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25,300 | Define the operations:
\( a \bigcirc b = a^{\log _{7} b} \),
\( a \otimes b = a^{\frac{1}{\log ^{6} b}} \), where \( a, b \in \mathbb{R} \).
A sequence \(\{a_{n}\} (n \geqslant 4)\) is given such that:
\[ a_{3} = 3 \otimes 2, \quad a_{n} = (n \otimes (n-1)) \bigcirc a_{n-1}. \]
Then, the integer closest to \(\log _{7} a_{2019}\) is: | 11 | 21.09375 |
25,301 | If
\[
\begin{pmatrix}
1 & 3 & b \\
0 & 1 & 5 \\
0 & 0 & 1
\end{pmatrix}^m = \begin{pmatrix}
1 & 33 & 6006 \\
0 & 1 & 55 \\
0 & 0 & 1
\end{pmatrix},
\]
then find $b + m.$ | 432 | 0 |
25,302 | Given that the broad money supply $\left(M2\right)$ balance was 2912000 billion yuan, express this number in scientific notation. | 2.912 \times 10^{6} | 0 |
25,303 | In a triangle, two angles measure 45 degrees and 60 degrees. The side opposite the 45-degree angle measures 8 units. Calculate the sum of the lengths of the other two sides. | 19.3 | 0 |
25,304 | 8. Shortening. There is a sequence of 2015 digits. All digits are chosen randomly from the set {0, 9} independently of each other. The following operation is performed on the resulting sequence. If several identical digits go in a row, they are replaced by one such digit. For example, if there was a fragment ...044566667..., then it becomes ...04567...
a) Find the probability that the sequence will shorten by exactly one digit.
b) Find the expected length of the new sequence. | 1813.6 | 1.5625 |
25,305 | Determine the value of $\sin {15}^{{}^\circ }+\cos {15}^{{}^\circ }$. | \frac{\sqrt{6}}{2} | 70.3125 |
25,306 | A cao has 6 legs, 3 on each side. A walking pattern for the cao is defined as an ordered sequence of raising and lowering each of the legs exactly once (altogether 12 actions), starting and ending with all legs on the ground. The pattern is safe if at any point, he has at least 3 legs on the ground and not all three legs are on the same side. Estimate $N$, the number of safe patterns.
An estimate of $E>0$ earns $\left\lfloor 20 \min (N / E, E / N)^{4}\right\rfloor$ points. | 1416528 | 0 |
25,307 | Let $ABCD$ be a parallelogram such that $\angle BAD = 60^{\circ}.$ Let $K$ and $L$ be the midpoints of $BC$ and $CD,$ respectively. Assuming that $ABKL$ is a cyclic quadrilateral, find $\angle ABD.$ | 75 | 0 |
25,308 | Given $\tan\alpha= \frac {1}{2}$ and $\tan(\alpha-\beta)=- \frac {2}{5}$, calculate the value of $\tan(2\alpha-\beta)$. | -\frac{1}{12} | 0 |
25,309 | Let \(\mathbf{A} = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\) and \(\mathbf{B} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\) be two matrices such that \(\mathbf{A} \mathbf{B} = \mathbf{B} \mathbf{A}\). Assuming \(4b \neq c\), find \(\frac{a - 2d}{c - 4b}\). | \frac{3}{10} | 0 |
25,310 | The same amount of steel used to create six solid steel balls, each with a radius of 2 inches, is used to create one larger steel ball. What is the radius of the larger ball? | 4\sqrt[3]{2} | 0 |
25,311 | Let $WXYZ$ be a rhombus with diagonals $WY = 20$ and $XZ = 24$. Let $M$ be a point on $\overline{WX}$, such that $WM = MX$. Let $R$ and $S$ be the feet of the perpendiculars from $M$ to $\overline{WY}$ and $\overline{XZ}$, respectively. Find the minimum possible value of $RS$. | \sqrt{244} | 0 |
25,312 | Given four points \( K, L, M, N \) that are not coplanar. A sphere touches the planes \( K L M \) and \( K L N \) at points \( M \) and \( N \) respectively. Find the surface area of the sphere, knowing that \( M L = 1 \), \( K M = 2 \), \( \angle M N L = 60^\circ \), and \( \angle K M L = 90^\circ \). | \frac{64\pi}{11} | 0 |
25,313 | The function $g : \mathbb{R} \to \mathbb{R}$ satisfies
\[g(x) + 3g(1 - x) = 2x^2 + 1\]for all $x.$ Find $g(5).$ | -9 | 0 |
25,314 |
