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stringlengths 10
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The arc of the cubic parabola $y=\frac{1}{3} x^{3}$, enclosed between the points $O(0,0)$ and $A(1,1 / 3)$, rotates around the $Ox$ axis. Find the surface area of the rotation. | \frac{\pi}{9}(2\sqrt{2}-1) | 3/8 |
Given the sets
$$
\begin{array}{c}
M=\{x, xy, \lg (xy)\} \\
N=\{0, |x|, y\},
\end{array}
$$
and that \( M = N \), determine the value of
$$
\left(x+\frac{1}{y}\right)+\left(x^2+\frac{1}{y^2}\right)+\left(x^3+\frac{1}{y^3}\right)+\cdots+\left(x^{2001}+\frac{1}{y^{2001}}\right).
$$ | -2 | 6/8 |
A linear function \( f(x) \) is given. It is known that the distance between the points of intersection of the graphs \( y = x^{2} \) and \( y = f(x) \) is \( 2 \sqrt{3} \), and the distance between the points of intersection of the graphs \( y = x^{2}-2 \) and \( y = f(x)+1 \) is \( \sqrt{60} \). Find the distance between the points of intersection of the graphs \( y = x^{2}-1 \) and \( y = f(x)+1 \). | 2 \sqrt{11} | 7/8 |
Given a set $S$ of $n$ points in the plane such that no three points from $S$ are collinear, show that the number of triangles of area 1 whose vertices are in $S$ is at most:
$$
\frac{2 n(n-1)}{3}
$$ | \frac{2n(n-1)}{3} | 6/8 |
Consider a polinomial $p \in \mathbb{R}[x]$ of degree $n$ and with no real roots. Prove that $$ \int_{-\infty}^{\infty}\frac{(p'(x))^2}{(p(x))^2+(p'(x))^2}dx $$ converges, and is less or equal than $n^{3/2}\pi.$ | n^{3/2}\pi | 1/8 |
It is given that $2^{333}$ is a 101-digit number whose first digit is 1. How many of the numbers $2^k$ , $1\le k\le 332$ have first digit 4? | 32 | 4/8 |
Given the complex number \( z = 2 \cos^2 \theta + \mathrm{i} + (\sin \theta + \cos \theta)^2 \mathrm{i} \), where \( \theta \in \left[ 0, \frac{\pi}{2} \right] \), find the maximum and minimum values of \( |z| \). | 2 | 6/8 |
Find the number of pairs of integers \((x, y)\) that satisfy the equation \(x^{2} + 7xy + 6y^{2} = 15^{50}\). | 4998 | 5/8 |
The ratio of $w$ to $x$ is $4:3$, of $y$ to $z$ is $3:2$ and of $z$ to $x$ is $1:6$. What is the ratio of $w$ to $y$? | 16:3 | 7/8 |
A circle of radius $r$ has chords $\overline{AB}$ of length $12$ and $\overline{CD}$ of length $9$. When $\overline{AB}$ and $\overline{CD}$ are extended through $B$ and $C$, respectively, they intersect at point $P$, which is outside of the circle. If $\angle{APD}=90^\circ$ and $BP=10$, determine $r^2$. | 221 | 1/8 |
In a debate competition with four students participating, the rules are as follows: Each student must choose one question to answer from two given topics, Topic A and Topic B. For Topic A, answering correctly yields 100 points and answering incorrectly results in a loss of 100 points. For Topic B, answering correctly yields 90 points and answering incorrectly results in a loss of 90 points. If the total score of the four students is 0 points, how many different scoring situations are there? | 36 | 7/8 |
A case contains 20 cassette tapes with disco music and 10 with techno music. A DJ randomly picks two tapes one after the other. What is the probability that
1) The first tape contains disco music;
2) The second tape also contains disco music.
Consider two scenarios:
a) The DJ returns the first tape to the case before picking the second tape;
b) The DJ does not return the first tape to the case. | \frac{19}{29} | 2/8 |
If $a,b,c,d$ are Distinct Real no. such that $a = \sqrt{4+\sqrt{5+a}}$ $b = \sqrt{4-\sqrt{5+b}}$ $c = \sqrt{4+\sqrt{5-c}}$ $d = \sqrt{4-\sqrt{5-d}}$ Then $abcd = $ | 11 | 7/8 |
Severus Snape, the potions professor, prepared three potions, each in an equal volume of 400 ml. The first potion makes the drinker smarter, the second makes them more beautiful, and the third makes them stronger. To ensure the effect of any potion, it is sufficient to drink at least 30 ml of that potion. Snape intended to drink the potions himself, but he was called to see the headmaster and had to leave, leaving the labeled potions on his desk in large jugs. Harry, Hermione, and Ron took advantage of his absence and began to taste the potions.
