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In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $B= \frac {\pi}{3}$ and $(a-b+c)(a+b-c)= \frac {3}{7}bc$.
(Ⅰ) Find the value of $\cos C$;
(Ⅱ) If $a=5$, find the area of $\triangle ABC$. | 10 \sqrt {3} | 6/8 |
Let $F$ be a finite field having an odd number $m$ of elements. Let $p(x)$ be an irreducible (i.e. nonfactorable) polynomial over $F$ of the form $$ x^2+bx+c, ~~~~~~ b,c \in F. $$ For how many elements $k$ in $F$ is $p(x)+k$ irreducible over $F$ ? | \frac{-1}{2} | 6/8 |
$A_1, A_2, ..., A_n$ are the subsets of $|S|=2019$ such that union of any three of them gives $S$ but if we combine two of subsets it doesn't give us $S$ . Find the maximum value of $n$ . | 64 | 5/8 |
Given positive integers \( a_{0}, a_{1}, a_{2}, \cdots, a_{99}, a_{100} \) such that
\[ a_{1} > a_{0}, \]
\[ a_{n} = 3a_{n-1} - 2a_{n-2}, \quad n = 2, 3, \cdots, 100. \]
Prove that \( a_{100} > 2^{99} \). | a_{100}>2^{99} | 7/8 |
Determine the sum of all distinct real values of $x$ such that $|||\cdots||x|+x|\cdots|+x|+x|=1$ where there are 2017 $x$ 's in the equation. | -\frac{2016}{2017} | 7/8 |
Points $A_1, A_2, \ldots, A_{2022}$ are chosen on a plane so that no three of them are collinear. Consider all angles $A_iA_jA_k$ for distinct points $A_i, A_j, A_k$ . What largest possible number of these angles can be equal to $90^\circ$ ?
*Proposed by Anton Trygub* | 2,042,220 | 1/8 |
The extensions of a telephone exchange have only 2 digits, from 00 to 99. Not all extensions are in use. By swapping the order of two digits of an extension in use, you either get the same number or the number of an extension not in use. What is the highest possible number of extensions in use?
(a) Less than 45
(b) 45
(c) Between 45 and 55
(d) More than 55
(e) 55 | 55 | 1/8 |
Each side of triangle \(ABC\) is divided into 8 equal segments. How many different triangles with vertices at the division points (points \(A, B, C\) cannot be the vertices of these triangles) exist such that none of their sides is parallel to any side of triangle \(ABC\)? | 216 | 1/8 |
Given the product sequence $\frac{5}{3} \cdot \frac{6}{5} \cdot \frac{7}{6} \cdot \ldots \cdot \frac{a}{b} = 12$, determine the sum of $a$ and $b$. | 71 | 7/8 |
Amy, Beth, and Claire each have some sweets. Amy gives one third of her sweets to Beth. Beth gives one third of all the sweets she now has to Claire. Then Claire gives one third of all the sweets she now has to Amy. All the girls end up having the same number of sweets.
Claire begins with 40 sweets. How many sweets does Beth have originally? | 50 | 7/8 |
The line $c$ is defined by the equation $y = 2x$. Points $A$ and $B$ have coordinates $A(2, 2)$ and $B(6, 2)$. On line $c$, find the point $C$ from which the segment $AB$ is seen at the largest angle. | (2,4) | 7/8 |
In the production of a steel cable, it was found that the cable has the same length as the curve defined by the system of equations:
$$
\left\{\begin{array}{l}
x + y + z = 8 \\
xy + yz + xz = -18
\end{array}\right.