A car license plate contains three letters and three digits, for example, A123BE. The allowed letters are А, В, Е, К, М, Н, О, Р, С, Т, У, Х (a total of 12 letters) and all digits except the combination 000. Kira considers a license plate lucky if the second letter is a vowel, the second digit is odd, and the third digit is even (other symbols have no restrictions). How many license plates does Kira consider lucky? | 359999 | 0 |
25,315 | If Ravi shortens the length of one side of a $5 \times 7$ index card by $1$ inch, the card would have an area of $24$ square inches. What is the area of the card in square inches if instead he shortens the length of the other side by $1$ inch? | 18 | 3.125 |
25,316 | Tina is trying to solve the equation by completing the square: $$25x^2+30x-55 = 0.$$ She needs to rewrite the equation in the form \((ax + b)^2 = c\), where \(a\), \(b\), and \(c\) are integers and \(a > 0\). What is the value of \(a + b + c\)? | -38 | 0 |
25,317 | Let $PQRS$ be an isosceles trapezoid with bases $PQ=120$ and $RS=25$. Suppose $PR=QS=y$ and a circle with center on $\overline{PQ}$ is tangent to segments $\overline{PR}$ and $\overline{QS}$. If $n$ is the smallest possible value of $y$, then $n^2$ equals what? | 2850 | 0 |
25,318 | Given the function $f(x)= \frac {1}{3}x^{3}+x^{2}+ax+1$, and the slope of the tangent line to the curve $y=f(x)$ at the point $(0,1)$ is $-3$.
$(1)$ Find the intervals of monotonicity for $f(x)$;
$(2)$ Find the extrema of $f(x)$. | -\frac{2}{3} | 7.8125 |
25,319 | Consider a modified sequence whose $n$th term is defined by $a_n = (-1)^n \cdot \lfloor \frac{3n+1}{2} \rfloor$. What is the average of the first $150$ terms of this sequence? | -\frac{1}{6} | 0 |
25,320 | Let $a,$ $b,$ $c,$ and $d$ be real numbers such that $ab = 2$ and $cd = 18.$ Find the minimum value of
\[(ac)^2 + (bd)^2.\] | 12 | 0 |
25,321 | The sides of $\triangle ABC$ measure 11,20, and 21 units. We fold it along $PQ,QR,RP$ where $P,Q,R$ are the midpoints of its sides until $A,B,C$ coincide. What is the volume of the resulting tetrahedron? | 45 | 0.78125 |
25,322 | The sum of 9 standard dice rolled has the same probability of occurring as the sum of 20. Find the value of the sum of 9 standard dice that shares this same probability. | 43 | 9.375 |
25,323 | Given a right triangle \( ABC \) with legs \( AC = 3 \) and \( BC = 4 \). Construct triangle \( A_1 B_1 C_1 \) by successively translating point \( A \) a certain distance parallel to segment \( BC \) to get point \( A_1 \), then translating point \( B \) parallel to segment \( A_1 C \) to get point \( B_1 \), and finally translating point \( C \) parallel to segment \( A_1 B_1 \) to get point \( C_1 \). If it turns out that angle \( A_1 B_1 C_1 \) is a right angle and \( A_1 B_1 = 1 \), what is the length of segment \( B_1 C_1 \)? | 12 | 0 |
25,324 | Compute $\arccos(\cos 9).$ All functions are in radians. | 9 - 2\pi | 10.9375 |
25,325 | Find the number of triples of natural numbers \( m, n, k \) that are solutions to the equation \( m + \sqrt{n+\sqrt{k}} = 2023 \). | 27575680773 | 0 |
25,326 | What is the nearest integer to $(3 + 2)^6$? | 9794 | 0 |
25,327 | Given triangle $ABC$ . Let $A_1B_1$ , $A_2B_2$ , $ ...$ , $A_{2008}B_{2008}$ be $2008$ lines parallel to $AB$ which divide triangle $ABC$ into $2009$ equal areas. Calculate the value of $$ \left\lfloor \frac{A_1B_1}{2A_2B_2} + \frac{A_1B_1}{2A_3B_3} + ... + \frac{A_1B_1}{2A_{2008}B_{2008}} \right\rfloor $$ | 29985 | 0 |
25,328 | The sequence $(a_n)$ is defined recursively by $a_0=1$, $a_1=\sqrt[17]{3}$, and $a_n=a_{n-1}a_{n-2}^2$ for $n\geq 2$. What is the smallest positive integer $k$ such that the product $a_1a_2\cdots a_k$ is an integer? | 11 | 0 |
25,329 | **polyhedral**
we call a $12$ -gon in plane good whenever:
first, it should be regular, second, it's inner plane must be filled!!, third, it's center must be the origin of the coordinates, forth, it's vertices must have points $(0,1)$ , $(1,0)$ , $(-1,0)$ and $(0,-1)$ .
