$, with $f(-1)=1$ and $f(0)=-2$, calculate the value of $f(1)+f(2)+f(3)+...+f(2016)$. | 0 | 84 | 1 |
math | Let $ABC$ be a triangle with $\angle BAC=60^\circ$ . Consider a point $P$ inside the triangle having $PA=1$ , $PB=2$ and $PC=3$ . Find the maximum possible area of the triangle $ABC$ . | \frac{3\sqrt{3}}{2} | 69 | 12 |
math | We use $\left[m\right]$ to represent the largest integer not greater than $m$, for example: $\left[2\right]=2$, $\left[4.1\right]=4$, $\left[3.99\right]=3$.
$(1) [\sqrt{2}]=$____;
$(2)$ If $[3+\sqrt{x}]=6$, then the range of $x$ is ____. | 9 \leq x < 16 | 91 | 9 |
math | The probability it will rain on Friday is $30\%$, the probability it will rain on Saturday is $50\%$, and the probability it will rain on Sunday is $40\%$. Assuming the probability that it rains on any day is independent of the other days except that if it rains on Saturday, the probability it will rain on Sunday incre... | 10.5\% | 98 | 6 |
math | Given functions $f(x)=\frac{a}{x}-e^{x}$ and $g(x)=x^{2}-2x-1$, if for any $x_{1}\in \left[\frac{1}{2},2\right]$, there exists $x_{2}\in \left[\frac{1}{2},2\right]$ such that $f(x_{1})-g(x_{2})\geqslant 1$, determine the range of real number $a$. | [2e^2 - 2, +\infty) | 106 | 14 |
math | Given a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (where $a > b > 0$), let $F$ be the left focus. A line passing through point $F$ with a slope of $1$ intersects each of the two asymptotes at points $A$ and $B$, respectively. If $\frac{|AF|}{|BF|} = \frac{1}{2}$, find the eccentricity of the hyperbola. | \sqrt{10} | 114 | 6 |
math | The volume of the tetrahedron \(ABCD\) is 5. A plane passes through the midpoints of edges \(AD\) and \(BC\) and intersects edge \(CD\) at point \(M\). The ratio of the length of segment \(DM\) to the length of segment \(CM\) is \(2/3\). Calculate the area of the cross-section of the tetrahedron formed by this plane, g... | 3 | 103 | 1 |
math | The final result of the following problems is represented by a number:
1. How many different five-digit even numbers can be formed using the digits 0, 1, 2, 3, and 4?
2. How many different five-digit numbers can be formed using the digits 1, 2, 3, 4, and 5 such that 2 and 3 are not adjacent?
3. How many different five... | 20 | 133 | 2 |
math | Evaluate the number of subsets of the set $\{1, 2, 3, \dots, 15\}$, including the empty subset, that are "spacy." A set is called "spacy" if it contains no more than one out of any three consecutive integers. | 406 | 60 | 3 |
math | In a rhombus $P Q R S$ with $P Q=Q R=R S=S P=S Q=6$ and $P T=R T=14$, what is the length of $S T$? | 10 | 47 | 2 |
math | Forty slips of paper numbered $1$ to $40$ are placed in a hat. Alice and Bob each draw one number from the hat without replacement, keeping their numbers hidden from each other. Alice says, "I can't tell who has the larger number." Then Bob says, "I know who has the larger number." Alice says, "You do? Is your number p... | 27 | 131 | 2 |
math | A tractor has $50L$ of oil in the tank before starting work. After starting work, it consumes $8L$ of oil per hour.
$(1)$ Write down the functional relationship between the remaining oil $W\left(L\right)$ in the tank and the working time $t\left(h\right)$.
