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Suppose $X$ is a random variable that takes on only nonnegative integer values, with $E[X]=1,$ $E[X^2]=2,$ and $E[X^3]=5.$ (Here $E[Y]$ denotes the expectation of the random variable $Y.$ ) Determine the smallest possible value of the probability of the event $X=0.$ | \frac{1}{3} | 5/8 |
Given $x_1$ and $x_2$ are the two real roots of the quadratic equation in $x$: $x^2 - 2(m+2)x + m^2 = 0$.
(1) When $m=0$, find the roots of the equation;
(2) If $(x_1 - 2)(x_2 - 2) = 41$, find the value of $m$;
(3) Given an isosceles triangle $ABC$ with one side length of 9, if $x_1$ and $x_2$ happen to be the lengths of the other two sides of $\triangle ABC$, find the perimeter of this triangle. | 19 | 5/8 |
Find the probability that the chord $\overline{AB}$ does not intersect with chord $\overline{CD}$ when four distinct points, $A$, $B$, $C$, and $D$, are selected from 2000 points evenly spaced around a circle. | \frac{2}{3} | 6/8 |
The number of solutions in positive integers of $2x+3y=763$ is:
$\textbf{(A)}\ 255 \qquad \textbf{(B)}\ 254\qquad \textbf{(C)}\ 128 \qquad \textbf{(D)}\ 127 \qquad \textbf{(E)}\ 0$ | \textbf{(D)}\127 | 1/8 |
Let $M$ be the intersection point of the medians of triangle $ABC$. Points $A_1$, $B_1$, and $C_1$ are taken on the perpendiculars dropped from $M$ to the sides $BC$, $AC$, and $AB$ respectively, such that $A_{1} B_{1} \perp M C$ and $A_{1} C_{1} \perp M B$. Prove that $M$ is also the intersection point of the medians in triangle $A_1 B_1 C_1$. | M | 5/8 |
A natural number \( n > 5 \) is called new if there exists a number that is not divisible by \( n \) but is divisible by all natural numbers less than \( n \). What is the maximum number of consecutive numbers that can be new? | 3 | 1/8 |
In a container, there is a mixture of equal masses of nitrogen $N_{2}$ and helium He under pressure $p$. The absolute temperature of the gas is doubled, and all nitrogen molecules dissociate into atoms. Find the pressure of the gas mixture at this temperature. The molar masses of the gases are $\mu_{\text{He}} = 4$ g/mol and $\mu_{N_{2}} = 28$ g/mol. Assume the gases are ideal. | \frac{9}{4}p | 7/8 |
Let $a$ , $b$ , $c$ , $d$ , $e$ , $f$ and $g$ be seven distinct positive integers not bigger than $7$ . Find all primes which can be expressed as $abcd+efg$ | 179 | 6/8 |
Let \( S = \{1, 2, \ldots, 2005\} \). If any set of \( n \) pairwise co-prime numbers in \( S \) always contains at least one prime number, what is the minimum value of \( n \)? | 16 | 6/8 |
In a city, from 7:00 to 8:00, is a peak traffic period, during which all vehicles travel at half their normal speed. Every morning at 6:50, two people, A and B, start from points A and B respectively and travel towards each other. They meet at a point 24 kilometers from point A. If person A departs 20 minutes later, they meet exactly at the midpoint of the route between A and B. If person B departs 20 minutes earlier, they meet at a point 20 kilometers from point A. What is the distance between points A and B in kilometers? | 48 | 1/8 |
The sum of the numerical coefficients in the complete expansion of $(x^2 - 2xy + y^2)^7$ is:
$\textbf{(A)}\ 0 \qquad \textbf{(B) }\ 7 \qquad \textbf{(C) }\ 14 \qquad \textbf{(D) }\ 128 \qquad \textbf{(E) }\ 128^2$ | \textbf{(A)}\0 | 1/8 |
Given a positive integer $n$, let $M(n)$ be the largest integer $m$ such that \[ \binom{m}{n-1} > \binom{m-1}{n}. \] Evaluate \[ \lim_{n \to \infty} \frac{M(n)}{n}. \] | \frac{3+\sqrt{5}}{2} | 7/8 |
Let $x_{1}=y_{1}=x_{2}=y_{2}=1$, then for $n \geq 3$ let $x_{n}=x_{n-1} y_{n-2}+x_{n-2} y_{n-1}$ and $y_{n}=y_{n-1} y_{n-2}- x_{n-1} x_{n-2}$. What are the last two digits of $\left|x_{2012}\right|$ ? | 84 | 1/8 |
On a plane with a given rectangular Cartesian coordinate system, a square is drawn with vertices at the points \((0, 0), (0, 65), (65, 65),\) and \((65, 0)\). Find the number of ways to choose two grid points inside this square (excluding its boundary) such that at least one of these points lies on one of the lines \(y = x\) or \(y = 65 - x\), but both selected points do not lie on any lines parallel to either of the coordinate axes. | 500032 | 1/8 |
If the direction vector of line $l$ is $\overrightarrow{d}=(1,\sqrt{3})$, then the inclination angle of line $l$ is ______. | \frac{\pi}{3} | 2/8 |
Extend the square pattern of 8 black and 17 white square tiles by attaching a border of black tiles around the square. What is the ratio of black tiles to white tiles in the extended pattern?
