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25,200 | Points $P$, $Q$, $R$, and $S$ lie in the plane of the square $EFGH$ so that $EPF$, $FQG$, $GRH$, and $HSE$ are equilateral triangles inside square $EFGH$. If the side length of square $EFGH$ is 8, find the area of quadrilateral $PQRS$. Assume that the vertices $P$, $Q$, $R$, and $S$ lie on the sides of square $EFGH$, respectively. | 48 | 2.34375 |
25,201 | When Julia divides her apples into groups of nine, ten, or eleven, she has two apples left over. Assuming Julia has more than two apples, what is the smallest possible number of apples in Julia's collection? | 200 | 0 |
25,202 | Given that the coefficient of $x^2$ in the expansion of $(1+ax)(1+x)^5$ is $5$, determine the value of $a$. | -1 | 95.3125 |
25,203 | The decimal $3.834834834\ldots$ can be written as a fraction. When reduced to lowest terms, find the sum of the numerator and denominator of this fraction. | 4830 | 0 |
25,204 | What is the smallest positive odd integer having the same number of positive divisors as 360? | 31185 | 0 |
25,205 | The diagram shows a square, its two diagonals, and two line segments, each of which connects two midpoints of the sides of the square. What fraction of the area of the square is shaded? | $\frac{1}{16}$ | 0 |
25,206 | In triangle $ABC$, $AX = XY = YB = \frac{1}{2}BC$ and $AB = 2BC$. If the measure of angle $ABC$ is 90 degrees, what is the measure of angle $BAC$? | 22.5 | 0 |
25,207 | Let set $A=\{x|\left(\frac{1}{2}\right)^{x^2-4}>1\}$, $B=\{x|2<\frac{4}{x+3}\}$
(1) Find $A\cap B$
(2) If the solution set of the inequality $2x^2+ax+b<0$ is $B$, find the values of $a$ and $b$. | -6 | 0 |
25,208 | Use Horner's method to find the value of the polynomial $f(x) = -6x^4 + 5x^3 + 2x + 6$ at $x=3$, denoted as $v_3$. | -115 | 0 |
25,209 | When a spring is stretched by 5 cm, 29.43 J of work is done. How far will the spring stretch if 9.81 J of work is done? | 0.029 | 0 |
25,210 | Let \(\mathbf{a} = \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}\), \(\mathbf{b} = \begin{pmatrix} 4 \\ 5 \\ -2 \end{pmatrix}\), and \(\mathbf{c} = \begin{pmatrix} 1 \\ 2 \\ 5 \end{pmatrix}\). Compute the following:
\[
((\mathbf{a} + \mathbf{c}) - \mathbf{b}) \cdot [(\mathbf{b} - \mathbf{c}) \times (\mathbf{c} - \mathbf{a})].
\] | 161 | 0 |
25,211 | Given 1985 sets, each consisting of 45 elements, and the union of any two sets contains exactly 89 elements. How many elements are in the union of all these 1985 sets? | 87341 | 6.25 |
25,212 | A line passing through any two vertices of a cube has a total of 28 lines. Calculate the number of pairs of skew lines among them. | 174 | 21.875 |
25,213 | The function $g(x)$ satisfies
\[g(3^x) + 2xg(3^{-x}) = 3\] for all real numbers $x$. Find $g(3)$. | -3 | 7.03125 |
25,214 | How many distinct four-digit positive integers are there such that the product of their digits equals 8? | 22 | 0 |
25,215 | In a two-day problem-solving competition, Gamma and Delta participated and attempted questions worth a total of 500 points. On the first day, Gamma scored 180 points out of 280 points attempted, and on the second day, he scored 120 points out of 220 points attempted. Delta, who also divided his attempts across the two days totaling 500 points, achieved a positive integer score each day. Delta's daily success ratio (points scored divided by points attempted) on each day was less than Gamma's on that day. Gamma's overall success ratio over the two days was $300/500 = 3/5$.
Determine the highest possible two-day success ratio that Delta could have achieved. | \frac{409}{500} | 0 |
25,216 | Angle bisectors $AA', BB'$ and $CC'$ are drawn in triangle $ABC$ with angle $\angle B= 120^o$ . Find $\angle A'B'C'$ . | 90 | 3.125 |
25,217 | Given the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with its right focus at $(\sqrt{3}, 0)$, and passing through the point $(-1, \frac{\sqrt{3}}{2})$. Point $M$ is on the $x$-axis, and the line $l$ passing through $M$ intersects the ellipse $C$ at points $A$ and $B$ (with point $A$ above the $x$-axis).