find the faces of the <u>massivest</u> polyhedral that it's image on every three plane $xy$ , $yz$ and $zx$ is a good $12$ -gon.
(it's obvios that centers of these three $12$ -gons are the origin of coordinates for three dimensions.)
time allowed for this question is 1 hour. | 36 | 0.78125 |
25,330 | In $\triangle ABC$, $AB= 400$, $BC=480$, and $AC=560$. An interior point $P$ is identified, and segments are drawn through $P$ parallel to the sides of the triangle. These three segments are of equal length $d$. Determine $d$. | 218\frac{2}{9} | 0 |
25,331 | In triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and $a$, $b$, $c$ form an arithmetic sequence in that order.
$(1)$ If the vectors $\overrightarrow{m}=(3,\sin B)$ and $\overrightarrow{n}=(2,\sin C)$ are collinear, find the value of $\cos A$;
$(2)$ If $ac=8$, find the maximum value of the area $S$ of $\triangle ABC$. | 2 \sqrt {3} | 0 |
25,332 | Given that the students are numbered from 01 to 70, determine the 7th individual selected by reading rightward starting from the number in the 9th row and the 9th column of the random number table. | 44 | 0 |
25,333 | $|2+i^{2}+2i^{2}|=$ | \sqrt{5} | 0 |
25,334 | Let $Q(x) = 0$ be the polynomial equation of the least possible degree, with rational coefficients, having $\sqrt[4]{13} + \sqrt[4]{169}$ as a root. Compute the product of all of the roots of $Q(x) = 0.$ | -13 | 9.375 |
25,335 | Compute
\[\frac{(10^4+400)(26^4+400)(42^4+400)(58^4+400)}{(2^4+400)(18^4+400)(34^4+400)(50^4+400)}.\] | 962 | 0 |
25,336 | What is the largest $2$-digit prime factor of the integer $n = {300 \choose 150}$? | 89 | 0 |
25,337 | Five distinct points are arranged on a plane such that the segments connecting them form lengths $a$, $a$, $a$, $a$, $a$, $b$, $b$, $2a$, and $c$. The shape formed by these points is no longer restricted to a simple polygon but could include one bend (not perfectly planar). What is the ratio of $c$ to $a$?