$$
Find the length of the cable. | 4\pi \sqrt{\frac{59}{3}} | 1/8 |
The base of a pyramid is a triangle with sides 6, 5, and 5. The lateral faces of the pyramid form $45^{\circ}$ angles with the plane of the base. Find the volume of the pyramid. | 6 | 7/8 |
A five-digit natural number $\overline{a_1a_2a_3a_4a_5}$ is considered a "concave number" if and only if $a_1 > a_2 > a_3$ and $a_3 < a_4 < a_5$, with each $a_i \in \{0,1,2,3,4,5\}$ for $i=1,2,3,4,5$. Calculate the number of possible "concave numbers". | 146 | 7/8 |
We have an empty equilateral triangle with length of a side $l$ . We put the triangle, horizontally, over a sphere of radius $r$ . Clearly, if the triangle is small enough, the triangle is held by the sphere. Which is the distance between any vertex of the triangle and the centre of the sphere (as a function of $l$ and $r$ )? | \sqrt{r^2+\frac{^2}{4}} | 3/8 |
Determine the range of $w(w + x)(w + y)(w + z)$, where $x, y, z$, and $w$ are real numbers such that
\[x + y + z + w = x^7 + y^7 + z^7 + w^7 = 0.\] | 0 | 1/8 |
The union of sets \( A \) and \( B \), \( A \cup B = \{a_1, a_2, a_3\} \), and \( A \neq B \). When \( (A, B) \) and \( (B, A) \) are considered as different pairs, how many such pairs \( (A, B) \) are there? | 27 | 1/8 |
Three squirrels typically eat porridge for breakfast: semolina (M), millet (P), oatmeal (O), buckwheat (G). No porridge is liked by all three squirrels, but for every pair of squirrels, there is at least one porridge that they both like. How many different tables can be made where each cell contains either a plus (if liked) or a minus (if not liked)?
| | M | P | O | G |
| :--- | :--- | :--- | :--- | :--- |
| Squirrel 1 | | | | |
| Squirrel 2 | | | | |
| Squirrel 3 | | | | | | 132 | 2/8 |
The radius of a sphere is \( r = 10 \text{ cm} \). Determine the volume of the spherical segment whose surface area is in the ratio 10:7 compared to the area of its base. | 288 \pi | 7/8 |
A shooter, in a shooting training session, has the probabilities of hitting the 10, 9, 8, and 7 rings as follows:
0.21, 0.23, 0.25, 0.28, respectively. Calculate the probability that the shooter in a single shot:
(1) Hits either the 10 or 9 ring; (2) Scores less than 7 rings. | 0.03 | 6/8 |
In a store, there are 21 white shirts and 21 purple shirts hanging in a row. Find the minimum number \( k \) such that, regardless of the initial order of the shirts, it is possible to take down \( k \) white shirts and \( k \) purple shirts so that the remaining white shirts are all hanging consecutively and the remaining purple shirts are also hanging consecutively. | 10 | 1/8 |
28 apples weigh 3 kilograms. If they are evenly divided into 7 portions, each portion accounts for $\boxed{\frac{1}{7}}$ of all the apples, and each portion weighs $\boxed{\frac{3}{7}}$ kilograms. | \frac{3}{7} | 7/8 |
The values of $a$ in the equation: $\log_{10}(a^2 - 15a) = 2$ are: | 20, -5 | 1/8 |
From the consecutive natural numbers \(1, 2, 3, \ldots, 2014\), select \( n \) numbers such that any two selected numbers do not have the property that one number is five times the other. Find the maximum value of \( n \) and provide a justification. | 1679 | 4/8 |
There are 10 cities in a state, and some pairs of cities are connected by 40 roads altogether. A city is called a "hub" if it is directly connected to every other city. What is the largest possible number of hubs? | 6 | 5/8 |
Given that $x$ is a multiple of $2520$, what is the greatest common divisor of $g(x) = (4x+5)(5x+2)(11x+8)(3x+7)$ and $x$? | 280 | 2/8 |
The side lengths of a triangle are integers. It is known that the altitude drawn to one of the sides divides it into integer segments, the difference of which is 7. For what minimum value of the length of this side, the altitude drawn to it is also an integer? | 25 | 1/8 |
Find all possible three-digit numbers that can be obtained by removing three digits from the number 112277. Sum them and write the result as the answer. | 1159 | 2/8 |