(I) Find the equation of the ellipse $C$;
(II) If $|AM| = 2|MB|$, and the line $l$ is tangent to the circle $O: x^2 + y^2 = \frac{4}{7}$ at point $N$, find the length of $|MN|$. | \frac{4\sqrt{21}}{21} | 0 |
25,218 | Calculate $\fbox{2,3,-1}$. | \frac{26}{3} | 0 |
25,219 | In a large box of ribbons, $\frac{1}{4}$ are yellow, $\frac{1}{3}$ are purple, $\frac{1}{6}$ are orange, and the remaining 40 are black. How many of the ribbons are orange? | 27 | 23.4375 |
25,220 | How many positive integers $n$ less than $1000$ have the property that the number of positive integers less than $n$ which are coprime to $n$ is exactly $\tfrac n3$ ? | 41 | 0 |
25,221 | Define a regular \(n\)-pointed star as described in the original problem, but with a modification: the vertex connection rule skips by \(m\) steps where \(m\) is coprime with \(n\) and \(m\) is not a multiple of \(3\). How many non-similar regular 120-pointed stars adhere to this new rule? | 15 | 14.84375 |
25,222 | Compute the multiplicative inverse of $185$ modulo $341$. Express your answer as an integer from $0$ to $340$. | 74466 | 0 |
25,223 | Let $T_i$ be the set of all integers $n$ such that $50i \leq n < 50(i + 1)$. For example, $T_4$ is the set $\{200, 201, 202, \ldots, 249\}$. How many of the sets $T_0, T_1, T_2, \ldots, T_{1999}$ do not contain a perfect square? | 1733 | 0 |
25,224 | In the Cartesian coordinate system $xoy$, given point $A(0,-2)$, point $B(1,-1)$, and $P$ is a moving point on the circle $x^{2}+y^{2}=2$, then the maximum value of $\dfrac{|\overrightarrow{PB}|}{|\overrightarrow{PA}|}$ is ______. | \dfrac{3\sqrt{2}}{2} | 0 |
25,225 | Given that Alice is 1.6 meters tall, can reach 50 centimeters above her head, and the ceiling is 3 meters tall, find the minimum height of the stool in centimeters needed for her to reach a ceiling fan switch located 15 centimeters below the ceiling. | 63 | 0 |
25,226 | Solve the following cryptarithm ensuring that identical letters correspond to identical digits:
$$
\begin{array}{r}
\text { К O Ш К A } \\
+ \text { К O Ш К A } \\
\text { К O Ш К A } \\
\hline \text { С О Б А К А }
\end{array}
$$ | 50350 | 0 |
25,227 | The quadratic $x^2 + 900x + 1800$ can be written in the form $(x+b)^2 + c$, where $b$ and $c$ are constants. What is $\frac{c}{b}$? | -446.\overline{2} | 0 |
25,228 | Given a fixed circle $\odot P$ with a radius of 1, the distance from the center $P$ to a fixed line $l$ is 2. Point $Q$ is a moving point on $l$, and circle $\odot Q$ is externally tangent to circle $\odot P$. Circle $\odot Q$ intersects $l$ at points $M$ and $N$. For any diameter $MN$, there is always a fixed point $A$ on the plane such that the angle $\angle MAN$ is a constant value. Find the degree measure of $\angle MAN$. | 60 | 14.84375 |
25,229 | Given that there are 5 balls in a pocket, among which there are 2 black balls and 3 white balls, calculate the probability that two randomly drawn balls of the same color are both white. | \frac{3}{4} | 0 |