Hermione was the first to try the potions: she approached the first jug with the intelligence potion and drank half of it, then poured the remaining potion into the second jug with the beauty potion, stirred the contents of the jug thoroughly, and drank half of it. Next, it was Harry's turn: he drank half of the third jug with the strength potion, poured the remaining potion into the second jug, stirred everything in this jug thoroughly, and drank half of it. Now all the contents are in the second jug, which went to Ron. What percentage of the contents of this jug does Ron need to drink to ensure that each of the three potions will have an effect on him? | 60 | 6/8 |
Given a positive integer \( n \geq 2 \), find the minimum value of \( |X| \) such that for any \( n \) binary subsets \( B_1, B_2, \ldots, B_n \) of set \( X \), there exists a subset \( Y \) of \( X \) satisfying:
1. \( |Y| = n \);
2. For \( i = 1, 2, \ldots, n \), \( |Y \cap B_i| \leq 1 \). | 2n-1 | 2/8 |
Through the centroid \( O \) of triangle \( ABC \), a line is drawn intersecting its sides at points \( M \) and \( N \). Prove that \( NO \leq 2 MO \). | NO\le2MO | 2/8 |
Suppose that a polynomial of the form $p(x)=x^{2010} \pm x^{2009} \pm \cdots \pm x \pm 1$ has no real roots. What is the maximum possible number of coefficients of -1 in $p$? | 1005 | 1/8 |
Given three sets, \( A = \{ x \mid x \in \mathbf{R} \text{ and } x > 1 \} \), \( B = \left\{ y \left\lvert y = \lg \frac{2}{x+1} , x \in A \right. \right\} \), \( C = \left\{ z \left\lvert z = \frac{m^{2} x - 1}{m x + 1}, x \in A \right. \right\} \). If \( C \subseteq B \), find the range of \( m \). | (-\infty,-1]\cup{0} | 4/8 |
In a plane, 100 points are marked. It turns out that 40 marked points lie on each of two different lines \( a \) and \( b \). What is the maximum number of marked points that can lie on a line that does not coincide with \( a \) or \( b \)? | 23 | 5/8 |
One line is described by
\[\begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix} + t \begin{pmatrix} 1 \\ 1 \\ -k \end{pmatrix}.\]Another line is described by
\[\begin{pmatrix} 1 \\ 4 \\ 5 \end{pmatrix} + u \begin{pmatrix} k \\ 2 \\ 1 \end{pmatrix}.\]If the lines are coplanar (i.e. there is a plane that contains both lines), then find all possible values of $k.$ | 0,-3 | 1/8 |
All vertices of a regular 2016-gon are initially white. What is the least number of them that can be painted black so that:
(a) There is no right triangle
(b) There is no acute triangle
having all vertices in the vertices of the 2016-gon that are still white? | 1008 | 3/8 |
Max repeatedly throws a fair coin in a hurricane. For each throw, there is a $4\%$ chance that the coin gets blown away. He records the number of heads $H$ and the number of tails $T$ before the coin is lost. (If the coin is blown away on a toss, no result is recorded for that toss.) What is the expected value of $|H-T|$ ?
*Proposed by Krit Boonsiriseth.* | \frac{24}{7} | 2/8 |
Define the sequence \(\{a_n\}\) as \(a_n = n^3 + 4\), where \(n \in \mathbb{N}_+\). Let \(d_n = (a_n, a_{n+1})\) which is the greatest common divisor of \(a_n\) and \(a_{n+1}\). What is the maximum value of \(d_n\)? | 433 | 5/8 |
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} | 1/8 |
Let $p, q, r, s, t, u, v, w$ be distinct elements in the set $\{-8, -6, -4, -1, 1, 3, 5, 14\}$. What is the minimum possible value of $(p+q+r+s)^2 + (t+u+v+w)^2$? | 10 | 3/8 |
Calculate the lengths of the arcs of the curves given by the equations in polar coordinates.
