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Given vectors $\overrightarrow{a}=(2\cos\omega x,-2)$ and $\overrightarrow{b}=(\sqrt{3}\sin\omega x+\cos\omega x,1)$, where $\omega\ \ \gt 0$, and the function $f(x)=\overrightarrow{a}\cdot\overrightarrow{b}+1$. The distance between two adjacent symmetric centers of the graph of $f(x)$ is $\frac{\pi}{2}$.
$(1)$ Find $\omega$;
$(2)$ Given $a$, $b$, $c$ are the opposite sides of the three internal angles $A$, $B$, $C$ of scalene triangle $\triangle ABC$, and $f(A)=f(B)=\sqrt{3}$, $a=\sqrt{2}$, find the area of $\triangle ABC$. | \frac{3-\sqrt{3}}{4} | 1/8 |
How many units are in the sum of the lengths of the two longest altitudes in a right triangle with sides $9$, $40$, and $41$? | 49 | 7/8 |
At the vertices of a regular $n$-gon, the numbers are arranged such that there are $n-1$ zeros and one one. It is allowed to increase by 1 all numbers at the vertices of any regular $k$-gon inscribed in this polygon. Is it possible to make all the numbers equal through these operations? | No | 2/8 |
If $\triangle PQR$ is right-angled at $P$ with $PR=12$, $SQ=11$, and $SR=13$, what is the perimeter of $\triangle QRS$? | 44 | 2/8 |
Let $z_1$ and $z_2$ be the complex roots of $z^2 + az + b = 0,$ where $a$ and $b$ are complex numbers. In the complex plane, 0, $z_1,$ and $z_2$ form the vertices of an equilateral triangle. Find $\frac{a^2}{b}.$ | 3 | 7/8 |
Given the complex number \( z \) satisfying
$$
\left|\frac{z^{2}+1}{z+\mathrm{i}}\right|+\left|\frac{z^{2}+4 \mathrm{i}-3}{z-\mathrm{i}+2}\right|=4,
$$
find the minimum value of \( |z - 1| \). | \sqrt{2} | 4/8 |
In acute triangle $\triangle ABC$, if $\sin A = 3\sin B\sin C$, then the minimum value of $\tan A\tan B\tan C$ is \_\_\_\_\_\_. | 12 | 7/8 |
The cells of a $5 \times 41$ rectangle are colored in two colors. Prove that it is possible to choose three rows and three columns such that all nine cells at their intersections will be of the same color. | 6 | 1/8 |
Each beach volleyball game has four players on the field. In a beach volleyball tournament, it is known that there are $n$ players who participated in a total of $n$ games, and any two players have played in at least one game together. Find the maximum value of $n$. | 13 | 7/8 |
A digit is inserted between the digits of a two-digit number to form a three-digit number. Some two-digit numbers, when a certain digit is inserted in between, become three-digit numbers that are $k$ times the original two-digit number (where $k$ is a positive integer). What is the maximum value of $k$? | 19 | 3/8 |
A solid right prism $PQRSTU$ has a height of 20, as shown. Its bases are equilateral triangles with side length 15. Points $M$, $N$, and $O$ are the midpoints of edges $PQ$, $QR$, and $RS$, respectively. Determine the perimeter of triangle $MNO$. | 32.5 | 1/8 |
How can you measure 15 minutes using a 7-minute hourglass and an 11-minute hourglass? | 15 | 1/8 |
The base nine numbers $125_9$ and $33_9$ need to be multiplied and the result expressed in base nine. What is the base nine sum of the digits of their product? | 16 | 2/8 |
The school organized a picnic with several participants. The school prepared many empty plates. Each attendee counts the empty plates and takes one empty plate to get food (each person can only take one plate, no more). The first attendee counts all the empty plates, the second attendee counts one less plate than the first attendee, and so on. The last attendee finds that there are 4 empty plates left. It is known that the total number of plates prepared by the school plus the number of attendees equals 2015. How many people attended the picnic? | 1006 | 2/8 |
Given the vectors $\overrightarrow{a} \cdot (\overrightarrow{a}+2\overrightarrow{b})=0$ and the magnitudes $|\overrightarrow{a}|=|\overrightarrow{b}|=2$, find the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{2\pi}{3} | 1/8 |