25,230 | In a slightly larger weekend softball tournament, five teams (A, B, C, D, E) are participating. On Saturday, Team A plays Team B, Team C plays Team D, and Team E will automatically advance to the semi-final round. On Sunday, the winners of A vs B and C vs D play each other (including E), resulting in one winner, while the remaining two teams (one from initial losers and Loser of semifinal of E's match) play for third and fourth places. The sixth place is reserved for the loser of the losers' game. One possible ranking of the teams from first place to sixth place at the end of this tournament is the sequence AECDBF. What is the total number of possible six-team ranking sequences at the end of the tournament? | 32 | 8.59375 |
25,231 | Given plane vectors $\vec{a}, \vec{b}, \vec{c}$ satisfying $|\vec{a}| = |\vec{b}|$, $\vec{c} = \lambda \vec{a} + \mu \vec{b}$, $|\vec{c}| = 1 + \vec{a} \cdot \vec{b}$, and $(\vec{a} + \vec{b}) \cdot \vec{c} = 1$, find the minimum possible value of $\frac{|\vec{a} - \vec{c}|}{|1 + \mu|}$. | 2 - \sqrt{2} | 0 |
25,232 | Find the area of a trapezoid, whose diagonals are 7 and 8, and whose bases are 3 and 6. | 4 : 3 | 0 |
25,233 | There are 7 cylindrical logs with a diameter of 5 decimeters each. They are tied together at two places using a rope. How many decimeters of rope are needed at least (excluding the rope length at the knots, and using $\pi$ as 3.14)? | 91.4 | 0 |
25,234 | To obtain the graph of the function $y=\cos \left( \frac{1}{2}x+ \frac{\pi}{6}\right)$, determine the necessary horizontal shift of the graph of the function $y=\cos \frac{1}{2}x$. | \frac{\pi}{6} | 0 |
25,235 | A circle of radius $r$ has chords $\overline{AB}$ of length $12$ and $\overline{CD}$ of length $9$. When $\overline{AB}$ and $\overline{CD}$ are extended through $B$ and $C$, respectively, they intersect at point $P$, which is outside of the circle. If $\angle{APD}=90^\circ$ and $BP=10$, determine $r^2$. | 221 | 9.375 |
25,236 | Determine the radius $r$ of a circle inscribed within three mutually externally tangent circles of radii $a = 5$, $b = 10$, and $c = 20$ using the formula:
\[
\frac{1}{r} = \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + 2 \sqrt{\frac{1}{ab} + \frac{1}{ac} + \frac{1}{bc}}.
\] | 1.381 | 0 |
25,237 | Find \(a\) in the following sequence: \(1, 8, 27, 64, a, 216, \ldots \ldots\)
\[1^{3}, 2^{3}, 3^{3}, 4^{3}, a, 6^{3}, \ldots \ldots\] | 16 | 0 |
25,238 | How many ways are there to arrange the $6$ permutations of the tuple $(1, 2, 3)$ in a sequence, such that each pair of adjacent permutations contains at least one entry in common?
For example, a valid such sequence is given by $(3, 2, 1) - (2, 3, 1) - (2, 1, 3) - (1, 2, 3) - (1, 3, 2) - (3, 1, 2)$ . | 144 | 0 |
25,239 | Let $f(x)$ be an even function defined on $\mathbb{R}$, which satisfies $f(x+1) = f(x-1)$ for any $x \in \mathbb{R}$. If $f(x) = 2^{x-1}$ for $x \in [0,1]$, then determine the correctness of the following statements:
(1) 2 is a period of the function $f(x)$.
(2) The function $f(x)$ is increasing on the interval (2, 3).
(3) The maximum value of the function $f(x)$ is 1, and the minimum value is 0.
(4) The line $x=2$ is an axis of symmetry for the function $f(x)$.
