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25,100 | Define the function $g(x) = x^{-2} + \frac{x^{-2}}{1+x^{-2}}$. Determine $g(g(3))$. | \frac{72596100}{3034921} | 0 |
25,101 | There are $n\leq 99$ people around a circular table. At every moment everyone can either be truthful (always says the truth) or a liar (always lies). Initially some of people (possibly none) are truthful and the rest are liars. At every minute everyone answers at the same time the question "Is your left neighbour truthful or a liar?" and then becomes the same type of person as his answer. Determine the largest $n$ for which, no matter who are the truthful people in the beginning, at some point everyone will become truthful forever. | 64 | 8.59375 |
25,102 | Given that $a$, $b$, and $c$ are the sides opposite the angles $A$, $B$, and $C$ in $\triangle ABC$ respectively, and the equation $\sqrt{3}b\sin A - a\cos B - 2a = 0$ holds, then the measure of $\angle B$ is ______. | \frac{2\pi}{3} | 78.125 |
25,103 | In a cone with a height of 4 and a slant height of 8, three spheres of radius \( r \) are placed. They touch each other (externally), the lateral surface of the cone, and the first two spheres touch the base of the cone. Find the maximum value of \( r \). | \frac{12}{5+2\sqrt{3}} | 0 |
25,104 | Given that $n$ is a positive integer, and $4^7 + 4^n + 4^{1998}$ is a perfect square, then one value of $n$ is. | 3988 | 1.5625 |
25,105 | Given $995 + 997 + 999 + 1001 + 1003 = 5100 - N$, determine $N$. | 100 | 19.53125 |
25,106 | In the right square prism $M-ABCD$, the base $ABCD$ is a rectangle, $MD \perp$ face $ABCD$ with $MD$ being an integer, and the lengths of $MA$, $MC$, and $MB$ are three consecutive even numbers. What is the volume of the right square prism $M-ABCD$? | $24 \sqrt{5}$ | 0 |
25,107 | Given $|\overrightarrow {a}|=4$, $|\overrightarrow {b}|=2$, and the angle between $\overrightarrow {a}$ and $\overrightarrow {b}$ is $120^{\circ}$, find:
1. $\left(\overrightarrow {a}-2\overrightarrow {b}\right)\cdot \left(\overrightarrow {a}+\overrightarrow {b}\right)$;
2. The projection of $\overrightarrow {a}$ onto $\overrightarrow {b}$;
3. The angle between $\overrightarrow {a}$ and $\overrightarrow {a}+\overrightarrow {b}$. | \dfrac{\pi}{6} | 0 |
25,108 | Estimate the probability of a rifle student hitting the target using a random simulation method by analyzing the ratio of the number of sets of three random numbers where exactly one represents a hit to the total number of sets of three random numbers. | \frac{9}{20} | 0 |
25,109 | Let M be a subst of {1,2,...,2006} with the following property: For any three elements x,y and z (x<y<z) of M, x+y does not divide z. Determine the largest possible size of M. Justify your claim. | 1004 | 1.5625 |
25,110 | The function $g(x)$ satisfies
\[g(x) - 2 g \left( \frac{1}{x} \right) = 3^x\] for all \( x \neq 0 \). Find $g(2)$. | -\frac{29}{9} | 0 |
25,111 | Given $x > 0$, $y > 0$, and the inequality $2\log_{\frac{1}{2}}[(a-1)x+ay] \leq 1 + \log_{\frac{1}{2}}(xy)$ always holds, find the minimum value of $4a$. | \sqrt{6}+\sqrt{2} | 0 |
25,112 | We call a natural number \( b \) lucky if for any natural \( a \) such that \( a^{5} \) is divisible by \( b^{2} \), the number \( a^{2} \) is divisible by \( b \).