Given that $\sqrt {3}\sin x+\cos x= \frac {2}{3}$, find the value of $\tan (x+ \frac {7\pi}{6})$. | \frac{\sqrt{2}}{4} | 1/8 |
Let $n$ be an integer greater than 1. If all digits of $97n$ are odd, find the smallest possible value of $n$ . | 35 | 6/8 |
Given a positive real number \( t \), and the sequence \(\{x_{n}\}\) defined by:
\[ x_{1} = 1, \quad 8 x_{n+1} = t + x_{n}^{2} \]
(1) Prove that when \( 7 < t \leqslant 12 \), for any \( n \in \mathbb{Z}_{+} \), it holds that \( 1 \leqslant x_{n} < x_{n+1} < 2 \);
(2) If for any \( n \in \mathbb{Z}_{+} \), it holds that \( x_{n} < 4 \), find the maximum value of \( t \). | 16 | 7/8 |
Given that the cube root of \( m \) is a number in the form \( n + r \), where \( n \) is a positive integer and \( r \) is a positive real number less than \(\frac{1}{1000}\). When \( m \) is the smallest positive integer satisfying the above condition, find the value of \( n \). | 19 | 6/8 |
In the expression $c \cdot a^b - d$, the values of $a$, $b$, $c$, and $d$ are 0, 1, 2, and 3, although not necessarily in that order. What is the maximum possible value of the result? | 9 | 6/8 |
Four brothers have together forty-eight Kwanzas. If the first brother's money were increased by three Kwanzas, if the second brother's money were decreased by three Kwanzas, if the third brother's money were triplicated and if the last brother's money were reduced by a third, then all brothers would have the same quantity of money. How much money does each brother have? | 6, 12, 3, 27 | 1/8 |
Three planes are drawn through the sides of an equilateral triangle, forming an angle $\alpha$ with the plane of the triangle, and intersecting at a point at a distance $d$ from the plane of the triangle. Find the radius of the circle inscribed in the given equilateral triangle. | \cot(\alpha) | 4/8 |
A function \( f: A \rightarrow A \) is called idempotent if \( f(f(x)) = f(x) \) for all \( x \in A \). Let \( I_{n} \) be the number of idempotent functions from \(\{1, 2, \ldots, n\}\) to itself. Compute
\[
\sum_{n=1}^{\infty} \frac{I_{n}}{n!}.
\] | e^1 | 7/8 |
In $\triangle ABC$, $\tan A= \frac {3}{4}$ and $\tan (A-B)=- \frac {1}{3}$, find the value of $\tan C$. | \frac {79}{3} | 7/8 |
How many times must a die be rolled so that the probability of the inequality
\[ \left| \frac{m}{n} - \frac{1}{6} \right| \leq 0.01 \]
is at least as great as the probability of the opposite inequality, where \( m \) is the number of times a specific face appears in \( n \) rolls of the die? | 632 | 2/8 |
A quadrilateral with angles of $120^\circ, 90^\circ, 60^\circ$, and $90^\circ$ is inscribed in a circle. The area of the quadrilateral is $9\sqrt{3}$ cm$^{2}$. Find the radius of the circle if the diagonals of the quadrilateral are mutually perpendicular. | 3 | 3/8 |
Let \( S = \{1, 2, \cdots, 98\} \). Find the smallest positive integer \( n \) such that, in any subset of \( S \) with \( n \) elements, it is always possible to select 10 numbers, and no matter how these 10 numbers are evenly divided into two groups, there will always be one number in one group that is relatively prime to the other 4 numbers in the same group, and one number in the other group that is not relatively prime to the other 4 numbers in that group. | 50 | 2/8 |
Given points \( A \) and \( B \) lie on the parabola \( y^2 = 6x \) and the circle \(\odot C: (x-4)^2 + y^2 = 1 \) respectively. Determine the range of values for \( |AB| \). | [\sqrt{15}-1,+\infty) | 1/8 |
The equilateral triangle \(ABC\) is divided into \(N\) convex polygons in such a way that each line intersects no more than 40 of them (a line is said to intersect a polygon if they have at least one point in common, for example, if the line passes through a vertex of the polygon). Can \(N\) be greater than a million? | Yes | 3/8 |
Let $S$ be the increasing sequence of positive integers whose binary representation has exactly $8$ ones. Let $N$ be the 1000th number in $S$. Find the remainder when $N$ is divided by $1000$.