[asy] filldraw((0,0)--(5,0)--(5,5)--(0,5)--cycle,white,black); filldraw((1,1)--(4,1)--(4,4)--(1,4)--cycle,mediumgray,black); filldraw((2,2)--(3,2)--(3,3)--(2,3)--cycle,white,black); draw((4,0)--(4,5)); draw((3,0)--(3,5)); draw((2,0)--(2,5)); draw((1,0)--(1,5)); draw((0,4)--(5,4)); draw((0,3)--(5,3)); draw((0,2)--(5,2)); draw((0,1)--(5,1)); [/asy]
$\textbf{(A) }8:17 \qquad\textbf{(B) }25:49 \qquad\textbf{(C) }36:25 \qquad\textbf{(D) }32:17 \qquad\textbf{(E) }36:17$ | \textbf{(D)}32:17 | 1/8 |
Let $ABCD$ be a cyclic quadrilateral, and $E$ be the intersection of its diagonals. If $m(\widehat{ADB}) = 22.5^\circ$ , $|BD|=6$ , and $|AD|\cdot|CE|=|DC|\cdot|AE|$ , find the area of the quadrilateral $ABCD$ . | 9\sqrt{2} | 1/8 |
Determine the smallest constant $n$, such that for any positive real numbers $x$, $y$, and $z$,
\[\sqrt{\frac{x}{y + 2z}} + \sqrt{\frac{y}{2x + z}} + \sqrt{\frac{z}{x + 2y}} > n.\] | \sqrt{3} | 1/8 |
A rhombus $ADEF$ is inscribed in triangle $ABC$ such that angle $A$ is common to both the triangle and the rhombus, and vertex $E$ lies on side $BC$. Find the side length of the rhombus, given that $AB = c$ and $AC = b$. | \frac{}{} | 7/8 |
There were seven boxes. Some of them were filled with seven more boxes each (not nested inside each other), and so on. In the end, there were 10 non-empty boxes.
How many boxes are there in total? | 77 | 1/8 |
How many rectangles can be formed where each of the four vertices are points on a 4x4 grid with points spaced evenly along the grid lines? | 36 | 5/8 |
Let the medians of the triangle $ABC$ meet at $G$ . Let $D$ and $E$ be different points on the line $BC$ such that $DC=CE=AB$ , and let $P$ and $Q$ be points on the segments $BD$ and $BE$ , respectively, such that $2BP=PD$ and $2BQ=QE$ . Determine $\angle PGQ$ . | 90 | 6/8 |
At a joint conference of the Parties of Liars and Truth-lovers, 32 people were elected to the presidium and seated in four rows of 8 people each. During the break, each member of the presidium claimed that among their neighbors there are representatives of both parties. It is known that liars always lie, and truth-lovers always tell the truth. What is the minimum number of liars in the presidium for the described situation to be possible? (Two members of the presidium are neighbors if one of them is seated to the left, right, in front, or behind the other). | 8 | 1/8 |
On the coordinate plane, we consider squares whose vertices all have natural number coordinates, and the center is at the point \((35, 65)\). Find the number of such squares. | 1190 | 1/8 |
Calculate: \(\left(2 \frac{2}{3} \times\left(\frac{1}{3}-\frac{1}{11}\right) \div\left(\frac{1}{11}+\frac{1}{5}\right)\right) \div \frac{8}{27} = 7 \underline{1}\). | 7 \frac{1}{2} | 4/8 |
Find all values of the positive integer $ m$ such that there exists polynomials $ P(x),Q(x),R(x,y)$ with real coefficient satisfying the condition: For every real numbers $ a,b$ which satisfying $ a^m-b^2=0$ , we always have that $ P(R(a,b))=a$ and $ Q(R(a,b))=b$ . | 1 | 1/8 |
There are $n$ people, and it is known that any two of them can make at most one phone call to each other. For any $n-2$ people among them, the total number of phone calls is equal and is $3^k$ (where $k$ is a positive integer). Find all possible values of $n$. | 5 | 3/8 |