Find the number of lucky natural numbers less than 2010. | 1961 | 0 |
25,113 | In $\Delta ABC$, it is known that $\overrightarrow{AB} \cdot \overrightarrow{AC} + 2\overrightarrow{BA} \cdot \overrightarrow{BC} = 3\overrightarrow{CA} \cdot \overrightarrow{CB}$. The minimum value of $\cos C$ is ______. | \dfrac{ \sqrt{2}}{3} | 7.03125 |
25,114 | If two distinct numbers from the set $\{ 5, 15, 21, 35, 45, 63, 70, 90 \}$ are randomly selected and multiplied, what is the probability that the product is a multiple of 105? Express your answer as a common fraction. | \frac{1}{4} | 0 |
25,115 | Each time you click a toggle switch, the switch either turns from *off* to *on* or from *on* to *off*. Suppose that you start with three toggle switches with one of them *on* and two of them *off*. On each move you randomly select one of the three switches and click it. Let $m$ and $n$ be relatively prime positive integers so that $\frac{m}{n}$ is the probability that after four such clicks, one switch will be *on* and two of them will be *off*. Find $m+n$ . | 61 | 0 |
25,116 | Find the number of positive integers $n \le 1500$ that can be expressed in the form
\[\lfloor x \rfloor + \lfloor 2x \rfloor + \lfloor 4x \rfloor = n\] for some real number $x.$ | 854 | 0 |
25,117 | In the coordinate plane, consider points $A = (0, 0)$, $B = (8, 0)$, and $C = (15, 0)$. Line $\ell_A$ has slope 1 and passes through $A$. Line $\ell_B$ is vertical and passes through $B$. Line $\ell_C$ has slope $-1$ and passes through $C$. Additionally, introduce point $D = (20, 0)$ with line $\ell_D$ having a slope of 2 and passing through $D$. All four lines $\ell_A$, $\ell_B$, $\ell_C$, and $\ell_D$ begin rotating clockwise about their respective points, $A$, $B$, $C$, and $D$, at the same angular rate. At any given time, the four lines form a quadrilateral. Determine the largest possible area of such a quadrilateral. | 110.5 | 0 |
25,118 | At the height \(BH\) of triangle \(ABC\), a certain point \(D\) is marked. Line \(AD\) intersects side \(BC\) at point \(E\), and line \(CD\) intersects side \(AB\) at point \(F\). Points \(G\) and \(J\) are the projections of points \(F\) and \(E\) respectively onto side \(AC\). The area of triangle \(H E J\) is twice the area of triangle \(H F G\). In what ratio does the height \(BH\) divide segment \(FE\)? | \sqrt{2}: 1 | 23.4375 |
25,119 | Let \(\{a, b, c, d\}\) be a subset of \(\{1, 2, \ldots, 17\}\). If 17 divides \(a - b + c - d\), then \(\{a, b, c, d\}\) is called a "good subset." Find the number of good subsets. | 476 | 1.5625 |
25,120 | Convert $102012_{(3)}$ to base 10. | 320 | 0 |
25,121 | Among the 8 vertices, the midpoints of the 12 edges, the centers of the 6 faces, and the center of a cube (totaling 27 points), how many groups of three collinear points are there? | 49 | 0.78125 |
25,122 |
1. Let $[x]$ denote the greatest integer less than or equal to the real number $x$. Given a sequence of positive integers $\{a_{n}\}$ such that $a_{1} = a$, and for any positive integer $n$, the sequence satisfies the recursion
$$
a_{n+1} = a_{n} + 2 \left[\sqrt{a_{n}}\right].
$$
(1) If $a = 8$, find the smallest positive integer $n$ such that $a_{n}$ is a perfect square.