$(2)$ After working for $4$ hours, how many liters of oil a... | 18 \text{ liters} | 92 | 7 |
math | Please write down the analytical expression of a function whose graph passes through the positive half-axis of the $y$-axis. | y = x + 1 | 25 | 6 |
math | Determine the probability that the yellow ball ends up in a higher-numbered bin than the blue ball. | \frac{2}{5} | 21 | 7 |
math | The sequence $\left\{a_{n}\right\}$ satisfies the conditions: $a_{1}=1$, $\frac{a_{2k}}{a_{2k-1}}=2$, $\frac{a_{2k+1}}{a_{2k}}=3$, for $k \geq 1$. What is the sum of the first 100 terms, $S_{100}$? | \frac{3}{5}(6^{50} - 1) | 92 | 16 |
math | A star-polygon is drawn on a clock face by drawing a chord from each number to the fifth number counted clockwise from that number. That is, chords are drawn from 12 to 5, from 5 to 10, from 10 to 3, and so on, ending back at 12. What is the degree measure of the angle at each vertex in the star-polygon? | 30 | 87 | 2 |
math | Let $\overline{PQ}$ be a diameter of a circle with diameter 2. Let $X$ and $Y$ be points on one of the semicircular arcs determined by $\overline{PQ}$ such that $X$ is the midpoint of the semicircle and $PY=\frac{4}{5}$. Point $Z$ lies on the other semicircular arc. Let $e$ be the length of the line segment whose endpo... | 13 + 6 + 5 = 24 | 176 | 12 |
math | Given the function $f(x)=|\frac{1+2x}{x}|+|\frac{1-2x}{x}|$ has a minimum value of $m$.
$(1)$ Find the value of $m$;
$(2)$ If $a \gt 0$, $b \gt 0$, and $a+b=m$, find the maximum value of $\frac{ab}{a+4b}$. | \frac{4}{9} | 90 | 7 |
math | In this addition problem, each letter stands for a different digit.
$\begin{array}{cccc}&T & W & O\\ +&T & W & O\\ \hline F& O & U & R\end{array}$
If T = 7 and the letter O represents an even number, what is the only possible value for W? | 3 | 73 | 1 |
math | In square $XYZW$, points $P$ and $S$ lie on $\overline{XZ}$ and $\overline{XW}$, respectively, such that $XP=XS=\sqrt{3}$. Points $Q$ and $R$ lie on $\overline{YZ}$ and $\overline{YW}$, respectively, and points $T$ and $U$ lie on $\overline{PS}$ so that $\overline{QT} \perp \overline{PS}$ and $\overline{RU} \perp \over... | 3 | 436 | 1 |
math | In the rectangular coordinate system, points A and B move along the x-axis and y-axis, respectively, and a circle C, with AB as its diameter, is tangent to the line $x + y - 4 = 0$. Calculate the minimum area of circle C. | 2π | 56 | 2 |
math | Given that $M$ is any point on the segment $AB$, $O$ is a point outside the line $AB$, $C$ is the symmetric point of $A$ with respect to point $O$, and $D$ is the symmetric point of $B$ with respect to point $C$. If $\overrightarrow{OM}=x\overrightarrow{OC}+y\overrightarrow{OD}$, then $x+3y=\_\_\_\_\_\_$. | -1 | 100 | 2 |
math | Let proposition p: For all $x \in \mathbb{R}$, $ax^2 > -ax - 1$ ($a \neq 0$) always holds; proposition q: The circle $x^2 + y^2 = a^2$ is externally disjoint from the circle $(x + 3)^2 + (y - 4)^2 = 4$. If the proposition “$p \lor q$” is true, and “$p \land q$” is false, find the range of the real number $a$. | (-3, 0] \cup [3, 4) | 119 | 14 |
math | Given that $f(x)$ is an even function, where $x \in \mathbb{R}$, if the graph of $f(x)$ is shifted one unit to the right and then becomes an odd function, and $f(2) = -1$, determine the value of $f(1) + f(2) + f(3) + \ldots + f(2010)$. | -1 | 87 | 2 |
math | A rectangular prism has dimensions 16 inches by 4 inches by 24 inches. If a cube has the same volume as this prism, what is the surface area of the cube, in square inches? | 798 \text{ square inches} | 43 | 9 |
math | An $11 \times 11 \times 11$ cube is made up of $11^{3}$ unit cubes. Calculate the maximum number of unit cubes that can be seen from one viewpoint. | 331 | 44 | 3 |
math | Given the price of an item increased by 30% in February, decreased by 10% in March, increased by 15% in April, and then decreased by $y\%$ in May, determine the percentage $y$ such that the price of the item at the end of May is the same as it had been at the beginning of February. | 26 | 76 | 2 |
math | In the number $2 * 0 * 1 * 6 * 0 *$, each of the 5 asterisks needs to be replaced with any digit from $0,1,2,3,4,5,6,7,8$ (digits can repeat) so that the resulting 10-digit number is divisible by 45. How many ways can this be done? | 1458 | 82 | 4 |
math | Given point A (-2, 1), the focus of the parabola $y^2 = -4x$ is F, and P is a point on the parabola $y^2 = -4x$. To minimize the value of $|PA| + |PF|$, the coordinates of point P are \_\_\_\_\_\_. | \left(-\frac{1}{4}, 1\right) | 74 | 15 |
math | A 10 by 10 checkerboard has alternating black and white squares. How many distinct squares, with sides on the grid lines of the checkerboard (horizontal and vertical) and containing at least 6 black squares, can be drawn on the checkerboard? | 140 | 54 | 3 |
math | Find all $(m,n) \in \mathbb{Z}^2$ that we can color each unit square of $m \times n$ with the colors black and white that for each unit square number of unit squares that have the same color with it and have at least one common vertex (including itself) is even. | (m, n) \in \mathbb{Z}^2 | 71 | 14 |
math | Calculate the volume of the tetrahedron with vertices at points \(A_{1}, A_{2}, A_{3}, A_{4}\) and its height dropped from vertex \(A_{4}\) onto the face \(A_{1} A_{2} A_{3}\).