In each row of a $100 \times n$ table, the numbers from 1 to 100 are arranged in some order, and the numbers within a row do not repeat (the table has $n$ rows and 100 columns). It is allowed to swap two numbers in a row if they differ by 1 and are not adjacent. It turns out that with such operations, it is not possible to obtain two identical rows. What is the maximum possible value of $n$? | 2^{99} | 1/8 |
The sequence 12, 15, 18, 21, 51, 81, $\ldots$ consists of all positive multiples of 3 that contain at least one digit that is a 1. What is the $50^{\mathrm{th}}$ term of the sequence? | 318 | 1/8 |
Given that sets \( A \) and \( B \) are sets of positive integers with \( |A| = 20 \) and \( |B| = 16 \). The set \( A \) satisfies the following condition: if \( a, b, m, n \in A \) and \( a + b = m + n \), then \( \{a, b\} = \{m, n\} \). Define
\[ A + B = \{a + b \mid a \in A, b \in B\}. \]
Determine the minimum value of \( |A + B| \). | 200 | 2/8 |
Given that $f(x+6) + f(x-6) = f(x)$ for all real $x$, determine the least positive period $p$ for these functions. | 36 | 7/8 |
The chord \( AB \) divides the circle into two arcs, with the smaller arc being \( 130^{\circ} \). The larger arc is divided by chord \( AC \) in the ratio \( 31:15 \) from point \( A \). Find the angle \( BAC \). | 37.5 | 7/8 |
Let \( k \) and \( n \) be integers, with \( n \geq k \geq 3 \). Consider \( n+1 \) points in the plane, none of which are collinear in sets of three. Each segment connecting two of these points is assigned a color from among \( k \) given colors.
An angle is said to be bicolored if it has one of the \( n+1 \) points as its vertex, and its sides are two of the segments mentioned above, each of a different color.
Prove that there exists a coloring such that the number of bicolored angles is strictly greater than
$$
n\left\lfloor\frac{n}{k}\right\rfloor^{2}\left(\begin{array}{c}
k \\
2
\end{array}\right)
$$
Note: \( \lfloor t \rfloor \) denotes the integer part of the real number \( t \), that is, the largest integer \( n \leq t \). | n\lfloor\frac{n}{k}\rfloor^2\binom{k}{2} | 1/8 |
Let $x$ , $y$ , $z$ be arbitrary positive numbers such that $xy+yz+zx=x+y+z$ .
Prove that $$ \frac{1}{x^2+y+1} + \frac{1}{y^2+z+1} + \frac{1}{z^2+x+1} \leq 1 $$ .
When does equality occur?
*Proposed by Marko Radovanovic* | 1 | 3/8 |
Let \( N \geqslant 2 \) be a natural number. What is the sum of all fractions of the form \( \frac{1}{mn} \), where \( m \) and \( n \) are coprime natural numbers such that \( 1 \leqslant m < n \leqslant N \) and \( m+n > N \)? | \frac{1}{2} | 3/8 |
In rectangle $ABCD$, $\overline{CE}$ bisects angle $C$ (no trisection this time), where $E$ is on $\overline{AB}$, $F$ is still on $\overline{AD}$, but now $BE=10$, and $AF=5$. Find the area of $ABCD$. | 200 | 1/8 |
Given an ellipse $C:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with left and right foci $F_{1}$ and $F_{2}$, and a point $P(1,\frac{{\sqrt{2}}}{2})$ on the ellipse, satisfying $|PF_{1}|+|PF_{2}|=2\sqrt{2}$.<br/>$(1)$ Find the standard equation of the ellipse $C$;<br/>$(2)$ A line $l$ passing through $F_{2}$ intersects the ellipse at points $A$ and $B$. Find the maximum area of $\triangle AOB$. | \frac{\sqrt{2}}{2} | 4/8 |
Given the sequence \(\{x_n\}\) with the sum of the first \(n\) terms \(S_n = 2a_n - 2\) (\(n \in \mathbf{N}^*\)).
1. Find the general term \(a_n\) of the sequence \(\{a_n\}\).
2. Let \(b_n = \frac{1}{a_n} - \frac{1}{n(n+1)}\), and \(T_n\) be the sum of the first \(n\) terms of the sequence \(\{b_n\}\). Find the positive integer \(k\) such that \(T_k \geq T_n\) holds for any \(n \in \mathbf{N}^*\).