(2) If $a = 2017$, find the smallest positive integer $n$ such that $a_{n}$ is a perfect square. | 82 | 50.78125 |
25,123 | Given that the diagonals of a rhombus are always perpendicular bisectors of each other, what is the area of a rhombus with side length $\sqrt{113}$ units and diagonals that differ by 10 units? | 72 | 0.78125 |
25,124 | Circle $C$ with radius 6 has diameter $\overline{AB}$. Circle $D$ is internally tangent to circle $C$ at $A$. Circle $E$ is internally tangent to circle $C$, externally tangent to circle $D$, and tangent to $\overline{AB}$. The radius of circle $D$ is twice the radius of circle $E$, and can be written in the form $\sqrt{p}-q$, where $p$ and $q$ are positive integers. Find $p+q$. | 186 | 0 |
25,125 | Suppose that \((x_{0}, y_{0})\) is a solution of the system:
\[
\begin{cases}
xy = 6 \\
x^2 y + xy^2 + x + y + c = 2
\end{cases}
\]
Find the value of \(d = x_{0}^{2} + y_{0}^{2}\). | 69 | 0 |
25,126 | Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$ and $Q$ are distinct points in the plane with nonnegative integer coordinates $(x,y)$ such that $51x + y = 2048$. Find the number of such distinct triangles whose area is a positive integer. | 400 | 0.78125 |
25,127 | Point $P$ is inside right triangle $\triangle ABC$ with $\angle B = 90^\circ$. Points $Q$, $R$, and $S$ are the feet of the perpendiculars from $P$ to $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$, respectively. Given that $PQ = 2$, $PR = 3$, and $PS = 4$, what is $BC$? | 6\sqrt{5} | 0 |
25,128 | Evaluate $(7 + 5 + 3) \div 3 - 2 \div 3$. | 4 \frac{1}{3} | 3.125 |
25,129 | 1. In the Cartesian coordinate system, given three points A(5, 4), B(k, 10), and C(12, -2), for what value of k are the vectors $$\overrightarrow {AB}$$ and $$\overrightarrow {BC}$$ collinear?
2. In the Cartesian coordinate system, given O as the origin, $$\overrightarrow {OA}=(-7, 6)$$, $$\overrightarrow {OB}=(3, k)$$, $$\overrightarrow {OC}=(5, 7)$$, for what value of k are the vectors $$\overrightarrow {AB}$$ and $$\overrightarrow {BC}$$ perpendicular? | 11 | 50 |
25,130 | Each square of a $33\times 33$ square grid is colored in one of the three colors: red, yellow or blue, such that the numbers of squares in each color are the same. If two squares sharing a common edge are in different colors, call that common edge a separating edge. Find the minimal number of separating edges in the grid. | 56 | 0 |
25,131 | Given positive numbers $a$ and $b$ satisfying $a+b=1$, determine the minimum value of $\sqrt{ab}$. | \frac{1}{2} | 50.78125 |
25,132 | If $x = 3$ and $y = 5$, what is the value of $\frac{3x^4 + 2y^2 + 10}{8}$? | 37 | 4.6875 |
25,133 | The angles of quadrilateral $PQRS$ satisfy $\angle P = 3\angle Q = 4\angle R = 6\angle S$. What is the degree measure of $\angle P$? | 206 | 0 |
25,134 | On a spherical surface with an area of $60\pi$, there are four points $S$, $A$, $B$, and $C$, and $\triangle ABC$ is an equilateral triangle. The distance from the center $O$ of the sphere to the plane $ABC$ is $\sqrt{3}$. If the plane $SAB$ is perpendicular to the plane $ABC$, then the maximum volume of the pyramid $S-ABC$ is \_\_\_\_\_\_. | 27 | 3.125 |
25,135 | Let $ABC$ be a triangle such that $AB=6,BC=5,AC=7.$ Let the tangents to the circumcircle of $ABC$ at $B$ and $C$ meet at $X.$ Let $Z$ be a point on the circumcircle of $ABC.$ Let $Y$ be the foot of the perpendicular from $X$ to $CZ.$ Let $K$ be the intersection of the circumcircle of $BCY$ with line $AB.$ Given that $Y$ is on the interior of segment $CZ$ and $YZ=3CY,$ compute $AK.$ | 147/10 | 0 |
25,136 | The diagram depicts a bike route through a park, along with the lengths of some of its segments in kilometers. What is the total length of the bike route in kilometers? | 52 | 0 |
25,137 | Tom, Sara, and Jim split $\$1200$ among themselves to be used in investments. Each starts with a different amount. At the end of one year, they have a total of $\$1800$. Sara triples her money by the end of the year, Jim doubles his money, whereas Tom loses $\$200$. What was Tom's original amount of money? | 400 | 11.71875 |
25,138 | Equilateral triangle $DEF$ has each side equal to $9$. A circle centered at $Q$ is tangent to side $DE$ at $D$ and passes through $F$. Another circle, centered at $R$, is tangent to side $DF$ at $F$ and passes through $E$. Find the magnitude of segment $QR$.