| 32 | 2/8 |
A and B depart from points A and B simultaneously, moving towards each other at a constant speed. When A and B meet at point C, C departs from point B, moving at a constant speed towards point A. When A and C meet at point D, A immediately turns around and reduces speed to 80% of the original speed. When A and C both arrive at point A, B is 720 meters away from point A. If the distance between C and D is 900 meters, what is the distance between A and B in meters? | 5265 | 1/8 |
Given that the ratio of the length, width, and height of a rectangular prism is $4: 3: 2$, and that a plane cuts through the prism to form a hexagonal cross-section (as shown in the diagram), with the minimum perimeter of such hexagons being 36, find the surface area of the rectangular prism. | 208 | 3/8 |
Determine all squarefree positive integers $n\geq 2$ such that \[\frac{1}{d_1}+\frac{1}{d_2}+\cdots+\frac{1}{d_k}\]is a positive integer, where $d_1,d_2,\ldots,d_k$ are all the positive divisors of $n$ . | 6 | 2/8 |
Write the first 10 prime numbers in a row. How to cross out 6 digits to get the largest possible number? | 7317192329 | 1/8 |
If $\log_{2x}216 = x$, where $x$ is real, then $x$ is:
$\textbf{(A)}\ \text{A non-square, non-cube integer}\qquad$
$\textbf{(B)}\ \text{A non-square, non-cube, non-integral rational number}\qquad$
$\textbf{(C)}\ \text{An irrational number}\qquad$
$\textbf{(D)}\ \text{A perfect square}\qquad$
$\textbf{(E)}\ \text{A perfect cube}$ | \textbf{(A)}\ | 1/8 |
Given that Liz had no money initially, and her friends gave her one-sixth, one-fifth, and one-fourth of their respective amounts, find the fractional part of the group's total money that Liz has. | \frac{1}{5} | 6/8 |
Given that point $A(-2,3)$ lies on the axis of parabola $C$: $y^{2}=2px$, and the line passing through point $A$ is tangent to $C$ at point $B$ in the first quadrant. Let $F$ be the focus of $C$. Then, $|BF|=$ _____ . | 10 | 1/8 |
There is one three-digit number and two two-digit numbers written on the board. The sum of the numbers containing the digit seven is 208. The sum of the numbers containing the digit three is 76. Find the sum of all three numbers. | 247 | 3/8 |
If a number eight times as large as $x$ is increased by two, then one fourth of the result equals | 2x + \frac{1}{2} | 7/8 |
Given that the weights (in kilograms) of 4 athletes are all integers, and they weighed themselves in pairs for a total of 5 times, obtaining weights of 99, 113, 125, 130, 144 kilograms respectively, and there are two athletes who did not weigh together, determine the weight of the heavier one among these two athletes. | 66 | 2/8 |
In triangle $ABC$ , let $P$ and $R$ be the feet of the perpendiculars from $A$ onto the external and internal bisectors of $\angle ABC$ , respectively; and let $Q$ and $S$ be the feet of the perpendiculars from $A$ onto the internal and external bisectors of $\angle ACB$ , respectively. If $PQ = 7, QR = 6$ and $RS = 8$ , what is the area of triangle $ABC$ ? | 84 | 1/8 |
Ben starts with an integer greater than $9$ and subtracts the sum of its digits from it to get a new integer. He repeats this process with each new integer he gets until he gets a positive $1$ -digit integer. Find all possible $1$ -digit integers Ben can end with from this process. | 9 | 6/8 |
Triangle $ABC$ satisfies $\tan A \cdot \tan B = 3$ and $AB = 5$ . Let $G$ and $O$ be the centroid and circumcenter of $ABC$ respectively. The maximum possible area of triangle $CGO$ can be written as $\frac{a\sqrt{b}}{c}$ for positive integers $a$ , $b$ , and $c$ with $a$ and $c$ relatively prime and $b$ not divisible by the square of any prime. Find $a + b + c$ . | 100 | 1/8 |
Let \( f \) be a function from \( \mathbb{R} \) to \( \mathbb{R} \), such that:
1. For any \( x, y \in \mathbb{R} \),
$$
f(x) + f(y) + 1 \geq f(x+y) \geq f(x) + f(y);
$$
2. For any \( x \in [0,1) \), \( f(0) \geq f(x) \);
3. \( -f(-1) = f(1) = 1 \).