Identify the correct statements. | \frac{1}{2} | 0 |
25,240 | The specific heat capacity of a body with mass \( m = 3 \) kg depends on the temperature in the following way: \( c = c_{0}(1 + \alpha t) \), where \( c_{0} = 200 \) J/kg·°C is the specific heat capacity at \( 0^{\circ} \mathrm{C} \), \( \alpha = 0.05 \,^{\circ} \mathrm{C}^{-1} \) is the temperature coefficient, and \( t \) is the temperature in degrees Celsius. Determine the amount of heat that needs to be transferred to this body to heat it from \( 30^{\circ} \mathrm{C} \) to \( 80^{\circ} \mathrm{C} \). | 112.5 | 0 |
25,241 | What is the minimum number of cells that need to be marked in a $7 \times 7$ grid so that in each vertical or horizontal $1 \times 4$ strip there is at least one marked cell? | 12 | 27.34375 |
25,242 | On the board, the product of the numbers $\overline{\text{IKS}}$ and $\overline{\text{KSI}}$ is written, where the letters correspond to different non-zero decimal digits. This product is a six-digit number and ends with S. Vasya erased all the zeros from the board, after which only IKS remained. What was written on the board? | 100602 | 17.1875 |
25,243 | There are many ways in which the list \(0,1,2,3,4,5,6,7,8,9\) can be separated into groups. For example, this list could be separated into the four groups \(\{0,3,4,8\}\), \(\{1,2,7\}\), \{6\}, and \{5,9\}. The sum of the numbers in each of these four groups is \(15\), \(10\), \(6\), and \(14\), respectively. In how many ways can the list \(0,1,2,3,4,5,6,7,8,9\) be separated into at least two groups so that the sum of the numbers in each group is the same? | 32 | 0 |
25,244 | Among the three-digit numbers formed by the digits 1, 2, 3, 4, 5, 6 with repetition allowed, how many three-digit numbers have exactly two different even digits (for example: 124, 224, 464, …)? (Answer with a number). | 72 | 7.03125 |
25,245 | Let $f(x) = x^4 + 20x^3 + 150x^2 + 500x + 625$. Let $z_1, z_2, z_3, z_4$ be the four roots of $f$. Find the smallest possible value of $|z_a^2 + z_b^2 + z_cz_d|$ where $\{a, b, c, d\} = \{1, 2, 3, 4\}$. | 1875 | 0 |
25,246 | 50 people, consisting of 30 people who all know each other, and 20 people who know no one, are present at a conference. Determine the number of handshakes that occur among the individuals who don't know each other. | 1170 | 0 |
25,247 | Compute $\tan 5105^\circ$. | 11.430 | 0 |
25,248 | Let $f(x)=\frac{\cos 2x-(a+2)\cos x+a+1}{\sin x}.$ For constant $a$ such that $\lim_{x\rightarrow 0} \frac{f(x)}{x}=\frac 12$ , evaluate $\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{1}{f(x)}dx.$ | \frac{1}{2} | 6.25 |
25,249 | Given that a flower bouquet contains pink roses, red roses, pink tulips, and red tulips, and that one fourth of the pink flowers are roses, one third of the red flowers are tulips, and seven tenths of the flowers are red, calculate the percentage of the flowers that are tulips. | 46\% | 0 |
25,250 | In the diagram, four circles with centers $P$, $Q$, $R$, and $S$ each have a radius of 2. These circles are tangent to one another and to the sides of $\triangle ABC$ as shown. The circles centered at $P$ and $Q$ are tangent to side $AB$, the circle at $R$ is tangent to side $BC$, and the circle at $S$ is tangent to side $AC$. Determine the perimeter of $\triangle ABC$ if $\triangle ABC$ is isosceles with $AB = AC$. | 36 | 1.5625 |
25,251 | What is the instantaneous velocity of the robot at the moment $t=2$ given the robot's motion equation $s = t + \frac{3}{t}$? | \frac{13}{4} | 0 |
25,252 | Compute \[\sum_{k=2}^{31} \log_3\left(1 + \frac{2}{k}\right) \log_k 3 \log_{k+1} 3.\] | \frac{0.5285}{3} | 0 |
25,253 | \(x, y\) are real numbers, \(z_{1}=x+\sqrt{11}+yi\), \(z_{6}=x-\sqrt{11}+yi\) (where \(i\) is the imaginary unit). Find \(|z_{1}| + |z_{6}|\). | 30(\sqrt{2} + 1) | 0 |
25,254 | Define the function $g$ on the set of integers such that \[g(n)= \begin{cases} n-4 & \mbox{if } n \geq 2000 \\ g(g(n+6)) & \mbox{if } n < 2000. \end{cases}\] Determine $g(172)$. | 2000 | 0 |
25,255 | Find the value of $\cos(-\pi - \alpha)$ given a point $P(-3, 4)$ on the terminal side of angle $\alpha$. | -\dfrac{3}{5} | 0 |