A) $12\sqrt{3}$
B) $9\sqrt{3}$
C) $15$
D) $18$
E) $9$ | 9\sqrt{3} | 14.0625 |
25,139 | How many binary strings of length $10$ do not contain the substrings $101$ or $010$ ? | 178 | 80.46875 |
25,140 | A, B, C, D, and E are five students who obtained the top 5 positions (no ties) in a math competition. When taking a photo, they stood in a line and each of them made the following statement:
A said: The two students next to me have rankings lower than mine;
B said: The two students next to me have rankings adjacent to mine;
C said: All students to my right (at least one) have higher rankings than mine;
D said: All students to my left (at least one) have lower rankings than mine;
E said: I am standing in second from the right.
Given that all their statements are true, determine the five-digit number $\overline{\mathrm{ABCDE}}$. | 23514 | 0 |
25,141 | We define a number as an ultimate mountain number if it is a 4-digit number and the third digit is larger than the second and fourth digit but not necessarily the first digit. For example, 3516 is an ultimate mountain number. How many 4-digit ultimate mountain numbers are there? | 204 | 0 |
25,142 | Two cars, Car A and Car B, start simultaneously from points A and B, respectively, and move towards each other. After 6 hours, the distance they have traveled is 3/5 of the distance between points A and B. Car A travels at a speed of 42 km per hour, which is 1/7 less than Car B's hourly speed. Find the distance between points A and B in meters. | 780 | 0 |
25,143 | The 79 trainees of the Animath workshop each choose an activity for the free afternoon among 5 offered activities. It is known that:
- The swimming pool was at least as popular as soccer.
- The students went shopping in groups of 5.
- No more than 4 students played cards.
- At most one student stayed in their room.
We write down the number of students who participated in each activity. How many different lists could we have written? | 3240 | 7.03125 |
25,144 | Compute $\binom{18}{6}$. | 18564 | 100 |
25,145 | A one-meter gas pipe has rusted in two places. Determine the probability that all three resulting parts can be used as connections to gas stoves, if the regulations stipulate that the stove should not be closer than 25 cm to the main gas pipe. | 9/16 | 0 |
25,146 | At the signal of the trainer, two ponies simultaneously started running uniformly along the outer circumference of the circus arena in opposite directions. The first pony ran slightly faster than the second and, by the time they met, had run 5 meters more than the second pony. Continuing to run, the first pony reached the trainer, who remained at the place from which the ponies started running, 9 seconds after meeting the second pony, while the second pony reached the trainer 16 seconds after their meeting. What is the diameter of the arena? | 6.25 | 0 |
25,147 | In triangle $PQR,$ $PQ = 25$ and $QR = 10.$ Find the largest possible value of $\tan B.$ | \frac{2}{\sqrt{21}} | 0 |
25,148 | Let $$Q(x) = \left(\frac{x^{24} - 1}{x - 1}\right)^2 - x^{23}$$ and this polynomial has complex zeros in the form $z_k = r_k[\cos(2\pi\alpha_k) + i\sin(2\pi\alpha_k)]$, for $k = 1, 2, ..., 46$, where $0 < \alpha_1 \le \alpha_2 \le \ldots \le \alpha_{46} < 1$ and $r_k > 0$. Find $\alpha_1 + \alpha_2 + \alpha_3 + \alpha_4 + \alpha_5.$ | \frac{121}{575} | 0 |
25,149 | The numbers \( x_1, x_2, x_3, y_1, y_2, y_3, z_1, z_2, z_3 \) are equal to the numbers \( 1, 2, 3, \ldots, 9 \) in some order. Find the smallest possible value of
\[ x_1 x_2 x_3 + y_1 y_2 y_3 + z_1 z_2 z_3. \] | 214 | 75.78125 |