3. Let \(c_n = \frac{a_{n+1}}{\left(1+a_n\right)\left(1+a_{n+1}\right)}\), and \(R_n\) be the sum of the first \(n\) terms of the sequence \(\{c_n\}\). If \(R_n < \lambda\) holds for any \(n \in \mathbf{N}^*\), find the minimum value of \(\lambda\). | \frac{2}{3} | 4/8 |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=1$, $(\overrightarrow{a}+\overrightarrow{b}) \perp \overrightarrow{a}$, and $(2\overrightarrow{a}+\overrightarrow{b}) \perp \overrightarrow{b}$, calculate the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{3\pi}{4} | 3/8 |
In triangle \(ABC\), the bisectors \(AM\) and \(BN\) intersect at the point \(O\). Vertex \(C\) lies on the circle passing through the points \(O, M, N\). Find \(OM\) and \(ON\) if \(MN = \sqrt{3}\). | 1 | 1/8 |
Calculate the volume of the tetrahedron with vertices at points \(A_{1}, A_{2}, A_{3}, A_{4}\) and its height dropped from vertex \(A_{4}\) to the face \(A_{1} A_{2} A_{3}\).
Given the coordinates:
\[A_{1}(-3, 4, -7)\]
\[A_{2}(1, 5, -4)\]
\[A_{3}(-5, -2, 0)\]
\[A_{4}(2, 5, 4)\] | \sqrt{\frac{151}{15}} | 2/8 |
In triangle \(ABC\), the angle bisector \(AL\) intersects the circumscribed circle at point \(M\). Find the area of triangle \(ACM\) if \(AC = 6\), \(BC = 3\sqrt{3}\), and \(\angle C = 30^\circ\). | \frac{9\sqrt{3}}{2} | 5/8 |
Find all injective functions $f: \mathbb R \rightarrow \mathbb R$ such that for every real number $x$ and every positive integer $n$ , $$ \left|\sum_{i=1}^n i\left(f(x+i+1)-f(f(x+i))\right)\right|<2016 $$ *(Macedonia)* | f(x)=x+1 | 1/8 |
Given that Bag A contains two red balls and three white balls, and Bag B contains three red balls and three white balls. If one ball is randomly drawn from Bag A and placed into Bag B, and then one ball is randomly drawn from Bag B and placed into Bag A, let $\xi$ represent the number of white balls in Bag A at that time. Find $\mathrm{E} \xi$. | \frac{102}{35} | 6/8 |
Let the base of the rectangular prism $A B C D-A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ be a rhombus with an area of $2 \sqrt{3}$ and $\angle ABC = 60^\circ$. Points $E$ and $F$ lie on edges $CC'$ and $BB'$, respectively, such that $EC = BC = 2FB$. What is the volume of the pyramid $A-BCFE$? | $\sqrt{3}$ | 2/8 |
Let the arithmetic sequence $\{a_n\}$ satisfy: the common difference $d\in \mathbb{N}^*$, $a_n\in \mathbb{N}^*$, and any two terms' sum in $\{a_n\}$ is also a term in the sequence. If $a_1=3^5$, then the sum of all possible values of $d$ is . | 364 | 7/8 |
In civil engineering, the stability of cylindrical columns under a load is often examined using the formula \( L = \frac{50T^3}{SH^2} \), where \( L \) is the load in newtons, \( T \) is the thickness of the column in centimeters, \( H \) is the height of the column in meters, and \( S \) is a safety factor. Given that \( T = 5 \) cm, \( H = 10 \) m, and \( S = 2 \), calculate the value of \( L \). | 31.25 | 3/8 |
What is the value of $\log_{10}{16} + 3\log_{5}{25} + 4\log_{10}{2} + \log_{10}{64} - \log_{10}{8}$? | 9.311 | 1/8 |
Suppose $P$ is a polynomial with integer coefficients such that for every positive integer $n$ , the sum of the decimal digits of $|P(n)|$ is not a Fibonacci number. Must $P$ be constant?
(A *Fibonacci number* is an element of the sequence $F_0, F_1, \dots$ defined recursively by $F_0=0, F_1=1,$ and $F_{k+2} = F_{k+1}+F_k$ for $k\ge 0$ .)
*Nikolai Beluhov* | yes | 1/8 |
Let $\mathcal{F}$ be the family of all nonempty finite subsets of $\mathbb{N} \cup \{0\}.$ Find all real numbers $a$ for which the series $$ \sum_{A \in \mathcal{F}} \frac{1}{\sum_{k \in A}a^k} $$ is convergent. | 2 | 1/8 |
On an $8 \times 8$ grid, 64 points are marked at the center of each square. What is the minimum number of lines needed to separate all of these points from each other? | 14 | 5/8 |
The cells of a $2021\times 2021$ table are filled with numbers using the following rule. The bottom left cell, which we label with coordinate $(1, 1)$ , contains the number $0$ . For every other cell $C$ , we consider a route from $(1, 1)$ to $C$ , where at each step we can only go one cell to the right or one cell up (not diagonally). If we take the number of steps in the route and add the numbers from the cells along the route, we obtain the number in cell $C$ . For example, the cell with coordinate $(2, 1)$ contains $1 = 1 + 0$ , the cell with coordinate $(3, 1)$ contains $3 = 2 + 0 + 1$ , and the cell with coordinate $(3, 2)$ contains $7 = 3 + 0 + 1 + 3$ . What is the last digit of the number in the cell $(2021, 2021)$ ?
| 5 | 4/8 |
The sum of two natural numbers is 11. What is the maximum possible product that can be obtained with these numbers?