**A)** $\sqrt{3}$
**B)** $2$
**C)** $2\sqrt{3}$
**D)** $3$
**E)** $\pi$ | 2\sqrt{3} | 1.5625 |
25,338 | Given that both $m$ and $n$ are non-negative integers, find the number of "simple" ordered pairs $(m, n)$ with a value of 2019. | 60 | 0 |
25,339 | Let $M$ be the number of ways to write $3050$ in the form $3050 = b_3 \cdot 10^3 + b_2 \cdot 10^2 + b_1 \cdot 10 + b_0$, where the $b_i$'s are integers, and $0 \le b_i \le 99$. Find $M$. | 306 | 25.78125 |
25,340 | Given a parallelogram \(ABCD\) with \(\angle B = 60^\circ\). Point \(O\) is the center of the circumcircle of triangle \(ABC\). Line \(BO\) intersects the bisector of the exterior angle \(\angle D\) at point \(E\). Find the ratio \(\frac{BO}{OE}\). | 1/2 | 2.34375 |
25,341 | Let \( A \subseteq \{0, 1, 2, \cdots, 29\} \) such that for any integers \( k \) and any numbers \( a \) and \( b \) (possibly \( a = b \)), the expression \( a + b + 30k \) is not equal to the product of two consecutive integers. Determine the maximum possible number of elements in \( A \). | 10 | 28.90625 |
25,342 | A, B, and C start from the same point on a circular track with a circumference of 360 meters: A starts first and runs in the counterclockwise direction; before A completes a lap, B and C start simultaneously and run in the clockwise direction; when A and B meet for the first time, C is exactly half a lap behind them; after some time, when A and C meet for the first time, B is also exactly half a lap behind them. If B’s speed is 4 times A’s speed, then how many meters has A run when B and C start? | 90 | 2.34375 |
25,343 | The polynomial \( x^{2n} + 1 + (x+1)^{2n} \) cannot be divided by \( x^2 + x + 1 \) under the condition that \( n \) is equal to: | 21 | 0 |
25,344 | Determine the area of the Crescent Gemini. | \frac{17\pi}{4} | 0 |
25,345 | In an acute-angled $\triangle ABC$, find the minimum value of $3 \tan B \tan C + 2 \tan A \tan C + \tan A \tan B$. | 6 + 2\sqrt{3} + 2\sqrt{2} + 2\sqrt{6} | 0 |
25,346 | In a bookshelf, there are four volumes of Astrid Lindgren's collected works in order, each containing 200 pages. A little worm living in these volumes burrowed a path from the first page of the first volume to the last page of the fourth volume. How many pages did the worm burrow through? | 400 | 0 |
25,347 | In triangle $DEF$, $\angle E = 45^\circ$, $DE = 100$, and $DF = 100 \sqrt{2}$. Find the sum of all possible values of $EF$. | \sqrt{30000 + 5000(\sqrt{6} - \sqrt{2})} | 0 |
25,348 | In right triangle $\triangle ABC$ with $\angle BAC = 90^\circ$, medians $\overline{AD}$ and $\overline{BE}$ are given such that $AD = 18$ and $BE = 24$. If $\overline{AD}$ is the altitude from $A$, find the area of $\triangle ABC$. | 432 | 7.8125 |
25,349 | Given that real numbers x and y satisfy x + y = 5 and x * y = -3, find the value of x + x^4 / y^3 + y^4 / x^3 + y. | 5 + \frac{2829}{27} | 0 |
25,350 | Let $x,$ $y,$ $z$ be real numbers such that $0 \leq x, y, z \leq 1$. Find the maximum value of
\[
\frac{1}{(2 - x)(2 - y)(2 - z)} + \frac{1}{(2 + x)(2 + y)(2 + z)}.
\] | \frac{12}{27} | 0 |
25,351 | Given that the probability of bus No. 3 arriving at the bus stop within 5 minutes is 0.20 and the probability of bus No. 6 arriving within 5 minutes is 0.60, calculate the probability that the passenger can catch the bus he needs within 5 minutes. | 0.80 | 52.34375 |
25,352 | The expression $\frac{\sqrt{3}\tan 12^{\circ} - 3}{(4\cos^2 12^{\circ} - 2)\sin 12^{\circ}}$ equals \_\_\_\_\_\_. | -4\sqrt{3} | 21.875 |
25,353 | Determine constants $\alpha$ and $\beta$ such that $\frac{x-\alpha}{x+\beta} = \frac{x^2 - 96x + 2210}{x^2 + 65x - 3510}$. What is $\alpha + \beta$? | 112 | 0 |
25,354 | If $\triangle ABC$ satisfies that $\cot A, \cot B, \cot C$ form an arithmetic sequence, then the maximum value of $\angle B$ is ______. | \frac{\pi}{3} | 35.9375 |
25,355 | Give an example of a function that, when \( x \) is equal to a known number, takes the form \( \frac{0}{0} \), but as \( x \) approaches this number, tends to a certain limit. | \frac{8}{7} | 0 |