25,150 | A robot invented a cipher for encoding words: it replaced certain letters of the alphabet with one-digit or two-digit numbers, using only the digits 1, 2, and 3 (different letters were replaced with different numbers). Initially, it encoded itself: ROBOT = 3112131233. After encoding the words CROCODILE and HIPPOPOTAMUS, it was surprised to find that the resulting numbers were exactly the same! Then, the robot encoded the word MATHEMATICS. Write down the number it obtained. Justify your answer.
| 2232331122323323132 | 0 |
25,151 | Suppose the 9-digit number $\overline{32 x 35717 y}$ is a multiple of 72, and $P = xy$. Find the value of $P$. | 144 | 0 |
25,152 | A dragon is tethered by a 25-foot golden rope to the base of a sorcerer's cylindrical tower whose radius is 10 feet. The rope is attached to the tower at ground level and to the dragon at a height of 7 feet. The dragon has pulled the rope taut, the end of the rope is 5 feet from the nearest point on the tower, and the length of the rope that is touching the tower is \(\frac{d-\sqrt{e}}{f}\) feet, where \(d, e,\) and \(f\) are positive integers, and \(f\) is prime. Find \(d+e+f.\) | 862 | 0 |
25,153 | A sphere is tangent to all the edges of the pyramid \( SABC \), specifically the lateral edges \( SA, SB, \) and \( SC \) at the points \( A', B' \), and \( C' \) respectively. Find the volume of the pyramid \( SA'B'C' \), given that \( AB = BC = SB = 5 \) and \( AC = 4 \). | \frac{2 \sqrt{59}}{15} | 0 |
25,154 | The side length of an equilateral triangle ABC is 2. Calculate the area of the orthographic (isometric) projection of triangle ABC. | \frac{\sqrt{6}}{4} | 0 |
25,155 | At Frank's Fruit Market, 6 bananas cost as much as 4 apples, and 5 apples cost as much as 3 oranges. In addition, 4 oranges cost as much as 7 pears. How many pears cost as much as 36 bananas? | 28 | 0 |
25,156 | Given the sequence 1, 2, 1, 2, 2, 1, 2, 2, 2, 1..., where each pair of 1s is separated by 2s, and there are n 2s between the nth pair of 1s, the sum of the first 1234 terms of this sequence is ______. | 2419 | 67.1875 |
25,157 | Every day, 12 tons of potatoes are delivered to the city using one type of transport from three collective farms. The price per ton is 4 rubles from the first farm, 3 rubles from the second farm, and 1 ruble from the third farm. To ensure timely delivery, the loading process for the required 12 tons should take no more than 40 minutes. It is known that the level of mechanization in the first farm allows loading 1 ton in 1 minute, in the second farm - 4 minutes, and in the third farm - 3 minutes. The production capacities of these farms are such that the first farm can supply no more than 10 tons per day, the second no more than 8 tons, and the third no more than 6 tons. How should the order for the supply of 12 tons be distributed among the farms so that the total cost of the potatoes delivered to the city is minimized? | 24.6667 | 0 |
25,158 | Mr. Fast eats a pizza slice in 5 minutes, Mr. Slow eats a pizza slice in 10 minutes, and Mr. Steady eats a pizza slice in 8 minutes. How long does it take for them to eat 24 pizza slices together? Express your answer in minutes. | 56 | 0.78125 |
25,159 | What is the smallest positive integer $x$ that, when multiplied by $450$, results in a product that is a multiple of $800$? | 32 | 0.78125 |
25,160 | Given a non-empty set of numbers, the sum of its maximum and minimum elements is called the "characteristic value" of the set. $A\_1$, $A\_2$, $A\_3$, $A\_4$, $A\_5$ each contain $20$ elements, and $A\_1∪A\_2∪A\_3∪A\_4∪A\_5={x∈N^⁎|x≤slant 100}$, find the minimum value of the sum of the "characteristic values" of $A\_1$, $A\_2$, $A\_3$, $A\_4$, $A\_5$. | 325 | 0 |