(a) 30
(b) 22
(c) 66
(d) 24
(e) 28 | 30 | 1/8 |
A circle with radius 10 is tangent to two adjacent sides $AB$ and $AD$ of the square $ABCD$. On the other two sides, the circle intersects, cutting off segments of 4 cm and 2 cm from the vertices, respectively. Find the length of the segment that the circle cuts off from vertex $B$ at the point of tangency. | 8 | 5/8 |
Show that at the beginning of the $k^{th}$ iteration in the main loop, the variables $u$ and $v$ are respectively equal to $u_k$ and $u_{2k}$. Deduce that if $p$ is a prime factor of $n$, the algorithm will enter the main loop at most $2p$ times, leading to Pollard's algorithm having a time complexity of $\mathcal{O}(p \log(n)^2)$. | \mathcal{O}(p\log(n)^2) | 5/8 |
Let $ABC$ be a right-angled triangle ( $\angle C = 90^\circ$ ) and $D$ be the midpoint of an altitude from C. The reflections of the line $AB$ about $AD$ and $BD$ , respectively, meet at point $F$ . Find the ratio $S_{ABF}:S_{ABC}$ .
Note: $S_{\alpha}$ means the area of $\alpha$ . | \frac{4}{3} | 1/8 |
\(ABCD\) is a rhombus with \(\angle B = 60^\circ\). \(P\) is a point inside \(ABCD\) such that \(\angle APC = 120^\circ\), \(BP = 3\) and \(DP = 2\). Find the difference between the lengths of \(AP\) and \(CP\). | \frac{\sqrt{21}}{3} | 1/8 |
Let's modify the problem slightly. Sara writes down four integers $a > b > c > d$ whose sum is $52$. The pairwise positive differences of these numbers are $2, 3, 5, 6, 8,$ and $11$. What is the sum of the possible values for $a$? | 19 | 1/8 |
The sultan gathered 300 court sages and proposed a trial. There are 25 different colors of hats, known in advance to the sages. The sultan informed them that each sage would be given one of these hats, and if they wrote down the number of hats for each color, all these numbers would be different. Each sage would see the hats of the other sages but not their own. Then all the sages would simultaneously announce the supposed color of their own hat. Can the sages agree in advance to act in such a way that at least 150 of them will correctly name their hat color? | 150 | 2/8 |
Point \(A\) on the plane is equidistant from all intersection points of the two parabolas given by the equations \(y = -3x^2 + 2\) and \(x = -4y^2 + 2\) in the Cartesian coordinate system on the plane. Find this distance. | \frac{\sqrt{697}}{24} | 1/8 |
Given \( n \) sticks. From any three, it is possible to form an obtuse triangle. What is the maximum possible value of \( n \)? | 4 | 1/8 |
(1) Evaluate: $\sin^2 120^\circ + \cos 180^\circ + \tan 45^\circ - \cos^2 (-330^\circ) + \sin (-210^\circ)$;
(2) Determine the monotonic intervals of the function $f(x) = \left(\frac{1}{3}\right)^{\sin x}$. | \frac{1}{2} | 5/8 |
What is the value of \[\left(\sum_{k=1}^{20} \log_{5^k} 3^{k^2}\right)\cdot\left(\sum_{k=1}^{100} \log_{9^k} 25^k\right)?\]
$\textbf{(A) }21 \qquad \textbf{(B) }100\log_5 3 \qquad \textbf{(C) }200\log_3 5 \qquad \textbf{(D) }2{,}200\qquad \textbf{(E) }21{,}000$ | \textbf{(E)}21{,}000 | 1/8 |
A particle moves through the first quadrant as follows. During the first minute it moves from the origin to $(1,0)$. Thereafter, it continues to follow the directions indicated in the figure, going back and forth between the positive x and y axes, moving one unit of distance parallel to an axis in each minute. At which point will the particle be after exactly 1989 minutes?