$$
\rho=\sqrt{2} e^{\varphi}, 0 \leq \varphi \leq \frac{\pi}{3}
$$ | 2(e^{\frac{\pi}{3}}-1) | 2/8 |
In $\triangle ABC,$ $AB=AC=30$ and $BC=28.$ Points $G, H,$ and $I$ are on sides $\overline{AB},$ $\overline{BC},$ and $\overline{AC},$ respectively, such that $\overline{GH}$ and $\overline{HI}$ are parallel to $\overline{AC}$ and $\overline{AB},$ respectively. What is the perimeter of parallelogram $AGHI$? | 60 | 5/8 |
Four points in the order \( A, B, C, D \) lie on a circle with the extension of \( AB \) meeting the extension of \( DC \) at \( E \) and the extension of \( AD \) meeting the extension of \( BC \) at \( F \). Let \( EP \) and \( FQ \) be tangents to this circle with points of tangency \( P \) and \( Q \) respectively. Suppose \( EP = 60 \) and \( FQ = 63 \). Determine the length of \( EF \). | 87 | 4/8 |
Steph Curry is playing the following game and he wins if he has exactly 5 points at some time. Flip a fair coin. If heads, shoot a 3-point shot which is worth 3 points. If tails, shoot a free throw which is worth 1 point. He makes \frac{1}{2} of his 3-point shots and all of his free throws. Find the probability he will win the game. (Note he keeps flipping the coin until he has exactly 5 or goes over 5 points) | \frac{140}{243} | 4/8 |
Find all solutions to \( x^{n+1} - (x + 1)^n = 2001 \) in positive integers \( x \) and \( n \). | (13,2) | 3/8 |
When three standard dice are tossed, the numbers $a, b, c$ are obtained. Find the probability that $abc = 72$. | \frac{1}{24} | 7/8 |
Let \( a, b, c \in \mathbf{R}^{+} \). Prove that
\[
\frac{(2a + b + c)^2}{2a^2 + (b + c)^2} + \frac{(2b + a + c)^2}{2b^2 + (c + a)^2} + \frac{(2c + a + b)^2}{2c^2 + (a + b)^2} \leq 8.
\] | 8 | 1/8 |
The function \( f: \mathbf{R}^{+} \rightarrow \mathbf{R}^{+} \) satisfies the condition: for any \( x, y, u, v \in \mathbf{R}^{+} \), the following inequality holds:
\[ f\left(\frac{x}{2 u}+\frac{y}{2 v}\right) \leq \frac{1}{2} u f(x) + \frac{1}{2} v f(y). \]
Find all such functions \( f \). | f(x)=\frac{}{x} | 1/8 |
Given the task of selecting 10 individuals to participate in a quality education seminar from 7 different schools, with the condition that at least one person must be chosen from each school, determine the total number of possible allocation schemes. | 84 | 1/8 |
In the diagram, triangles $ABC$ and $CBD$ are isosceles. The perimeter of $\triangle CBD$ is $19,$ the perimeter of $\triangle ABC$ is $20,$ and the length of $BD$ is $7.$ What is the length of $AB?$ [asy]
size(7cm);
defaultpen(fontsize(11));
pair b = (0, 0);
pair d = 7 * dir(-30);
pair a = 8 * dir(-140);
pair c = 6 * dir(-90);
draw(a--b--d--c--cycle);
draw(b--c);
label("$y^\circ$", a, 2 * (E + NE));
label("$y^\circ$", b, 2 * (S + SW));
label("$x^\circ$", b, 2 * (S + SE));
label("$x^\circ$", d, 2 * (2 * W));
label("$A$", a, W);
label("$B$", b, N);
label("$D$", d, E);
label("$C$", c, S);
[/asy] | 8 | 3/8 |
Compute the number of real solutions $(x,y,z,w)$ to the system of equations:
\begin{align*}
x &= z+w+zwx, \\
y &= w+x+wxy, \\
z &= x+y+xyz, \\
w &= y+z+yzw.
\end{align*} | 5 | 1/8 |
In the adjoining figure, $CD$ is the diameter of a semicircle with center $O$. Point $A$ lies on the extension of $DC$ past $C$; point $E$ lies on the semicircle, and $B$ is the point of intersection (distinct from $E$) of line segment $AE$ with the semicircle. If length $AB$ equals length $OD$, and the measure of $\angle EOD$ is $45^\circ$, then find the measure of $\angle BAO$, in degrees.
[asy]
import graph;
unitsize(2 cm);
pair O, A, B, C, D, E;
O = (0,0);
C = (-1,0);
D = (1,0);
E = dir(45);
B = dir(165);
A = extension(B,E,C,D);
draw(arc(O,1,0,180));
draw(D--A--E--O);
label("$A$", A, W);
label("$B$", B, NW);
label("$C$", C, S);
label("$D$", D, S);
label("$E$", E, NE);
label("$O$", O, S);
[/asy] | 15^\circ | 4/8 |
Let $A B C D E$ be a convex pentagon such that $\angle A B C=\angle A C D=\angle A D E=90^{\circ}$ and $A B=B C=C D=D E=1$. Compute $A E$. | 2 | 3/8 |
Let the function \( f:(0,1) \rightarrow \mathbf{R} \) be defined as
$$
f(x)=\left\{\begin{array}{l}
x, \text{ if } x \text{ is irrational; } \\
\frac{p+1}{q}, \text{ if } x=\frac{p}{q}, \text{ where } (p, q)=1 \text{ and } 0<p<q.