25,356 | The diagram below shows part of a city map. The small rectangles represent houses, and the spaces between them represent streets. A student walks daily from point $A$ to point $B$ on the streets shown in the diagram, and can only walk east or south. At each intersection, the student has an equal probability ($\frac{1}{2}$) of choosing to walk east or south (each choice is independent of others). What is the probability that the student will walk through point $C$? | $\frac{21}{32}$ | 0 |
25,357 | Alex is thinking of a number that is divisible by all of the positive integers 1 through 200 inclusive except for two consecutive numbers. What is the smaller of these numbers? | 128 | 1.5625 |
25,358 | For an arithmetic sequence $\{a_n\}$ with a non-zero common difference, some terms $a_{k_1}$, $a_{k_2}$, $a_{k_3}$, ... form a geometric sequence $\{a_{k_n}\}$, and it is given that $k_1 \neq 1$, $k_2 \neq 2$, $k_3 \neq 6$. Find the value of $k_4$. | 22 | 1.5625 |
25,359 | Each twin from the first 4 sets shakes hands with all twins except his/her sibling and with one-third of the triplets; the remaining 8 sets of twins shake hands with all twins except his/her sibling but does not shake hands with any triplet; and each triplet shakes hands with all triplets except his/her siblings and with one-fourth of all twins from the first 4 sets only. | 394 | 0 |
25,360 | We draw diagonals in some of the squares on a chessboard in such a way that no two diagonals intersect at a common point. What is the maximum number of diagonals that can be drawn this way? | 36 | 0 |
25,361 | A right square pyramid with base edges of length $12$ units each and slant edges of length $15$ units each is cut by a plane that is parallel to its base and $4$ units above its base. What is the volume, in cubic units, of the top pyramid section that is cut off by this plane? | \frac{1}{3} \times \left(\frac{(144 \cdot (153 - 8\sqrt{153}))}{153}\right) \times (\sqrt{153} - 4) | 0 |
25,362 | A wooden cube with edges of length $3$ meters has square holes, of side one meter, centered in each face, cut through to the opposite face. Find the entire surface area, including the inside, of this cube in square meters. | 72 | 9.375 |
25,363 | Find the number of permutations of \( n \) distinct elements \( a_1, a_2, \cdots, a_n \) (where \( n \geqslant 2 \)) such that \( a_1 \) is not in the first position and \( a_2 \) is not in the second position. | 32,527,596 | 0 |
25,364 | A triangular box is to be cut from an equilateral triangle of length 30 cm. Find the largest possible volume of the box (in cm³). | 500 | 1.5625 |
25,365 | There are two distinguishable flagpoles, and there are $17$ flags, of which $11$ are identical red flags, and $6$ are identical white flags. Determine the number of distinguishable arrangements using all of the flags such that each flagpole has at least one flag and no two white flags on either pole are adjacent. Compute the remainder when this number is divided by $1000$. | 164 | 0 |
25,366 | Given that the center of circle $M$ lies on the $y$-axis, the radius is $1$, and the chord intercepted by line $l: y = 2x + 2$ on circle $M$ has a length of $\frac{4\sqrt{5}}{5}$. Additionally, the circle center $M$ is located below line $l$.
(1) Find the equation of circle $M$;
(2) Let $A(t, 0), B(t + 5, 0) \, (-4 \leqslant t \leqslant -1)$, if $AC, BC$ are tangent lines to circle $M$, find the minimum value of the area of $\triangle ABC$. | \frac{125}{21} | 0 |
25,367 | Find all real numbers \( p \) such that the cubic equation \( 5x^{3} - 5(p+1)x^{2} + (71p - 1)x + 1 = 66p \) has three roots, all of which are positive integers. | 76 | 78.90625 |
25,368 | Tom has a scientific calculator. Unfortunately, all keys are broken except for one row: 1, 2, 3, + and -.
Tom presses a sequence of $5$ random keystrokes; at each stroke, each key is equally likely to be pressed. The calculator then evaluates the entire expression, yielding a result of $E$ . Find the expected value of $E$ .
(Note: Negative numbers are permitted, so 13-22 gives $E = -9$ . Any excess operators are parsed as signs, so -2-+3 gives $E=-5$ and -+-31 gives $E = 31$ . Trailing operators are discarded, so 2++-+ gives $E=2$ . A string consisting only of operators, such as -++-+, gives $E=0$ .)