[asy] import graph; Label f; f.p=fontsize(6); xaxis(0,3.5,Ticks(f, 1.0)); yaxis(0,4.5,Ticks(f, 1.0)); draw((0,0)--(1,0)--(1,1)--(0,1)--(0,2)--(2,2)--(2,0)--(3,0)--(3,3)--(0,3)--(0,4)--(1.5,4),blue+linewidth(2)); arrow((2,4),dir(180),blue); [/asy] | (44,35) | 1/8 |
For the equation \( x^{2} - mx + 2m - 2 = 0 \) to have a solution within the interval \(\left[0, \frac{3}{2}\right]\), find the range of the real number \( m \). | [-\frac{1}{2},4-2\sqrt{2}] | 6/8 |
The Incredible Hulk can double the distance he jumps with each succeeding jump. If his first jump is 1 meter, the second jump is 2 meters, the third jump is 4 meters, and so on, then on which jump will he first be able to jump more than 1 kilometer (1,000 meters)?
$\textbf{(A)}\ 9^\text{th} \qquad \textbf{(B)}\ 10^\text{th} \qquad \textbf{(C)}\ 11^\text{th} \qquad \textbf{(D)}\ 12^\text{th} \qquad \textbf{(E)}\ 13^\text{th}$ | \textbf{(C)}\11^{ | 1/8 |
On a $300 \times 300$ board, several rooks are placed that beat the entire board. Within this case, each rook beats no more than one other rook. At what least $k$ , it is possible to state that there is at least one rook in each $k\times k$ square ? | 201 | 1/8 |
Find the lateral surface area of a regular triangular pyramid if its height is 4 and its slant height is 8. | 288 | 4/8 |
Given that \( a, b, c \) are three distinct real numbers. Consider the quadratic equations:
$$
\begin{array}{l}
x^{2}+a x+b=0, \\
x^{2}+b x+c=0, \\
x^{2}+c x+a=0
\end{array}
$$
where any two of these equations share exactly one common root. Find the value of \( a^{2}+b^{2}+c^{2} \). | 6 | 1/8 |
Given a circle, inscribe an isosceles triangle with an acute angle. Above one of the legs of this triangle, construct another isosceles triangle whose apex is also on the circumference of the circle. Continue this process and determine the limit of the angles at the vertices of the triangles. | 60 | 6/8 |
Given an equilateral triangle $DEF$ with $DE = DF = EF = 8$ units and a circle with radius $4$ units tangent to line $DE$ at $E$ and line $DF$ at $F$, calculate the area of the circle passing through vertices $D$, $E$, and $F$. | \frac{64\pi}{3} | 2/8 |
Suppose $a_{1}, a_{2}, \cdots, a_{n}, \cdots$ is a non-decreasing sequence of positive integers. For $m \geq 1$, define $b_{m}=\min \left\{n \mid a_{n} \geq m\right\}$, which means $b_{m}$ is the smallest $n$ such that $a_{n} \geq m$. Given $a_{19}=85$, find the maximum value of
$$
a_{1}+a_{2}+\cdots+a_{19}+b_{1}+b_{2}+\cdots+b_{85}
$$
(1985 USAMO problem). | 1700 | 3/8 |
Find all prime numbers whose representation in a base-14 numeral system has the form 101010...101 (alternating ones and zeros). | 197 | 5/8 |
When \( x, y, z \) are positive numbers, the maximum value of \( \frac{4xz + yz}{x^2 + y^2 + z^2} \) is ___. | \frac{\sqrt{17}}{2} | 6/8 |
Trapezoid $EFGH$ has sides $EF=105$, $FG=45$, $GH=21$, and $HE=80$, with $EF$ parallel to $GH$. A circle with center $Q$ on $EF$ is drawn tangent to $FG$ and $HE$. Find the exact length of $EQ$ using fractions. | \frac{336}{5} | 1/8 |
Given $\sin x_{1}=\sin x_{2}=\frac{1}{3}$ and $0 \lt x_{1} \lt x_{2} \lt 2\pi$, find $\cos |\overrightarrow{a}|$. | -\frac{7}{9} | 7/8 |
Kelvin and 15 other frogs are in a meeting, for a total of 16 frogs. During the meeting, each pair of distinct frogs becomes friends with probability $\frac{1}{2}$. Kelvin thinks the situation after the meeting is cool if for each of the 16 frogs, the number of friends they made during the meeting is a multiple of 4. Say that the probability of the situation being cool can be expressed in the form $\frac{a}{b}$, where $a$ and $b$ are relatively prime. Find $a$. | 1167 | 1/8 |
Let $x_1, x_2, ... , x_6$ be non-negative real numbers such that $x_1 +x_2 +x_3 +x_4 +x_5 +x_6 =1$, and $x_1 x_3 x_5 +x_2 x_4 x_6 \ge {\frac{1}{540}}$. Let $p$ and $q$ be positive relatively prime integers such that $\frac{p}{q}$ is the maximum possible value of $x_1 x_2 x_3 + x_2 x_3 x_4 +x_3 x_4 x_5 +x_4 x_5 x_6 +x_5 x_6 x_1 +x_6 x_1 x_2$. Find $p+q$. | 559 | 1/8 |
In an isosceles trapezoid \(M N K L\) with bases \(ML\) and \(NK\), the diagonals are perpendicular to the sides \(MN\) and \(KL\), and they intersect at an angle of \(22.5^\circ\). Find the height of the trapezoid if the length \(NQ = 3\), where \(Q\) is the midpoint of the longer base. | \frac{3\sqrt{2-\sqrt{2}}}{2} | 1/8 |
On the coordinate plane is given the square with vertices $T_1(1,0),T_2(0,1),T_3(-1,0),T_4(0,-1)$ . For every $n\in\mathbb N$ , point $T_{n+4}$ is defined as the midpoint of the segment $T_nT_{n+1}$ . Determine the coordinates of the limit point of $T_n$ as $n\to\infty$ , if it exists. | (0,0) | 1/8 |
In the sequence $\{a_n\}$,
$$
a_1 = 1, \quad a_2 = \frac{1}{4},
$$
and $a_{n+1} = \frac{(n-1) a_n}{n - a_n}$ for $n = 2, 3, \cdots$.