25,256 | In space, there are four spheres with radii 2, 2, 3, and 3. Each sphere is externally tangent to the other three spheres. Additionally, there is a smaller sphere that is externally tangent to all four of these spheres. Find the radius of this smaller sphere. | \frac{6}{2 - \sqrt{26}} | 0 |
25,257 | Given that the focus of the parabola $y^{2}=ax$ coincides with the left focus of the ellipse $\frac{x^{2}}{6}+ \frac{y^{2}}{2}=1$, find the value of $a$. | -16 | 0 |
25,258 | Let \(a, b\), and \(c\) be the roots of the cubic polynomial \(3x^3 - 4x^2 + 100x - 3\). Compute \[(a+b+2)^2 + (b+c+2)^2 + (c+a+2)^2.\] | 119.888... | 0 |
25,259 | Let (a,b,c,d) be an ordered quadruple of not necessarily distinct integers, each one of them in the set {0,1,2,3,4}. Determine the number of such quadruples that make the expression $a \cdot d - b \cdot c + 1$ even. | 136 | 0 |
25,260 | A store sold an air conditioner for 2000 yuan and a color TV for 2000 yuan. The air conditioner made a 30% profit, while the color TV incurred a 20% loss. Could you help the store owner calculate whether the store made a profit or a loss on this transaction, and by how much? | 38.5 | 0 |
25,261 | George is trying to find the Fermat point $P$ of $\triangle ABC$, where $A$ is at the origin, $B$ is at $(10,2)$, and $C$ is at $(5,4)$. He guesses that the point is at $P = (3,1)$. Compute the sum of the distances from $P$ to the vertices of $\triangle ABC$. If he obtains $x + y\sqrt{z}$, where $x$, $y$, and $z$ are integers, what is $x + y + z$? | 16 | 0 |
25,262 | In a \(10 \times 10\) grid (where the sides of the cells have a unit length), \(n\) cells are selected, and a diagonal is drawn in each of them with an arrow pointing in one of two directions. It turns out that for any two arrows, either the end of one coincides with the beginning of the other, or the distance between their ends is at least 2. What is the largest possible \(n\)? | 48 | 0 |
25,263 | If $a$ and $b$ are positive integers such that $ab - 7a + 6b = 559$, what is the minimal possible value of $|a - b|$? | 587 | 46.875 |
25,264 | Seven frogs are sitting in a row. They come in four colors: two green, two red, two yellow, and one blue. Green frogs refuse to sit next to red frogs, and yellow frogs refuse to sit next to blue frogs. In how many ways can the frogs be positioned respecting these restrictions? | 16 | 1.5625 |
25,265 | During a survey of 500 people, it was found that $46\%$ of the respondents like strawberry ice cream, $71\%$ like vanilla ice cream, and $85\%$ like chocolate ice cream. Are there at least six respondents who like all three types of ice cream? | 10 | 0 |
25,266 | In the coordinate plane, a point is called a $\text{lattice point}$ if both of its coordinates are integers. Let $A$ be the point $(12,84)$ . Find the number of right angled triangles $ABC$ in the coordinate plane $B$ and $C$ are lattice points, having a right angle at vertex $A$ and whose incenter is at the origin $(0,0)$ . | 18 | 0 |
25,267 | In isosceles $\triangle ABC$ with $AB=AC$, let $D$ be the midpoint of $AC$ and $BD=1$. Find the maximum area of $\triangle ABC$. | \frac{2\sqrt{2}}{3} | 0 |
25,268 | The minimum value of $\frac{b^{2}+1}{\sqrt{3}a}$ is what occurs when the eccentricity of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ is 2. | \frac {4 \sqrt {3}}{3} | 0 |
25,269 | Compute $\tan(-405^\circ)$. | -1 | 100 |
25,270 | If the surface area of a cone is $3\pi$, and its lateral surface unfolds into a semicircle, then the diameter of the base of the cone is ___. | \sqrt{6} | 3.125 |
25,271 | The net change in the population over these four years is a 20% increase, then a 30% decrease, then a 20% increase, and finally a 30% decrease. Calculate the net change in the population over these four years. | -29 | 0 |
25,272 | Let a $9$ -digit number be balanced if it has all numerals $1$ to $9$ . Let $S$ be the sequence of the numerals which is constructed by writing all balanced numbers in increasing order consecutively. Find the least possible value of $k$ such that any two subsequences of $S$ which has consecutive $k$ numerals are different from each other. | 17 | 0 |
25,273 | Find the largest prime divisor of \( 16^2 + 81^2 \). | 53 | 0 |