There are very many symmetrical dice. They are thrown simultaneously. With a certain probability \( p > 0 \), it is possible to get a sum of 2022 points. What is the smallest sum of points that can fall with the same probability \( p \)? | 337 | 7/8 |
Solve for the range of real numbers \( k \) such that the equation \( k \cdot 9^{x} - k \cdot 3^{x+1} + 6(k-5) = 0 \) has solutions for \( x \) in the interval \([0, 2]\). | [\frac{1}{2},8] | 6/8 |
Let $[x]$ denote the greatest integer not exceeding the real number $x$, and $\{x\} = x - [x]$. Find the solutions to the equation
$$
[x]^{4}+\{x\}^{4}+x^{4}=2048
$$ | -3-\sqrt{5} | 2/8 |
If \( z_{1}, z_{2}, \cdots, z_{n} \in \mathbf{C}, \left|z_{1}\right|=\left|z_{2}\right|=\cdots=\left|z_{n}\right|=r \neq 0, \) and \( T_{s} \) represents the sum of all possible products of \( s \) out of these \( n \) numbers, find \( \left|\frac{T_{s}}{T_{n-s}}\right| \) in terms of \( r \) given that \( T_{n-s} \neq 0 \). | r^{2s-n} | 4/8 |
Egor, Nikita, and Innokentiy took turns playing chess with each other (two play, one watches). After each game, the loser gave up their spot to the spectator (there were no draws). It turned out that Egor participated in 13 games, and Nikita in 27 games. How many games did Innokentiy play? | 14 | 1/8 |
Given a regular quadrilateral pyramid. The side of the base is 12 and the length of the lateral edge is 10. Sphere \( Q_{1} \) is inscribed in the pyramid. Sphere \( Q_{2} \) touches \( Q_{1} \) and all lateral faces of the pyramid. Find the radius of sphere \( Q_{2} \). | \frac{6\sqrt{7}}{49} | 5/8 |
The teacher gave the children a task to multiply the number written on the board by three and add seven to the result. Kamča solved the task correctly. Her friend Růža also calculated correctly, but a different example: she multiplied the given number by seven and added three to the result. Růža's result was 84 more than Kamča's result.
What number was written on the board? | 22 | 7/8 |
Horizontal parallel segments $A B=10$ and $C D=15$ are the bases of trapezoid $A B C D$. Circle $\gamma$ of radius 6 has center within the trapezoid and is tangent to sides $A B, B C$, and $D A$. If side $C D$ cuts out an arc of $\gamma$ measuring $120^{\circ}$, find the area of $A B C D$. | \frac{225}{2} | 2/8 |
Sixteen 6-inch wide square posts are evenly spaced with 6 feet between them to enclose a square field. What is the outer perimeter, in feet, of the fence? | 106 | 2/8 |
A circle with radius \( u_a \) is inscribed in angle \( A \) of triangle \( ABC \), and a circle with radius \( u_b \) is inscribed in angle \( B \); these circles touch each other externally. Prove that the radius of the circumscribed circle of the triangle with sides \( a_1 = \sqrt{u_a \cot \frac{\alpha}{2}}, b_1 = \sqrt{u_b \cot \frac{\beta}{2}}, \quad c_1=\sqrt{c} \) is equal to \( \frac{\sqrt{p}}{2} \), where \( p \) is the semiperimeter of triangle \( ABC \). | \frac{\sqrt{p}}{2} | 1/8 |
A school is hosting a Mathematics Culture Festival, and it was recorded that on that day, there were more than 980 (at least 980 and less than 990) students visiting. Each student visits the school for a period of time and then leaves, and once they leave, they do not return. Regardless of how these students schedule their visit, we can always find \( k \) students such that either all \( k \) students are present in the school at the same time, or at any time, no two of them are present in the school simultaneously. Find the maximum value of \( k \). | 32 | 6/8 |
In the triangular pyramid \(ABCD\) with base \(ABC\), the lateral edges are pairwise perpendicular, \(DA = DB = 5, DC = 1\). A ray of light is emitted from a point on the base. After reflecting exactly once from each lateral face (the ray does not reflect from the edges), the ray hits a point on the pyramid's base. What is the minimum distance the ray could travel? | \frac{10 \sqrt{3}}{9} | 2/8 |