Find all functions \( f \) that satisfy these conditions. | f(x)=\lfloorx\rfloor | 1/8 |
The store has 89 gold coins with numbers ranging from 1 to 89, each priced at 30 yuan. Among them, only one is a "lucky coin." Feifei can ask an honest clerk if the number of the lucky coin is within a chosen subset of numbers. If the answer is "Yes," she needs to pay a consultation fee of 20 yuan. If the answer is "No," she needs to pay a consultation fee of 10 yuan. She can also choose not to ask any questions and directly buy some coins. What is the minimum amount of money (in yuan) Feifei needs to pay to guarantee she gets the lucky coin? | 130 | 1/8 |
Find the maximal possible finite number of roots of the equation $|x-a_1|+\dots+|x-a_{50}|=|x-b_1|+\dots+|x-b_{50}|$ , where $a_1,\,a_2,\,\dots,a_{50},\,b_1,\dots,\,b_{50}$ are distinct reals. | 49 | 1/8 |
On the coordinate plane, we consider squares where all vertices have non-negative integer coordinates and the center is at the point $(50 ; 30)$. Find the number of such squares. | 930 | 1/8 |
Continue the sequence of numbers: 1, 11, 21, 1112, 3112, 211213, 312213, 212223, 114213... | 31121314 | 4/8 |
The roots of $(x^{2}-3x+2)(x)(x-4)=0$ are: | 0, 1, 2 and 4 | 1/8 |
The volume of a cylinder circumscribed around a sphere with radius $r$ is $V_{1}$, and the volume of a cone circumscribed around the same sphere is $V_{2}$. What is the minimum value of the ratio $V_{2} / V_{1}$? | 4/3 | 5/8 |
Among the four-digit numbers formed by the digits 0, 1, 2, ..., 9 without repetition, determine the number of cases where the absolute difference between the units digit and the hundreds digit equals 8. | 210 | 7/8 |
In triangle \(ABC\), point \(K\) on side \(AB\) and point \(M\) on side \(AC\) are positioned such that \(AK:KB = 3:2\) and \(AM:MC = 4:5\). Determine the ratio in which the line through point \(K\) parallel to side \(BC\) divides segment \(BM\). | 18/7 | 6/8 |
The point of tangency of the incircle of a right triangle divides the hypotenuse into segments of lengths $\boldsymbol{m}$ and $n$. Prove that the area of the triangle $S = mn$. Find the area of the rectangle inscribed in the given triangle such that one of its vertices coincides with the vertex of the right angle, and the opposite vertex coincides with the point of tangency of the incircle with the hypotenuse. | \frac{2m^2n^2}{(+n)^2} | 2/8 |
Given a triangle $K L M$ with $\angle K M L = 121^{\circ}$. Points $S$ and $N$ on side $K L$ are such that $K S = S N = N L$. It is known that $M N > K S$. Prove that $M S < N L$. | MS<NL | 3/8 |
Let \( a_{0}=1, a_{1}=2 \), and \( n(n+1) a_{n+1}=n(n-1) a_{n}-(n-2) a_{n-1}, n=1,2,3, \cdots \). Find the value of \( \frac{a_{0}}{a_{1}}+\frac{a_{1}}{a_{2}}+\frac{a_{2}}{a_{3}}+\cdots+\frac{a_{50}}{a_{51}} \). | 1326 | 1/8 |
Let the function \( f: \{1, 2, \cdots\} \rightarrow \{2, 3, \cdots\} \) satisfy, for all positive integers \( m \) and \( n \),