25,161 | What is the positive difference of the solutions of $\dfrac{s^2 - 4s - 22}{s + 3} = 3s + 8$? | \frac{27}{2} | 0 |
25,162 | A circle is inscribed in quadrilateral \( ABCD \), tangent to \( \overline{AB} \) at \( P \) and to \( \overline{CD} \) at \( Q \). Given that \( AP=15 \), \( PB=35 \), \( CQ=45 \), and \( QD=25 \), find the square of the radius of the circle. | 160 | 0 |
25,163 | In a given isosceles right triangle, a square is inscribed such that its one vertex touches the right angle vertex of the triangle and its two other vertices touch the legs of the triangle. If the area of this square is found to be $784 \text{cm}^2$, determine the area of another square inscribed in the same triangle where the square fits exactly between the hypotenuse and the legs of the triangle. | 784 | 3.125 |
25,164 | A triangle $H$ is inscribed in a regular hexagon $S$ such that one side of $H$ is parallel to one side of $S$. What is the maximum possible ratio of the area of $H$ to the area of $S$? | 3/8 | 0 |
25,165 | Given a triangle whose three sides are all positive integers, with only one side length equal to 5 and not the shortest side, find the number of such triangles. | 10 | 6.25 |
25,166 | On graph paper (1 cell = 1 cm), two equal triangles ABC and BDE are depicted.
Find the area of their common part. | 0.8 | 0 |
25,167 | A semicircle of diameter 3 sits at the top of a semicircle of diameter 4, as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a $\textit{lune}$. Determine the area of this lune. Express your answer in terms of $\pi$ and in simplest radical form. | \frac{11}{24}\pi | 0 |
25,168 | A sealed bottle, which contains water, has been constructed by attaching a cylinder of radius \(1 \text{ cm}\) to a cylinder of radius \(3 \text{ cm}\). When the bottle is right side up, the height of the water inside is \(20 \text{ cm}\). When the bottle is upside down, the height of the liquid is \(28 \text{ cm}\). What is the total height, in \(\text{cm}\), of the bottle? | 29 | 3.125 |
25,169 | Suppose that 9 boys and 15 girls line up in a row, but the arrangement must start with a boy and end with a girl. Let $T$ be the number of places in the row where a boy and a girl are standing next to each other. Calculate the average value of $T$ when all possible orders of these 24 people under the given conditions are considered. | 12 | 3.125 |
25,170 | The sum of the ages of three people A, B, and C, denoted as \(x, y, z\), is 120, with \(x, y, z \in (20, 60)\). How many ordered triples \((x, y, z)\) satisfy this condition? | 1198 | 0 |
25,171 | Calculate the area of a rectangle which remains unchanged when it is made $3 \frac{1}{2}$ inches longer and $1 \frac{1}{3}$ inch narrower, or when it is made $3 \frac{1}{2}$ inches shorter and $1 \frac{2}{3}$ inch wider. | 35 | 0 |
25,172 | For positive integers $N$ and $k$ define $N$ to be $k$-nice if there exists a positive integer $a$ such that $a^k$ has exactly $N$ positive divisors. Determine the quantity of positive integers smaller than $1500$ that are neither $9$-nice nor $10$-nice. | 1199 | 0.78125 |
25,173 | What is the smallest positive integer with exactly 18 positive divisors? | 288 | 0 |
25,174 | Given vectors $\overrightarrow {OA} = (2, -3)$, $\overrightarrow {OB} = (-5, 4)$, $\overrightarrow {OC} = (1-\lambda, 3\lambda+2)$:
1. If $\triangle ABC$ is a right-angled triangle and $\angle B$ is the right angle, find the value of the real number $\lambda$.