The midpoints of the sides \(PQ\), \(QR\), and \(RP\) of triangle \(PQR\) are \(A\), \(B\), and \(C\) respectively. The triangle \(ABC\) is enlarged from its centroid \(S\) by a factor of \(k\), where \(1 < k < 4\). The sides of the resulting enlarged triangle intersect the sides of triangle \(PQR\) at points \(D_{1}D_{2}\), \(E_{1}E_{2}\), and \(F_{1}F_{2}\) respectively. For which value of \(k\) will the area of the hexagon \(D_{1}D_{2}E_{1}E_{2}F_{1}F_{2}\) be twice the area of triangle \(ABC\)? | 4-\sqrt{6} | 1/8 |
The product $(8)(888\dots8)$, where the second factor has $k$ digits, is an integer whose digits have a sum of $1000$. What is $k$? | 991 | 7/8 |
Let \( F(x) \) and \( G(x) \) be polynomials of degree 2021. It is known that for all real \( x \), \( F(F(x)) = G(G(x)) \) holds, and there exists a real number \( k \neq 0 \) such that for all real \( x \), \( F(k F(F(x))) = G(k G(G(x))) \) holds. Find the degree of the polynomial \( F(x) - G(x) \). | 0 | 2/8 |
Given that $f(x)$ is an odd function on $\mathbb{R}$, when $x\geqslant 0$, $f(x)= \begin{cases} \log _{\frac {1}{2}}(x+1),0\leqslant x < 1 \\ 1-|x-3|,x\geqslant 1\end{cases}$. Find the sum of all the zeros of the function $y=f(x)+\frac {1}{2}$. | \sqrt {2}-1 | 7/8 |
Xiao Ming has multiple 1 yuan, 2 yuan, and 5 yuan banknotes. He wants to buy a kite priced at 18 yuan using no more than 10 of these banknotes and must use at least two different denominations. How many different ways can he pay? | 11 | 5/8 |
Given a sequence $\left\{a_{n}\right\}$, where $a_{1}=a_{2}=1$, $a_{3}=-1$, and $a_{n}=a_{n-1} a_{n-3}$, find $a_{1964}$. | -1 | 3/8 |
The image shows a grid consisting of 25 small equilateral triangles. How many rhombuses can be formed from two adjacent small triangles? | 30 | 6/8 |
Let $P(x)=x^3+ax^2+bx+c$ be a polynomial where $a,b,c$ are integers and $c$ is odd. Let $p_{i}$ be the value of $P(x)$ at $x=i$ . Given that $p_{1}^3+p_{2}^{3}+p_{3}^{3}=3p_{1}p_{2}p_{3}$ , find the value of $p_{2}+2p_{1}-3p_{0}.$ | 18 | 7/8 |
In space, there are four spheres with radii of 2, 2, 3, and 3. Each sphere is externally tangent to the other three spheres. There is an additional small sphere that is externally tangent to all four of these spheres. Find the radius of this small sphere. | \frac{6}{11} | 5/8 |
In a cone with a base radius of 2 and a slant height of \( c \), three spheres with radius \( r \) are placed. These spheres touch each other externally, touch the lateral surface of the cone, and the first two spheres touch the base of the cone. Find the maximum value of \( r \). | \sqrt{3}-1 | 1/8 |
The number of ounces of water needed to reduce $9$ ounces of shaving lotion containing $50$ % alcohol to a lotion containing $30$ % alcohol is:
$\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7$ | \textbf{(D)}6 | 1/8 |
In a household, when someone is at home, the probability of a phone call being answered at the first ring is 0.1, at the second ring is 0.3, at the third ring is 0.4, and at the fourth ring is 0.1. Calculate the probability of the phone call being answered within the first four rings. | 0.9 | 4/8 |
Let \( S \) be a set of 9 distinct integers all of whose prime factors are at most 3. Prove that \( S \) contains 3 distinct integers such that their product is a perfect cube. | 3 | 4/8 |
Find maximum of the expression $(a -b^2)(b - a^2)$ , where $0 \le a,b \le 1$ . | \frac{1}{16} | 4/8 |
Ed has five identical green marbles, and a large supply of identical red marbles. He arranges the green marbles and some of the red ones in a row and finds that the number of marbles whose right hand neighbor is the same color as themselves is equal to the number of marbles whose right hand neighbor is the other color. An example of such an arrangement is GGRRRGGRG. Let $m$ be the maximum number of red marbles for which such an arrangement is possible, and let $N$ be the number of ways he can arrange the $m+5$ marbles to satisfy the requirement. Find the remainder when $N$ is divided by $1000$.