25,274 | Regular decagon `ABCDEFGHIJ` has its center at `K`. Each of the vertices and the center are to be associated with one of the digits `1` through `10`, with each digit used once, in such a way that the sums of the numbers on the lines `AKE`, `BKF`, `CKG`, `DLH` and `EJI` are all equal. In how many ways can this be done? | 3840 | 0 |
25,275 | Let $a,$ $b,$ and $c$ be positive real numbers such that $ab^2c^3 = 256$. Find the minimum value of
\[a^2 + 8ab + 16b^2 + 2c^5.\] | 768 | 0 |
25,276 | Consider a convex pentagon $FGHIJ$ where $\angle F = \angle G = 100^\circ$. Let $FI = IJ = JG = 3$ and $GH = HF = 5$. Calculate the area of pentagon $FGHIJ$. | \frac{9\sqrt{3}}{4} + 24.62 | 0 |
25,277 | Let $\triangle PQR$ have side lengths $PQ=13$, $PR=15$, and $QR=14$. Inside $\angle QPR$ are two circles: one is tangent to rays $\overline{PQ}$, $\overline{PR}$, and segment $\overline{QR}$; the other is tangent to the extensions of $\overline{PQ}$ and $\overline{PR}$ beyond $Q$ and $R$, and also tangent to $\overline{QR}$. Compute the distance between the centers of these two circles. | 5\sqrt{13} | 0 |
25,278 | The equation of one of the axes of symmetry for the graph of the function $f(x)=\sin \left(x- \frac {\pi}{4}\right)$ $(x\in\mathbb{R})$ can be found. | -\frac{\pi}{4} | 0.78125 |
25,279 | In acute triangle \( ABC \), \( M \) and \( N \) are the midpoints of sides \( AB \) and \( BC \), respectively. The tangents to the circumcircle of triangle \( BMN \) at \( M \) and \( N \) meet at \( P \). Suppose that \( AP \) is parallel to \( BC \), \( AP = 9 \), and \( PN = 15 \). Find \( AC \). | 20\sqrt{2} | 0 |
25,280 | Quadrilateral $ABCD$ is a parallelogram with an area of $50$ square units. Points $P$ and $Q$ are located on sides $AB$ and $CD$ respectively, such that $AP = \frac{1}{3}AB$ and $CQ = \frac{2}{3}CD$. What is the area of triangle $APD$? | 16.67 | 0 |
25,281 | In the decimal representation of the even number \( M \), only the digits \( 0, 2, 4, 5, 7, \) and \( 9 \) participate, and digits may repeat. It is known that the sum of the digits of the number \( 2M \) is 31, and the sum of the digits of the number \( M / 2 \) is 28. What values can the sum of the digits of the number \( M \) have? List all possible answers. | 29 | 38.28125 |
25,282 | A hyperbola has its center at the origin O, with its foci on the x-axis and two asymptotes denoted as l₁ and l₂. A line perpendicular to l₁ passes through the right focus F intersecting l₁ and l₂ at points A and B, respectively. It is known that the magnitudes of vectors |OA|, |AB|, and |OB| form an arithmetic sequence, and the vectors BF and FA are in the same direction. Determine the eccentricity of the hyperbola. | \frac{\sqrt{5}}{2} | 8.59375 |
25,283 | Given \( a_{n} = \mathrm{C}_{200}^{n} \cdot (\sqrt[3]{6})^{200-n} \cdot \left( \frac{1}{\sqrt{2}} \right)^{n} \) for \( n = 1, 2, \ldots, 95 \), find the number of integer terms in the sequence \(\{a_{n}\}\). | 15 | 3.90625 |
25,284 | A truncated right circular cone has a large base radius of 10 cm and a small base radius of 5 cm. The height of the truncated cone is 10 cm. Calculate the volume of this solid. | 583.33\pi | 0 |
25,285 | Let \(x\), \(y\), and \(z\) be positive real numbers such that \(x + y + z = 3.\) Find the maximum value of \(x^3 y^3 z^2.\) | \frac{4782969}{390625} | 0 |
25,286 | How many integers between 1 and 3015 are either multiples of 5 or 7 but not multiples of 35? | 948 | 0 |
25,287 | Given the plane vectors $\overrightarrow{a}=(1,0)$ and $\overrightarrow{b}=\left( -\frac{1}{2}, \frac{\sqrt{3}}{2} \right)$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{a}+ \overrightarrow{b}$. | \frac{\pi}{3} | 85.15625 |
25,288 | In triangle $ABC,$ $\angle A = 45^\circ,$ $\angle B = 75^\circ,$ and $AC = 6.$ Find $BC$. | 6\sqrt{3} - 6 | 28.125 |
25,289 | Given $a = 1 + 2\binom{20}{1} + 2^2\binom{20}{2} + \ldots + 2^{20}\binom{20}{20}$, and $a \equiv b \pmod{10}$, determine the possible value(s) for $b$. | 2011 | 0 |
25,290 | The numbers \(2, 3, 4, 5, 6, 7, 8\) are to be placed, one per square, in the diagram shown so that the sum of the four numbers in the horizontal row equals 21 and the sum of the four numbers in the vertical column also equals 21. In how many different ways can this be done? | 12 | 0.78125 |
25,291 | In triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\frac{c}{2} = b - a\cos C$,
(1) Determine the measure of angle $A$.