(1) Find the general term formula for the sequence $\{a_n\}$;
(2) Prove that for all $n \in \mathbf{Z}_+$,
$$
\sum_{k=1}^{n} a_{k}^{2} < \frac{7}{6}.
$$ | \sum_{k=1}^{n}a_k^2<\frac{7}{6} | 1/8 |
Suppose $a$ , $b$ , $c$ , and $d$ are positive real numbers which satisfy the system of equations \[\begin{aligned} a^2+b^2+c^2+d^2 &= 762, ab+cd &= 260, ac+bd &= 365, ad+bc &= 244. \end{aligned}\] Compute $abcd.$ *Proposed by Michael Tang* | 14400 | 7/8 |
Show that the characteristic function $\varphi=\varphi(t)$ of any absolutely continuous distribution with density $f=f(x)$ can be represented in the form
$$
\varphi(t)=\int_{\mathbb{R}} \phi(t+s) \overline{\phi(s)} \, d s, \quad t \in \mathbb{R}
$$
for some complex-valued function $\phi$ that satisfies the condition $\int_{\mathbb{R}}|\phi(s)|^{2} \, d s=1$. Assume for simplicity that
$$
\int_{\mathbb{R}} \sqrt{f(x)} \, d x<\infty, \quad \int_{\mathbb{R}} d t \left|\int_{\mathbb{R}} e^{i t x} \sqrt{f(x)} \, d x\right|<\infty.
$$ | \varphi()=\int_{\mathbb{R}}\phi(+)\overline{\phi()}\, | 2/8 |
Three cockroaches run along a circle in the same direction. They start simultaneously from a point $S$ . Cockroach $A$ runs twice as slow than $B$ , and thee times as slow than $C$ . Points $X, Y$ on segment $SC$ are such that $SX = XY =YC$ . The lines $AX$ and $BY$ meet at point $Z$ . Find the locus of centroids of triangles $ZAB$ . | (0,0) | 1/8 |
Let $ABCD$ be a square with side length $1$ . How many points $P$ inside the square (not on its sides) have the property that the square can be cut into $10$ triangles of equal area such that all of them have $P$ as a vertex?
*Proposed by Josef Tkadlec - Czech Republic* | 16 | 4/8 |
In a plane, three lines form an acute-angled, non-equilateral triangle. Fedya, who has a compass and a straightedge, wants to construct all the altitudes of this triangle. Vanya, with an eraser, tries to hinder him. On each turn, Fedya can either draw a straight line through two marked points or draw a circle with its center at one marked point and passing through another marked point. After that, Fedya can mark any number of points (points of intersection of the drawn lines, random points on the drawn lines, and random points in the plane). On his turn, Vanya erases no more than three marked points. (Fedya cannot use erased points in his constructions until he re-marks them). They take turns, with Fedya going first. Initially, no points are marked on the plane. Can Fedya construct the altitudes? | Yes | 5/8 |
Given the function \( y=\left(a \cos ^{2} x-3\right) \sin x \) has a minimum value of -3, determine the range of values for the real number \( a \). | [-\frac{3}{2},12] | 1/8 |
Professor Rackbrain would like to know the sum of all the numbers that can be composed of nine digits (excluding 0), using each digit exactly once in each number. | 201599999798400 | 2/8 |
Anton sold rabbits, an equal number every day. On the first day, 20% of all the rabbits were bought by Grandma #1, 1/4 of the remaining rabbits were taken by Man #1, then an unknown person X bought some rabbits, and Anton had 7 rabbits left. On the second day, 25% of all the rabbits were bought by Grandma #2, 1/3 of the remaining rabbits were taken by Man #2, then the unknown person X bought twice the number of rabbits he bought on the first day, and Anton had no rabbits left. How many rabbits did the unknown person X buy on the first day? | 5 | 4/8 |
In a magic square, the sum of the three entries in any row, column, or diagonal is the same value. The figure below shows four of the entries of a magic square. Find $x$.