\[ f(m+n) \mid (f(m) + f(n)). \]
Prove that there exists a positive integer \( c > 1 \) such that \( c \) divides all values of \( f \). | c | 3/8 |
Find the rank of the matrix \( A \) using elementary row operations:
$$
A=\left(\begin{array}{ccccc}
5 & 7 & 12 & 48 & -14 \\
9 & 16 & 24 & 98 & -31 \\
14 & 24 & 25 & 146 & -45 \\
11 & 12 & 24 & 94 & -25
\end{array}\right)
$$ | 3 | 6/8 |
You have six blocks in a row, labeled 1 through 6, each with weight 1. Call two blocks \( x \leq y \) connected when, for all \( x \leq z \leq y \), block \( z \) has not been removed. While there is still at least one block remaining, you choose a remaining block uniformly at random and remove it. The cost of this operation is the sum of the weights of the blocks that are connected to the block being removed, including itself. Compute the expected total cost of removing all the blocks. | \frac{163}{10} | 1/8 |
Let's call a "median" of a system of $2n$ points on a plane a line that passes exactly through two of them, with an equal number of points of this system on either side. What is the minimum number of medians that a system of $2n$ points, no three of which are collinear, can have? | n | 2/8 |
This spring, three Hungarian women's handball teams reached the top eight in the EHF Cup. The teams were paired by drawing lots. All three Hungarian teams were paired with foreign opponents. What was the probability of this happening? | \frac{4}{7} | 7/8 |
Determine the value of $x$ for which $9^{x+6} = 5^{x+1}$ can be expressed in the form $x = \log_b 9^6$. Find the value of $b$. | \frac{5}{9} | 1/8 |
There are 4 male and 2 female volunteers, a total of 6 volunteers, and 2 elderly people standing in a row for a group photo. The photographer requests that the two elderly people stand next to each other in the very center, with the two female volunteers standing immediately to the left and right of the elderly people. The number of different ways they can stand is: | 96 | 4/8 |
How many positive odd integers greater than 1 and less than $200$ are square-free? | 80 | 6/8 |
On the segment \([0, 2002]\), its ends and a point with coordinate \(d\), where \(d\) is a number coprime with 1001, are marked. It is allowed to mark the midpoint of any segment with ends at marked points if its coordinate is an integer. Is it possible, by repeating this operation several times, to mark all integer points on the segment? | Yes | 1/8 |
In a quarter circle \( A O B \), let \( A_{1} \) and \( B_{1} \) be the first trisection points of the radii from point \( O \). Calculate the radius of a circle that is tangent to the semicircles with diameters \( A A_{1} \) and \( B B_{1} \), and the arc \( A B \). Use the segment \( O A_{1} \) as the unit of measurement. Additionally, construct the center of the circle. | 0.6796 | 1/8 |
The sequence $(x_n)$ is determined by the conditions: $x_0=1992,x_n=-\frac{1992}{n} \cdot \sum_{k=0}^{n-1} x_k$ for $n \geq 1$ .