| 3 | 1/8 |
The deli has four kinds of bread, six kinds of meat, and five kinds of cheese. A sandwich consists of one type of bread, one type of meat, and one type of cheese. Ham, chicken, cheddar cheese, and white bread are each offered at the deli. If Al never orders a sandwich with a ham/cheddar cheese combination nor a sandwich with a white bread/chicken combination, how many different sandwiches could Al order? | 111 | 3/8 |
How many ordered triples \((x, y, z)\) satisfy the following conditions:
\[ x^2 + y^2 + z^2 = 9, \]
\[ x^4 + y^4 + z^4 = 33, \]
\[ xyz = -4? \] | 12 | 6/8 |
Masha talked a lot on the phone with her friends, and the charged battery discharged exactly after a day. It is known that the charge lasts for 5 hours of talk time or 150 hours of standby time. How long did Masha talk with her friends? | 126/29 | 7/8 |
The degrees of polynomials $P$ and $Q$ with real coefficients do not exceed $n$ . These polynomials satisfy the identity
\[ P(x) x^{n + 1} + Q(x) (x+1)^{n + 1} = 1. \]
Determine all possible values of $Q \left( - \frac{1}{2} \right)$ . | 2^n | 2/8 |
Six regular hexagons surround a regular hexagon of side length $1$ as shown. What is the area of $\triangle{ABC}$? | $3\sqrt{3}$ | 1/8 |
Find the largest integer $n$ which equals the product of its leading digit and the sum of its digits. | 48 | 1/8 |
Let $f(x)=x^{3}+3 x-1$ have roots $a, b, c$. Given that $$\frac{1}{a^{3}+b^{3}}+\frac{1}{b^{3}+c^{3}}+\frac{1}{c^{3}+a^{3}}$$ can be written as $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n)=1$, find $100 m+n$. | 3989 | 7/8 |
In a particular game, each of $4$ players rolls a standard $6$-sided die. The winner is the player who rolls the highest number. If there is a tie for the highest roll, those involved in the tie will roll again and this process will continue until one player wins. Hugo is one of the players in this game. What is the probability that Hugo's first roll was a $5,$ given that he won the game? | \frac{41}{144} | 1/8 |
Let \(\Gamma\) be a circle in the plane and \(S\) be a point on \(\Gamma\). Mario and Luigi drive around the circle \(\Gamma\) with their go-karts. They both start at \(S\) at the same time. They both drive for exactly 6 minutes at constant speed counterclockwise around the track. During these 6 minutes, Luigi makes exactly one lap around \(\Gamma\) while Mario, who is three times as fast, makes three laps.
While Mario and Luigi drive their go-karts, Princess Daisy positions herself such that she is always exactly in the middle of the chord between them. When she reaches a point she has already visited, she marks it with a banana.
How many points in the plane, apart from \(S\), are marked with a banana by the end of the race? | 3 | 1/8 |
The famous skater Tony Hawk is riding a skateboard (segment $A B$) in a ramp, which is a semicircle with a diameter $P Q$. Point $M$ is the midpoint of the skateboard, and $C$ is the foot of the perpendicular dropped from point $A$ to the diameter $P Q$. What values can the angle $\angle A C M$ take, if it is known that the arc measure of $A B$ is $24^\circ$? | 12 | 3/8 |
Given the system of equations for positive numbers \( x, y, z \):
\[
\begin{cases}
x^2 + xy + y^2 = 27 \\
y^2 + yz + z^2 = 9 \\
z^2 + xz + x^2 = 36
\end{cases}
\]
find the value of the expression \( xy + yz + xz \). | 18 | 7/8 |
\( 427 \div 2.68 \times 16 \times 26.8 \div 42.7 \times 16 \) | 25600 | 7/8 |
A gambling student tosses a fair coin. She gains $1$ point for each head that turns up, and gains $2$ points for each tail that turns up. Prove that the probability of the student scoring *exactly* $n$ points is $\frac{1}{3}\cdot\left(2+\left(-\frac{1}{2}\right)^{n}\right)$ . | \frac{1}{3}(2+(-\frac{1}{2})^n) | 7/8 |
Given the function $f(x)=\frac{{1-x}}{{2x+2}}$.