[asy]
size(2cm);
for (int i=0; i<=3; ++i) draw((i,0)--(i,3)^^(0,i)--(3,i));
label("$x$",(0.5,2.5));label("$19$",(1.5,2.5));
label("$96$",(2.5,2.5));label("$1$",(0.5,1.5));
[/asy] | 200 | 6/8 |
In a store where all items cost an integer number of rubles, there are two special offers:
1) A customer who buys at least three items simultaneously can choose one item for free, whose cost does not exceed the minimum of the prices of the paid items.
2) A customer who buys exactly one item costing at least $N$ rubles receives a 20% discount on their next purchase (regardless of the number of items).
A customer, visiting this store for the first time, wants to purchase exactly four items with a total cost of 1000 rubles, where the cheapest item costs at least 99 rubles. Determine the maximum $N$ for which the second offer is more advantageous than the first. | 504 | 1/8 |
The diagonal \( BD \) of quadrilateral \( ABCD \) is the diameter of the circle circumscribed around this quadrilateral. Find the diagonal \( AC \) if \( BD = 2 \), \( AB = 1 \), and \( \angle ABD : \angle DBC = 4 : 3 \). | \frac{\sqrt{2} + \sqrt{6}}{2} | 2/8 |
Adams plans a profit of $10$ % on the selling price of an article and his expenses are $15$ % of sales. The rate of markup on an article that sells for $ $5.00$ is:
$\textbf{(A)}\ 20\% \qquad \textbf{(B)}\ 25\% \qquad \textbf{(C)}\ 30\% \qquad \textbf{(D)}\ 33\frac {1}{3}\% \qquad \textbf{(E)}\ 35\%$ | \textbf{(D)}\33\frac{1}{3} | 1/8 |
In the new clubroom, there were only chairs and a table. Each chair had four legs, and the table had three legs. Scouts came into the clubroom. Each sat on their own chair, two chairs remained unoccupied, and the total number of legs in the room was 101.
Determine how many chairs were in the clubroom. | 17 | 2/8 |
Find $d$, given that $\lfloor d\rfloor$ is a solution to \[3x^2 + 19x - 70 = 0\] and $\{d\} = d - \lfloor d\rfloor$ is a solution to \[4x^2 - 12x + 5 = 0.\] | -8.5 | 1/8 |
Let \( ABC \) be a triangle with centroid \( G \), and let \( E \) and \( F \) be points on side \( BC \) such that \( BE = EF = FC \). Points \( X \) and \( Y \) lie on lines \( AB \) and \( AC \), respectively, so that \( X, Y \), and \( G \) are not collinear. If the line through \( E \) parallel to \( XG \) and the line through \( F \) parallel to \( YG \) intersect at \( P \neq G \), prove that \( GP \) passes through the midpoint of \( XY \). | GPpassesthroughthemidpointofXY | 4/8 |
A random simulation method is used to estimate the probability of a shooter hitting the target at least 3 times out of 4 shots. A calculator generates random integers between 0 and 9, where 0 and 1 represent missing the target, and 2 through 9 represent hitting the target. Groups of 4 random numbers represent the results of 4 shots. After randomly simulating, 20 groups of random numbers were generated:
7527 0293 7140 9857 0347 4373 8636 6947 1417 4698
0371 6233 2616 8045 6011 3661 9597 7424 7610 4281
Estimate the probability that the shooter hits the target at least 3 times out of 4 shots based on the data above. | 0.75 | 7/8 |
The measure of each exterior angle of a regular polygon is \(20^\circ\). What is the sum of the measures of the interior angles and the total number of diagonals of this polygon? | 135 | 7/8 |
When a swing is stationary, with the footboard one foot off the ground, pushing it forward two steps (in ancient times, one step was considered as five feet) equals 10 feet, making the footboard of the swing the same height as a person who is five feet tall, determine the length of the rope when pulled straight at this point. | 14.5 | 3/8 |
Given that Mr. A initially owns a home worth $\$15,000$, he sells it to Mr. B at a $20\%$ profit, then Mr. B sells it back to Mr. A at a $15\%$ loss, then Mr. A sells it again to Mr. B at a $10\%$ profit, and finally Mr. B sells it back to Mr. A at a $5\%$ loss, calculate the net effect of these transactions on Mr. A. | 3541.50 | 1/8 |
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