Find $\sum_{n=0}^{1992} 2^nx_n$ . | 1992 | 7/8 |
The incenter of the triangle \(ABC\) is \(K\). The midpoint of \(AB\) is \(C_1\) and that of \(AC\) is \(B_1\). The lines \(C_1 K\) and \(AC\) meet at \(B_2\), and the lines \(B_1 K\) and \(AB\) meet at \(C_2\). If the areas of the triangles \(AB_2C_2\) and \(ABC\) are equal, what is the measure of angle \(\angle CAB\)? | 60 | 3/8 |
Given the sets \( A_{1}, A_{2}, \cdots, A_{n} \) are different subsets of the set \( \{1, 2, \cdots, n\} \), satisfying the following conditions:
(i) \( i \notin A_{i} \) and \( \operatorname{Card}(A_{i}) \geqslant 3 \) for \( i = 1, 2, \cdots, n \);
(ii) \( i \in A_{j} \) if and only if \( j \notin A_{i} \) for \( i \neq j \) and \( i, j = 1, 2, \cdots, n \).
Answer the following questions:
1. Find \( \sum_{i=1}^{n} \operatorname{Card}(A_{i}) \).
2. Find the minimum value of \( n \). | 7 | 7/8 |
Let $A B C D E F G H$ be an equilateral octagon with $\angle A \cong \angle C \cong \angle E \cong \angle G$ and $\angle B \cong \angle D \cong \angle F \cong$ $\angle H$. If the area of $A B C D E F G H$ is three times the area of $A C E G$, then $\sin B$ can be written as $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n)=1$. Find $100 m+n$. | 405 | 1/8 |
In a square \(ABCD\) with side length 2, \(E\) is the midpoint of \(AB\). The triangles \(\triangle AED\) and \(\triangle BEC\) are folded along \(ED\) and \(EC\) respectively so that \(EA\) and \(EB\) coincide, forming a tetrahedron. What is the volume of this tetrahedron? | \frac{\sqrt{3}}{3} | 3/8 |
The four points $A(-1,2), B(3,-4), C(5,-6),$ and $D(-2,8)$ lie in the coordinate plane. Compute the minimum possible value of $PA + PB + PC + PD$ over all points P . | 23 | 3/8 |
The numbers \( a_{1}, a_{2}, \cdots, a_{1987} \) are any permutation of the natural numbers \( 1, 2, \cdots, 1987 \). Each number \( a_{k} \) is multiplied by its index \( k \) to obtain the product \( k a_{k} \). Prove that among the resulting 1987 products, the maximum value is at least \( 994^{2} \). | 994^2 | 5/8 |
An abstract animal lives in groups of two and three.
In a forest, there is one group of two and one group of three. Each day, a new animal arrives in the forest and randomly chooses one of the inhabitants. If the chosen animal belongs to a group of three, that group splits into two groups of two; if the chosen animal belongs to a group of two, they form a group of three. What is the probability that the $n$-th arriving animal will join a group of two? | 4/7 | 1/8 |
In isosceles trapezoid \(ABCD\), \(AB \parallel DC\). The diagonals intersect at \(O\), and \(\angle AOB = 60^\circ\). Points \(P\), \(Q\), and \(R\) are the midpoints of \(OA\), \(BC\), and \(OD\), respectively. Prove that \(\triangle PQR\) is an equilateral triangle. | \trianglePQR | 5/8 |
In $\triangle ABC$, $\angle B = \angle C = 36^\circ$, the height from $A$ to $BC$ is $h$. The lengths of the internal angle trisector and the external angle quarterlater from $A$ are $p_1$ and $p_2$ respectively. Prove that:
$$
\frac{1}{p_1^2} + \frac{1}{p_2^2} = \frac{1}{h^2}
$$ | \frac{1}{p_1^2}+\frac{1}{p_2^2}=\frac{1}{^2} | 1/8 |
Among the first hundred elements of the arithmetic progression \(3, 7, 11, \ldots\), find those that are also elements of the arithmetic progression \(2, 9, 16, \ldots\). Provide the sum of the found numbers in your answer. | 2870 | 5/8 |
Function $f(n), n \in \mathbb N$ , is defined as follows:
Let $\frac{(2n)!}{n!(n+1000)!} = \frac{A(n)}{B(n)}$ , where $A(n), B(n)$ are coprime positive integers; if $B(n) = 1$ , then $f(n) = 1$ ; if $B(n) \neq 1$ , then $f(n)$ is the largest prime factor of $B(n)$ . Prove that the values of $f(n)$ are finite, and find the maximum value of $f(n).$ | 1999 | 5/8 |
Plane M is parallel to plane N. There are 3 different points on plane M and 4 different points on plane N. The maximum number of tetrahedrons with different volumes that can be determined by these 7 points is ____. | 34 | 1/8 |