*Proposed by Lewis Chen* | 1866 | 0 |
25,369 | Given that the largest four-digit number $M$ has digits with a product of $72$, find the sum of its digits. | 17 | 0.78125 |
25,370 | Simplify $$\frac{13!}{10! + 3\cdot 9!}$$ | 1320 | 70.3125 |
25,371 | Three of the four endpoints of the axes of an ellipse are, in some order, \[(10, -3), \; (15, 7), \; (25, -3).\] Find the distance between the foci of the ellipse. | 11.18 | 0 |
25,372 | The maximum number that can be formed by the digits 0, 3, 4, 5, 6, 7, 8, 9, given that there are only thirty 2's and twenty-five 1's. | 199 | 0 |
25,373 | Find a whole number, $M$, such that $\frac{M}{5}$ is strictly between 9.5 and 10.5. | 51 | 0 |
25,374 | If the difference between each number in a row and the number immediately to its left in the given diagram is the same, and the quotient of each number in a column divided by the number immediately above it is the same, then $a + b \times c =\quad$ | 540 | 0 |
25,375 | 9 judges each award 20 competitors a rank from 1 to 20. The competitor's score is the sum of the ranks from the 9 judges, and the winner is the competitor with the lowest score. For each competitor, the difference between the highest and lowest ranking (from different judges) is at most 3. What is the highest score the winner could have obtained? | 24 | 0 |
25,376 | Let $p = 2017,$ a prime number. Let $N$ be the number of ordered triples $(a,b,c)$ of integers such that $1 \le a,b \le p(p-1)$ and $a^b-b^a=p \cdot c$ . Find the remainder when $N$ is divided by $1000000.$ *Proposed by Evan Chen and Ashwin Sah*
*Remark:* The problem was initially proposed for $p = 3,$ and $1 \le a, b \le 30.$ | 2016 | 0.78125 |
25,377 | 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 number of elements in a $\text{minimally friendly}$ set $\mathcal{F}$ . | 72 | 0 |
25,378 | What is the minimum number of cells required to mark on a chessboard so that each cell of the board (marked or unmarked) is adjacent by side to at least one marked cell? | 20 | 0 |
25,379 | The function $f(x) = x(x - m)^2$ reaches its maximum value at $x = -2$. Determine the value of $m$. | -6 | 7.8125 |
25,380 | Cindy wants to arrange her coins into $X$ piles, each consisting of the same number of coins, $Y$. Each pile will have more than one coin and no pile will have all the coins. If there are 16 possible values for $Y$ given all of the restrictions, what is the smallest number of coins she could have? | 131072 | 0 |
25,381 | Find the value of \(\log _{2}\left[2^{3} 4^{4} 8^{5} \cdots\left(2^{20}\right)^{22}\right]\).
Choose one of the following options:
(a) 3290
(b) 3500
(c) 3710
(d) 4172 | 5950 | 0 |
25,382 | In the cells of an $8\times 8$ board, marbles are placed one by one. Initially there are no marbles on the board. A marble could be placed in a free cell neighboring (by side) with at least three cells which are still free. Find the greatest possible number of marbles that could be placed on the board according to these rules. | 36 | 2.34375 |
25,383 | An assembly line produces an average of $85\%$ first-grade products. How many products need to be selected so that with a probability of 0.997, the deviation of the frequency of first-grade products from the probability $p=0.85$ does not exceed $0.01$ in absolute value? | 11171 | 0 |
25,384 | On graph paper, large and small triangles are drawn (all cells are square and of the same size). How many small triangles can be cut out from the large triangle? Triangles cannot be rotated or flipped (the large triangle has a right angle in the bottom left corner, the small triangle has a right angle in the top right corner). | 12 | 3.125 |