\(A_{1}(0 ;-3 ; 1)\)
\(A_{2}(-4 ; 1 ; 2)\)
\(A_{3}(2 ;-1 ; 5)\)
\(A_{4}(3 ; 1 ;-4)\) | \sqrt{\frac{97}{2}} | 114 | 10 |
math | In the number \(2 * 0 * 1 * 6 * 07 *\), each of the 5 asterisks needs to be replaced with any of the digits \(0, 2, 4, 5, 6, 7\) (digits may be repeated) so that the resulting 11-digit number is divisible by 75. In how many ways can this be done? | 432 | 85 | 3 |
math | Convert the decimal number 2015 into binary. | 11111011111_{(2)} | 12 | 15 |
math | In two boxes, A and B, there are four balls each, labeled with the numbers \\(1\\), \\(2\\), \\(3\\), and \\(4\\). Now, one ball is drawn from each of the two boxes, and each ball has an equal chance of being drawn.
\\((1)\\) Calculate the probability that the numbers on the two drawn balls are consecutive integers.
... | \dfrac{5}{16} | 112 | 8 |
math | A regular triangular prism \( ABC A_1 B_1 C_1 \) is inscribed in a sphere. The base of the prism is \(ABC \) and the lateral edges are \(AA_1, BB_1, CC_1\). Segment \(CD\) is the diameter of this sphere, and points \(K\) and \(L\) are the midpoints of edge \(AA_1\) and \(AB\) respectively. Find the volume of the prism ... | 4 | 116 | 1 |
math | Given that the coefficients $p$ and $q$ are integers and the roots $\alpha_{1}$ and $\alpha_{2}$ are irrational, a quadratic trinomial $x^{2} + px + q$ is called an irrational quadratic trinomial. Determine the minimum sum of the absolute values of the roots among all irrational quadratic trinomials. | \sqrt{5} | 74 | 5 |
math | Given the number of balls and bins, calculate $\frac{p}{q}$ where $p$ is the probability that one bin ends up with 2 balls, two bins with 4 balls each, and the remaining three bins with 5 balls each, and $q$ is the probability that two bins end up with 3 balls each and the remaining four with 4 balls each. | 6 | 79 | 1 |
math | A 10x10 arrangement of alternating black and white squares has a black square $R$ in the second-bottom row and a white square $S$ in the top-most row. Given that a marker is initially placed at $R$ and can move to an immediately adjoining white square on the row above either to the left or right, and the path must cons... | 70 | 95 | 2 |
math | Given that one root of the equation $x^2+mx+3=0$ is 1, find the other root and the value of $m$. | -4 | 33 | 2 |
math | Given $z_{1}$, $z_{2} \in \mathbb{C}$ and $z_{1} = i\overline{{z_2}}$ (where $i$ is the imaginary unit), satisfying $|z_{1}-1| = 1$, then the range of $|z_{1}-z_{2}|$ is ____. | [0, 2 + \sqrt{2}] | 78 | 11 |
math | Given that point $P$ lies on the line $y=2x$, if there exist two points $A$ and $B$ on the circle $C(x-3)^2+y^2=4$ such that $PA \perp PB$, then the range of the abscissa $x_{0}$ of point $P$ is _______. | \left[ \frac{1}{5},1\right] | 74 | 14 |
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