Given the point \( P \) on the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \left(a>b>0, c=\sqrt{a^{2}-b^{2}}\right)\), and the equation of the line \( l \) is \(x=-\frac{a^{2}}{c}\), and the coordinate of the point \( F \) is \((-c, 0)\). Draw \( PQ \perp l \) at point \( Q \). If the points \( P \), \( Q \), and \( F \) form an isosceles triangle, what is the eccentricity of the ellipse? | \frac{\sqrt{2}}{2} | 1/8 |
Define a T-grid to be a $3\times3$ matrix which satisfies the following two properties:
Exactly five of the entries are $1$'s, and the remaining four entries are $0$'s.
Among the eight rows, columns, and long diagonals (the long diagonals are $\{a_{13},a_{22},a_{31}\}$ and $\{a_{11},a_{22},a_{33}\}$, no more than one of the eight has all three entries equal.
Find the number of distinct T-grids. | 68 | 1/8 |
Compute the remainder when
${2007 \choose 0} + {2007 \choose 3} + \cdots + {2007 \choose 2007}$
is divided by 1000.
| 42 | 7/8 |
Two teachers and 4 students need to be divided into 2 groups, each consisting of 1 teacher and 2 students. Calculate the number of different arrangements. | 12 | 2/8 |
Let $R$ be the region in the Cartesian plane of points $(x, y)$ satisfying $x \geq 0, y \geq 0$, and $x+y+\lfloor x\rfloor+\lfloor y\rfloor \leq 5$. Determine the area of $R$. | \frac{9}{2} | 7/8 |
A triangle \( \triangle ABC \) is inscribed in a circle of radius 1, with \( \angle BAC = 60^\circ \). Altitudes \( AD \) and \( BE \) of \( \triangle ABC \) intersect at \( H \). Find the smallest possible value of the length of the segment \( AH \). | 1 | 2/8 |
Let $a, b, c, d, e, f$ be integers selected from the set $\{1,2, \ldots, 100\}$, uniformly and at random with replacement. Set $M=a+2 b+4 c+8 d+16 e+32 f$. What is the expected value of the remainder when $M$ is divided by 64? | \frac{63}{2} | 5/8 |
Represent in the rectangular coordinate system those pairs of real numbers \((a ; b)\) for which the two-variable polynomial
$$
x(x+4) + a\left(y^2 - 1\right) + 2by
$$
can be factored into the product of two first-degree polynomials. | (2)^2+b^2=4 | 7/8 |
In a parlor game, the magician asks one of the participants to think of a three digit number $(abc)$ where $a$, $b$, and $c$ represent digits in base $10$ in the order indicated. The magician then asks this person to form the numbers $(acb)$, $(bca)$, $(bac)$, $(cab)$, and $(cba)$, to add these five numbers, and to reveal their sum, $N$. If told the value of $N$, the magician can identify the original number, $(abc)$. Play the role of the magician and determine $(abc)$ if $N= 3194$. | 358 | 7/8 |
The numbers $1,2,...,2n-1,2n$ are divided into two disjoint sets, $a_1 < a_2 < ... < a_n$ and $b_1 > b_2 > ... > b_n$ . Prove that $$ |a_1 - b_1| + |a_2 - b_2| + ... + |a_n - b_n| = n^2. $$ | n^2 | 3/8 |
In triangle \(ABC\) with area \(S\), medians \(AK\) and \(BE\) are drawn, intersecting at point \(O\). Find the area of quadrilateral \(CKOE\). | \frac{S}{3} | 7/8 |
Determine all infinite sequences \((a_1, a_2, \ldots)\) of positive integers satisfying
\[ a_{n+1}^2 = 1 + (n + 2021) a_n \]
for all \( n \geq 1 \). | a_n=n+2019 | 1/8 |
Three pairwise non-intersecting circles \(\omega_{x}, \omega_{y}, \omega_{z}\) with radii \(r_{x}, r_{y}, r_{z}\) respectively lie on one side of the line \(t\) and touch it at points \(X, Y, Z\) respectively. It is known that \(Y\) is the midpoint of the segment \(XZ\), \(r_{x} = r_{z} = r\), and \(r_{y} > r\). Let \(p\) be one of the common internal tangents to the circles \(\omega_{x}\) and \(\omega_{y}\), and \(q\) be one of the common internal tangents to the circles \(\omega_{y}\) and \(\omega_{z}\). A scalene triangle is formed at the intersection of the lines \(p, q,\) and \(t\). Prove that the radius of its inscribed circle is equal to \(r\).