(2) If $a=\sqrt{15}$ and $b=4$, find the length of side $c$. | 2 - \sqrt{3} | 14.84375 |
25,292 | Given the sequences $\{a_{n}\}$ and $\{b_{n}\}$, where $a_{n}$ represents the $n$-th digit after the decimal point of $\sqrt{2}=1.41421356237⋯$ (for example, $a_{1}=4$, $a_{6}=3)$, and $b_{1}=a_{1}$, with ${b_{n+1}}={a_{{b_n}}}$ for all $n\in N^{*}$, find all values of $n$ that satisfy $b_{n}=n-2022$. | 675 | 0 |
25,293 | If Samuel has a $3 \times 7$ index card and shortening the length of one side by $1$ inch results in an area of $15$ square inches, determine the area of the card in square inches if instead he shortens the length of the other side by $1$ inch. | 10 | 3.125 |
25,294 | Given that point $P$ is inside isosceles triangle $ABC$, with $AB = BC$ and $\angle BPC = 108^{\circ}$. Let $D$ be the midpoint of side $AC$, and let $BD$ intersect $PC$ at point $E$. If $P$ is the incenter of $\triangle ABE$, find $\angle PAC$. | 48 | 0 |
25,295 | Let $y=f(x)$ be a quadratic function, and the equation $f(x)=0$ has two equal real roots. Also, $f'(x)=2x+2$.
1. Find the expression for $y=f(x)$.
2. Find the area of the shape enclosed by the graph of $y=f(x)$ and the two coordinate axes.
3. If the line $x=-t$ ($0<t<1$) divides the area enclosed by the graph of $y=f(x)$ and the two coordinate axes into two equal parts, find the value of $t$. | 1-\frac{1}{32} | 0 |
25,296 | If the digits of a natural number can be divided into two groups such that the sum of the digits in each group is equal, the number is called a "balanced number". For example, 25254 is a "balanced number" because $5+2+2=4+5$. If two adjacent natural numbers are both "balanced numbers", they are called a pair of "twin balanced numbers". What is the sum of the smallest pair of "twin balanced numbers"? | 1099 | 2.34375 |
25,297 | By multiplying a natural number by the number that is one greater than it, the product takes the form $ABCD$, where $A, B, C, D$ are different digits. Starting with the number that is 3 less, the product takes the form $CABD$. Starting with the number that is 30 less, the product takes the form $BCAD$. Determine these numbers. | 8372 | 0 |
25,298 | Note that there are exactly three ways to write the integer $4$ as a sum of positive odd integers where the order of the summands matters:
\begin{align*}
1+1+1+1&=4,
1+3&=4,
3+1&=4.
\end{align*}
Let $f(n)$ be the number of ways to write a natural number $n$ as a sum of positive odd integers where the order of the summands matters. Find the remainder when $f(2008)$ is divided by $100$ . | 71 | 0 |
25,299 | Given the two-digit integer $MM$, where both digits are equal, when multiplied by the one-digit integer $K$ (different from $M$ and only $1\leq K \leq 9$), it results in a three-digit number $NPK$. Identify the digit pairs $(M, K)$ that yield the highest value of $NPK$. | 891 | 3.125 |
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