$(1)$ Find the solution set of the inequality $f(2^{x})-2^{x+1}+2 \gt 0$;
$(2)$ If the function $g(x)$ satisfies $2f(2^{x})\cdot g(x)=2^{-x}-2^{x}$, and for any $x$ $(x\neq 0)$, the inequality $g(2x)+3\geqslant k\cdot [g(x)-2]$ always holds, find the maximum value of the real number $k$. | \frac{7}{2} | 7/8 |
If $a,b,c$ are positive integers less than $10$, then $(10a + b)(10a + c) = 100a(a + 1) + bc$ if:
$\textbf{(A) }b+c=10$
$\qquad\textbf{(B) }b=c$
$\qquad\textbf{(C) }a+b=10$
$\qquad\textbf {(D) }a=b$
$\qquad\textbf{(E) }a+b+c=10$ | \textbf{(A)} | 1/8 |
In a certain city, the payment scheme for metro rides using a card is as follows: the first ride costs 50 rubles, and each following ride costs either the same as the previous one or 1 ruble less. Petya spent 345 rubles on several rides, and then spent another 365 rubles on several subsequent rides. How many rides did he have? | 15trips | 1/8 |
A horse stands at the corner of a chessboard, on a white square. With each jump, the horse can move either two squares horizontally and one vertically or two vertically and one horizontally, like a knight moves. The horse earns two carrots every time it lands on a black square, but it must pay a carrot in rent to the rabbit who owns the chessboard for every move it makes. When the horse reaches the square on which it began, it can leave. What is the maximum number of carrots the horse can earn without touching any square more than twice? | 0 | 1/8 |
The diameter of the semicircle $AB=4$, with $O$ as the center, and $C$ is any point on the semicircle different from $A$ and $B$. Find the minimum value of $(\vec{PA}+ \vec{PB})\cdot \vec{PC}$. | -2 | 2/8 |
Vasisualiy Lokhankin has 8 identical cubes. Engineer Ptiburduk took a cube and wrote all the numbers from 1 to 6 on its faces, one number on each face; then he did the same with the remaining cubes. Is it necessarily possible for Vasisualiy to arrange these cubes into a $2 \times 2 \times 2$ cube so that the sum of the visible numbers equals 80? | No | 1/8 |
After the implementation of the "double reduction" policy, schools have attached importance to extended services and increased the intensity of sports activities in these services. A sports equipment store seized the opportunity and planned to purchase 300 sets of table tennis rackets and badminton rackets for sale. The number of table tennis rackets purchased does not exceed 150 sets. The purchase price and selling price are as follows:
| | Purchase Price (元/set) | Selling Price (元/set) |
|------------|-------------------------|------------------------|
| Table Tennis Racket | $a$ | $50$ |
| Badminton Racket | $b$ | $60$ |
It is known that purchasing 2 sets of table tennis rackets and 1 set of badminton rackets costs $110$ yuan, and purchasing 4 sets of table tennis rackets and 3 sets of badminton rackets costs $260$ yuan.
$(1)$ Find the values of $a$ and $b$.
$(2)$ Based on past sales experience, the store decided to purchase a number of table tennis rackets not less than half the number of badminton rackets. Let $x$ be the number of table tennis rackets purchased, and $y$ be the profit obtained after selling these sports equipment.
① Find the functional relationship between $y$ and $x$, and write down the range of values for $x$.