(P. Kozhevnikov) | r | 1/8 |
How many different three-letter sets of initials are possible using the letters $A$ through $J$, where no letter is repeated in any set? | 720 | 7/8 |
In the triangular pyramid \(ABCD\), it is known that \(AB = a\) and \(\angle ACB = \angle ADB = 90^\circ\). Find the radius of the sphere circumscribed around this pyramid. | \frac{}{2} | 5/8 |
A necklace consists of 100 beads of red, blue, and green colors. It is known that among any five consecutive beads, there is at least one blue bead, and among any seven consecutive beads, there is at least one red bead. What is the maximum number of green beads that can be in this necklace? (The beads in the necklace are arranged cyclically, meaning the last one is adjacent to the first one.) | 65 | 1/8 |
The area enclosed by the curve defined by the equation \( |x-1| + |y-1| = 1 \) is:
A. 1
B. 2
C. \( \pi \)
D. 4 | 2 | 1/8 |
Given that the complex number \( z \) satisfies the condition that the real and imaginary parts of \( \frac{z}{10} \) and \( \frac{10}{\bar{z}} \) are positive numbers less than 1, find the area of the region on the complex plane that corresponds to \( z \). | 75-\frac{25}{2}\pi | 4/8 |
The graph of the function $y=\sin 2x-\sqrt{3}\cos 2x$ can be obtained by shifting the graph of the function $y=\sin 2x+\sqrt{3}\cos 2x$ to the right by $\frac{\pi}{3}$ units. | \frac{\pi}{3} | 7/8 |
There are $n$ mathematicians seated around a circular table with $n$ seats numbered $1,$ $2,$ $3,$ $...,$ $n$ in clockwise order. After a break they again sit around the table. The mathematicians note that there is a positive integer $a$ such that
($1$) for each $k,$ the mathematician who was seated in seat $k$ before the break is seated in seat $ka$ after the break (where seat $i + n$ is seat $i$);
($2$) for every pair of mathematicians, the number of mathematicians sitting between them after the break, counting in both the clockwise and the counterclockwise directions, is different from either of the number of mathematicians sitting between them before the break.
Find the number of possible values of $n$ with $1 < n < 1000.$ | 332 | 2/8 |
Evaluate the expression:
\[4(1+4(1+4(1+4(1+4(1+4(1+4(1+4(1))))))))\] | 87380 | 4/8 |
Two friends planned a hunting trip. One lives 46 km from the hunting base, while the other, who has a car, lives 30 km from the base between the latter and his friend's house. They started their journey at the same time, with the car owner driving to meet his friend, who was walking. After meeting, they traveled together to the base and arrived one hour after setting out from home. If the walker had left home 2 hours and 40 minutes earlier than the car owner, they would have met 11 km from the walker's home. What is the speed of the car? | 60\, | 1/8 |
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