2. If points A, B, and C can form a triangle, determine the condition that the real number $\lambda$ must satisfy. | -2 | 0 |
25,175 | A set \( \mathcal{S} \) of distinct positive integers has the property that for every integer \( x \) in \( \mathcal{S}, \) the arithmetic mean of the set of values obtained by deleting \( x \) from \( \mathcal{S} \) is an integer. Given that 1 belongs to \( \mathcal{S} \) and that 2310 is the largest element of \( \mathcal{S}, \) and also \( n \) must be a prime, what is the greatest number of elements that \( \mathcal{S} \) can have? | 20 | 3.125 |
25,176 | Let $a,$ $b,$ $c,$ $d,$ $e$ be positive real numbers such that $a^2 + b^2 + c^2 + d^2 + e^2 = 100.$ Let $N$ be the maximum value of
\[ac + 3bc + 4cd + 8ce,\]and let $a_N,$ $b_N$, $c_N,$ $d_N,$ $e_N$ be the values of $a,$ $b,$ $c,$ $d,$ $e,$ respectively, that produce the maximum value of $N.$ Find $N + a_N + b_N + c_N + d_N + e_N.$ | 16 + 150\sqrt{10} + 5\sqrt{2} | 0 |
25,177 | Let \( f \) be the function defined by \( f(x) = -3 \sin(\pi x) \). How many values of \( x \) such that \(-3 \le x \le 3\) satisfy the equation \( f(f(f(x))) = f(x) \)? | 79 | 0 |
25,178 | Simplify the following expressions:
(1) $\sin^{4}{\alpha} + \tan^{2}{\alpha} \cdot \cos^{4}{\alpha} + \cos^{2}{\alpha}$
(2) $\frac{\cos{(180^{\circ} + \alpha)} \cdot \sin{(\alpha + 360^{\circ})}}{\sin{(-\alpha - 180^{\circ})} \cdot \cos{(-180^{\circ} - \alpha)}}$ | -1 | 5.46875 |
25,179 | As shown in the diagram, square ABCD and square EFGH have their corresponding sides parallel to each other. Line CG is extended to intersect with line BD at point I. Given that BD = 10, the area of triangle BFC is 3, and the area of triangle CHD is 5, what is the length of BI? | 15/4 | 9.375 |
25,180 | Given the cubic equation $10x^3 - 25x^2 + 8x - 1 = 0$, whose roots are $p$, $q$, and $s$, all of which are positive and less than 1. Calculate the sum of
\[\frac{1}{1-p} + \frac{1}{1-q} + \frac{1}{1-s}.\] | 0.5 | 0.78125 |
25,181 | When the square of four times a positive integer is decreased by twice the integer, the result is $8066$. What is the integer? | 182 | 0 |
25,182 | Given the numbers 1, 3, 5 and 2, 4, 6, calculate the total number of different three-digit numbers that can be formed when arranging these numbers on three cards. | 48 | 1.5625 |
25,183 | Compute $\dbinom{20}{5}$. | 11628 | 0 |
25,184 | How many times does the digit 8 appear in the list of all integers from 1 to 800? | 161 | 1.5625 |
25,185 | Rationalize the denominator: $$\frac{1}{\sqrt[3]{3}+\sqrt[3]{27}}$$ | \frac{\sqrt[3]{9}}{12} | 0.78125 |
25,186 | For every $0 < \alpha < 1$ , let $R(\alpha)$ be the region in $\mathbb{R}^2$ whose boundary is the convex pentagon of vertices $(0,1-\alpha), (\alpha, 0), (1, 0), (1,1)$ and $(0, 1)$ . Let $R$ be the set of points that belong simultaneously to each of the regions $R(\alpha)$ with $0 < \alpha < 1$ , that is, $R =\bigcap_{0<\alpha<1} R(\alpha)$ .