② When the store actually makes purchases, coinciding with the "Double Eleven" shopping festival, the purchase price of table tennis rackets is reduced by $a$ yuan per set $(0 < a < 10)$, while the purchase price of badminton rackets remains the same. Given that the selling price remains unchanged, and all the sports equipment can be sold out, how should the store make purchases to maximize profit? | 150 | 2/8 |
50 businessmen - Japanese, Koreans, and Chinese - are sitting at a round table. It is known that between any two nearest Japanese, there are exactly as many Chinese as there are Koreans at the table. How many Chinese can be at the table? | 32 | 1/8 |
Find the radius of the circle if an inscribed angle subtended by an arc of \(120^{\circ}\) has sides with lengths 1 and 2. | 1 | 7/8 |
Find the set of all attainable values of $\frac{ab+b^{2}}{a^{2}+b^{2}}$ for positive real $a, b$. | \left(0, \frac{1+\sqrt{2}}{2}\right] | 3/8 |
Given $2\cos(2\alpha)=\sin\left(\alpha-\frac{\pi}{4}\right)$ and $\alpha\in\left(\frac{\pi}{2},\pi\right)$, calculate the value of $\cos 2\alpha$. | \frac{\sqrt{15}}{8} | 7/8 |
Let $f(x)=(x^2+3x+2)^{\cos(\pi x)}$. Find the sum of all positive integers $n$ for which
\[\left |\sum_{k=1}^n\log_{10}f(k)\right|=1.\] | 21 | 7/8 |
The distances from the midpoint of the height of a regular triangular pyramid to a lateral face and to a lateral edge are 2 and $\sqrt{13}$, respectively. Find the volume of the pyramid. If necessary, round the answer to two decimal places. | 432.99 | 1/8 |
Compute
$$
\lim _{h \rightarrow 0} \frac{\sin \left(\frac{\pi}{3}+4 h\right)-4 \sin \left(\frac{\pi}{3}+3 h\right)+6 \sin \left(\frac{\pi}{3}+2 h\right)-4 \sin \left(\frac{\pi}{3}+h\right)+\sin \left(\frac{\pi}{3}\right)}{h^{4}}
$$ | \frac{\sqrt{3}}{2} | 4/8 |
Given a moving point $M$ whose distance to the point $F(0,1)$ is equal to its distance to the line $y=-1$, the trajectory of point $M$ is denoted as $C$.
(1) Find the equation of the trajectory $C$.
(2) Let $P$ be a point on the line $l: x-y-2=0$. Construct two tangents $PA$ and $PB$ from point $P$ to the curve $C$.
(i) When the coordinates of point $P$ are $\left(\frac{1}{2},-\frac{3}{2}\right)$, find the equation of line $AB$.
(ii) When point $P(x_{0},y_{0})$ moves along the line $l$, find the minimum value of $|AF|\cdot|BF|$. | \frac{9}{2} | 7/8 |
What is the largest $n$ for which $n$ distinct numbers can be arranged in a circle so that each number is equal to the product of its two neighbors? | 6 | 6/8 |
Given \( x \cdot y \in \left[-\frac{\pi}{4}, \frac{\pi}{4}\right] \), \( a \in \mathbf{R} \), and the system of equations \( \left\{ \begin{array}{l} x^{3} + \sin x - 2a = 0 \\ 4y^{3} + \frac{1}{2} \sin 2y - a = 0 \end{array} \right. \), find \( \cos(x + 2y) \). | 1 | 3/8 |
Compute the smallest positive integer \( n \) for which
\[
\sqrt{100+\sqrt{n}} + \sqrt{100-\sqrt{n}}
\]
is an integer. | 6156 | 4/8 |
A gymnastics team consists of 48 members. To form a square formation, they need to add at least ____ people or remove at least ____ people. | 12 | 2/8 |
The function \( f(x) \) is defined on the set of real numbers, and satisfies the equations \( f(2+x) = f(2-x) \) and \( f(7+x) = f(7-x) \) for all real numbers \( x \). Let \( x = 0 \) be a root of \( f(x) = 0 \). Denote the number of roots of \( f(x) = 0 \) in the interval \(-1000 \leq x \leq 1000 \) by \( N \). Find the minimum value of \( N \). | 401 | 6/8 |
Given positive integers \(a, b, k\) satisfying \(\frac{a^{2} + b^{2}}{a b - 1} = k\), prove that \(k = 5\). | 5 | 3/8 |
Given \( \theta_{1}, \theta_{2}, \theta_{3}, \theta_{4} \in \mathbf{R}^{+} \) and \( \theta_{1} + \theta_{2} + \theta_{3} + \theta_{4} = \pi \), find the minimum value of \( \left(2 \sin^{2} \theta_{1} + \frac{1}{\sin^{2} \theta_{1}}\right)\left(2 \sin^{2} \theta_{2} + \frac{1}{\sin^{2} \theta_{2}}\right)\left(2 \sin^{2} \theta_{3} + \frac{1}{\sin^{2} \theta_{3}}\right)\left(2 \sin^{2} \theta_{4} + \frac{1}{\sin^{2} \theta_{1}}\right) \). | 81 | 3/8 |
Find the number of different numbers of the form $\left\lfloor\frac{i^2}{2015} \right\rfloor$ , with $i = 1,2, ..., 2015$ . | 2016 | 1/8 |
Mary is about to pay for five items at the grocery store. The prices of the items are $7.99$, $4.99$, $2.99$, $1.99$, and $0.99$. Mary will pay with a twenty-dollar bill. Which of the following is closest to the percentage of the $20.00$ that she will receive in change?