25,385 | Given that line $l\_1$ passes through points $A(m,1)$ and $B(-3,4)$, and line $l\_2$ passes through points $C(1,m)$ and $D(-1,m+1)$, find the values of the real number $m$ when $l\_1$ is parallel to $l\_2$ or $l\_1$ is perpendicular to $l\_2$. | -\frac{9}{2} | 18.75 |
25,386 | A flag in the shape of a square has a symmetrical cross of uniform width, featuring a center decorated with a green square, and set against a yellow background. The entire cross (comprising both the arms and the green center) occupies 49% of the total area of the flag. What percentage of the flag's area is occupied by the green square? | 25\% | 2.34375 |
25,387 | A light flashes green every 3 seconds. Determine the number of times the light has flashed green after 671 seconds. | 154 | 0 |
25,388 | Let $x$ be a real number such that
\[ x^2 + 8 \left( \frac{x}{x-3} \right)^2 = 53. \]
Find all possible values of $y = \frac{(x - 3)^3 (x + 4)}{2x - 5}.$ | \frac{17000}{21} | 0 |
25,389 | Given that angles $\angle A, \angle B, \angle C$ are the interior angles of triangle $ABC$, and vector $\alpha=\left(\cos \frac{A-B}{2}, \sqrt{3} \sin \frac{A+B}{2}\right)$ with $|\alpha|=\sqrt{2}$. If when $\angle C$ is maximized, there exists a moving point $M$ such that $|MA|, |AB|, |MB|$ form an arithmetic sequence, the maximum value of $|AB|$ is ____. | \frac{2\sqrt{3} + \sqrt{2}}{4} | 0 |
25,390 | If $p$, $q$, $r$, $s$, $t$, and $u$ are integers for which $512x^3 + 64 = (px^2 + qx +r)(sx^2 + tx + u)$ for all x, find the value of $p^2 + q^2 + r^2 + s^2 + t^2 + u^2$. | 5472 | 2.34375 |
25,391 | Given that \(1 \leqslant a_{1} \leqslant a_{2} \leqslant a_{3} \leqslant a_{4} \leqslant a_{5} \leqslant a_{6} \leqslant 64\), find the minimum value of \(Q = \frac{a_{1}}{a_{2}} + \frac{a_{3}}{a_{4}} + \frac{a_{5}}{a_{6}}\). | 3/2 | 15.625 |
25,392 | Let $a, b, c$, and $d$ be positive real numbers such that
\[
\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}
a^2 + b^2 &=& c^2 + d^2 &=& 1458, \\
ac &=& bd &=& 1156.
\end{array}
\]
If $S = a + b + c + d$, compute the value of $\lfloor S \rfloor$. | 77 | 0.78125 |
25,393 | Find the area bounded by the graph of $y = \arcsin(\cos x)$ and the $x$-axis on the interval $0 \le x \le 2\pi.$ | \frac{\pi^2}{4} | 1.5625 |
25,394 | Calculate the sum of the series:
\[
\sum_{n=1}^\infty \frac{3^n}{1 + 3^n + 3^{n+1} + 3^{2n+1}}.
\] | \frac{1}{4} | 4.6875 |
25,395 | In a park, 10,000 trees are planted in a square grid pattern (100 rows of 100 trees). What is the maximum number of trees that can be cut down such that if one stands on any stump, no other stumps are visible? (Trees can be considered thin enough for this condition.)
| 2500 | 6.25 |
25,396 | Given that \( x - \frac{1}{x} = \sqrt{3} \), find \( x^{2048} - \frac{1}{x^{2048}} \). | 277526 | 0 |
25,397 | A calculator has digits from 0 to 9 and signs of two operations. Initially, the display shows the number 0. You can press any keys. The calculator performs operations in the sequence of keystrokes. If an operation sign is pressed several times in a row, the calculator remembers only the last press. A distracted Scientist pressed many buttons in a random sequence. Find approximately the probability that the result of the resulting sequence of actions is an odd number? | \frac{1}{3} | 0.78125 |
25,398 | Given lines $l_{1}$: $(3+a)x+4y=5-3a$ and $l_{2}$: $2x+(5+a)y=8$, find the value of $a$ such that the lines are parallel. | -5 | 0 |
25,399 | Given that the volume of the parallelepiped formed by vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ is 4, find the volume of the parallelepiped formed by the vectors $\mathbf{2a} + \mathbf{b}$, $\mathbf{b} + 4\mathbf{c}$, and $\mathbf{c} - 5\mathbf{a}$. | 232 | 0 |
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