Determine the area of $R$ . | 2/3 | 0 |
25,187 | Find the result of $(1011101_2 + 1101_2) \times 101010_2 \div 110_2$. Express your answer in base 2. | 1110111100_2 | 0 |
25,188 | Let $N$ be the number of positive integers that are less than or equal to $5000$ and whose base-$3$ representation has more $1$'s than any other digit. Find the remainder when $N$ is divided by $1000$. | 379 | 0 |
25,189 | In quadrilateral \(ABCD\), \(AB = BC\), \(\angle A = \angle B = 20^{\circ}\), \(\angle C = 30^{\circ}\). The extension of side \(AD\) intersects \(BC\) at point... | 30 | 2.34375 |
25,190 | Given $\begin{vmatrix} x & y \\ z & w \end{vmatrix} = 3,$ find $\begin{vmatrix} 3x & 3y \\ 3z & 3w \end{vmatrix}.$ | 27 | 99.21875 |
25,191 | Complex numbers $p,$ $q,$ and $r$ are zeros of a polynomial $Q(z) = z^3 + sz^2 + tz + u,$ and $|p|^2 + |q|^2 + |r|^2 = 360.$ The points corresponding to $p,$ $q,$ and $r$ in the complex plane are the vertices of a right triangle with hypotenuse $k.$ Find $k^2.$ | 540 | 0.78125 |
25,192 | Given the function $y=\sin (2x+1)$, determine the direction and magnitude of the horizontal shift required to obtain this graph from the graph of the function $y=\sin 2x$. | \frac{1}{2} | 17.1875 |
25,193 | A barn with a roof is rectangular in shape, $12$ yd. wide, $15$ yd. long, and $6$ yd. high. Calculate the total area to be painted. | 828 | 0 |
25,194 | Find the distance between the foci of the hyperbola given by the equation \(x^2 - 4x - 12y^2 + 24y = -36.\) | \frac{2\sqrt{273}}{3} | 0 |
25,195 | The angle bisectors of triangle \( A B C \) intersect at point \( I \), and the external angle bisectors of angles \( B \) and \( C \) intersect at point \( J \). The circle \( \omega_{b} \) with center at point \( O_{b} \) passes through point \( B \) and is tangent to line \( C I \) at point \( I \). The circle \( \omega_{c} \) with center at point \( O_{c} \) passes through point \( C \) and is tangent to line \( B I \) at point \( I \). The segments \( O_{b} O_{c} \) and \( I J \) intersect at point \( K \). Find the ratio \( I K / K J \). | 1/3 | 0 |
25,196 | Given a unit square region $R$ and an integer $n \geq 4$, determine how many points are $80$-ray partitional but not $50$-ray partitional. | 7062 | 0 |
25,197 | Given the function $f(x)=x\ln x-x$, find the monotonic intervals and the extreme values of the function $f(x)$. | -1 | 9.375 |
25,198 | Determine the required distance between the pins and the length of the string in order to draw an ellipse with a length of 12 cm and a width of 8 cm. Find a simple rule that allows for constructing an ellipse of predetermined dimensions. | 24 | 0 |
25,199 | What is the sum of all positive integers $n$ that satisfy $$\mathop{\text{lcm}}[n,120] = \gcd(n,120) + 600~?$$ | 2520 | 0.78125 |
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