$\text {(A) } 5 \qquad \text {(B) } 10 \qquad \text {(C) } 15 \qquad \text {(D) } 20 \qquad \text {(E) } 25$ | (A)5 | 1/8 |
Starting from a positive integer \( a \), we formed the numbers \( b = 2a^2 \), \( c = 2b^2 \), and \( d = 2c^2 \). What could the number \( a \) have been if the decimal representations of \( a \), \( b \), and \( c \) written consecutively in this order form exactly the decimal representation of \( d \)? | 1 | 4/8 |
Let $a_1,$ $a_2,$ $a_3,$ $\dots$ be a sequence of real numbers satisfying
\[a_n = a_{n - 1} a_{n + 1}\]for all $n \ge 2.$ If $a_1 = 1 + \sqrt{7}$ and $a_{1776} = 13 + \sqrt{7},$ then determine $a_{2009}.$ | -1 + 2 \sqrt{7} | 5/8 |
Find the maximum natural number which is divisible by 30 and has exactly 30 different positive divisors. | 11250 | 7/8 |
There are positive integers that have these properties:
$\bullet$ I. The sum of the squares of their digits is $50,$ and
$\bullet$ II. Each digit is larger than the one on its left.
What is the product of the digits of the largest integer with both properties? | 36 | 7/8 |
Let the three-digit number \( n = abc \). If \( a, b, \) and \( c \) as the lengths of the sides can form an isosceles (including equilateral) triangle, then how many such three-digit numbers \( n \) are there? | 165 | 4/8 |
From a certain airport, 100 planes (1 command plane and 99 supply planes) take off at the same time. When their fuel tanks are full, each plane can fly 1000 kilometers. During the flight, planes can refuel each other, and after completely transferring their fuel, planes can land as planned. How should the flight be arranged to enable the command plane to fly as far as possible? | 100000 | 1/8 |
Determine the maximum value of the positive integer \( n \) such that for any simple graph of order \( n \) with vertices \( v_1, v_2, \ldots, v_n \), one can always find \( n \) subsets \( A_1, A_2, \ldots, A_n \) of the set \( \{1, 2, \ldots, 2020\} \) which satisfy: \( A_i \cap A_j \neq \varnothing \) if and only if \( v_i \) and \( v_j \) are adjacent. | 89 | 1/8 |
Quadrilateral $ABCD$ is a square, and segment $AE$ is perpendicular to segment $ED$. If $AE = 8$ units and $DE = 6$ units, what is the area of pentagon $AEDCB$, in square units? [asy]
size(150);
pair A, B, C, D, E;
A=(0,10);
B=(0,0);
C=(10,0);
D=(10,10);
E=(6.4,5.2);
draw(A--B--C--D--A);
draw(A--E--D);
label("A", A, NW);
label("B", B, SW);
label("C", C, SE);
label("D", D, NE);
label("E", E, S);
[/asy] | 76 | 1/8 |
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