\end{array}\right.
$$
Find the maximum value of \( f(x) \) on the interval \( \left(\frac{7}{8}, \frac{8}{9}\right) \). | \frac{16}{17} | 2/8 |
The ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (where $a > b > 0$) has an eccentricity of $e = \frac{2}{3}$. Points A and B lie on the ellipse and are not symmetrical with respect to the x-axis or the y-axis. The perpendicular bisector of segment AB intersects the x-axis at point P(1, 0). Let the midpoint of AB be C($x_0$, $y_0$). Find the value of $x_0$. | \frac{9}{4} | 6/8 |
When simplified, $\log{8} \div \log{\frac{1}{8}}$ becomes:
$\textbf{(A)}\ 6\log{2} \qquad \textbf{(B)}\ \log{2} \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 0\qquad \textbf{(E)}\ -1$ | \textbf{(E)} | 1/8 |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=2$, and $|\overrightarrow{a}-\overrightarrow{b}|=\sqrt{7}$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{2\pi}{3} | 1/8 |
The least common multiple of $a$ and $b$ is $20$, and the least common multiple of $b$ and $c$ is $21$. Find the least possible value of the least common multiple of $a$ and $c$. | 420 | 6/8 |
If $\log_3 (x+5)^2 + \log_{1/3} (x - 1) = 4,$ compute $x.$ | \frac{71 + \sqrt{4617}}{2} | 1/8 |
Five cards labeled A, B, C, D, and E are placed consecutively in a row. How many ways can they be re-arranged so that no card is moved more than one position away from where it started? | 8 | 6/8 |
The sum of the squares of four consecutive positive integers is 9340. What is the sum of the cubes of these four integers? | 457064 | 1/8 |
What is the smallest positive value of $m$ so that the equation $10x^2 - mx + 420 = 0$ has integral solutions? | 130 | 7/8 |
Starting with the display "1," calculate the fewest number of keystrokes needed to reach "240" using only the keys [+1] and [x2]. | 10 | 3/8 |
A kite-shaped field is planted uniformly with wheat. The sides of the kite are 120 m and 80 m, with angles between the unequal sides being \(120^\circ\) and the other two angles being \(60^\circ\) each. At harvest, the wheat at any point in the field is brought to the nearest point on the field's perimeter. Determine the fraction of the crop that is brought to the longest side of 120 m. | \frac{1}{2} | 1/8 |
In how many ways can eight out of the nine digits \(1, 2, 3, 4, 5, 6, 7, 8,\) and \(9\) be placed in a \(4 \times 2\) table (4 rows, 2 columns) such that the sum of the digits in each row, starting from the second, is 1 more than in the previous row? | 64 | 6/8 |
Each cell of an \( m \times n \) board is filled with some nonnegative integer. Two numbers in the filling are said to be adjacent if their cells share a common side. The filling is called a garden if it satisfies the following two conditions:
(i) The difference between any two adjacent numbers is either 0 or 1.
(ii) If a number is less than or equal to all of its adjacent numbers, then it is equal to 0.
Determine the number of distinct gardens in terms of \( m \) and \( n \). | 2^{mn}-1 | 1/8 |
A trapezoid has side lengths 3, 5, 7, and 11. The sum of all the possible areas of the trapezoid can be written in the form of $r_1\sqrt{n_1}+r_2\sqrt{n_2}+r_3$, where $r_1$, $r_2$, and $r_3$ are rational numbers and $n_1$ and $n_2$ are positive integers not divisible by the square of any prime. What is the greatest integer less than or equal to $r_1+r_2+r_3+n_1+n_2$? | 63 | 1/8 |
Let \( A_{12} \) denote the answer to problem 12. There exists a unique triple of digits \( (B, C, D) \) such that \( 10 > A_{12} > B > C > D > 0 \) and
\[ \overline{A_{12} B C D} - \overline{D C B A_{12}} = \overline{B D A_{12} C}, \]
where \( \overline{A_{12} B C D} \) denotes the four-digit base 10 integer. Compute \( B + C + D \). | 11 | 7/8 |
How many of the first $1000$ positive integers can be expressed in the form
\[\lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor\]where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$? | 600 | 1/8 |
Given the harmonic mean of the first n terms of the sequence $\left\{{a}_{n}\right\}$ is $\dfrac{1}{2n+1}$, and ${b}_{n}= \dfrac{{a}_{n}+1}{4}$, find the value of $\dfrac{1}{{b}_{1}{b}_{2}}+ \dfrac{1}{{b}_{2}{b}_{3}}+\ldots+ \dfrac{1}{{b}_{10}{b}_{11}}$. | \dfrac{10}{11} | 1/8 |
Determine the area of triangle \(ABC\) if \(A(1, 2)\), \(B(-2, 5)\), and \(C(4, -2)\). | 1.5 | 1/8 |
Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(2017,0),(2017,2018),$ and $(0,2018)$. What is the probability that $x > 9y$? Express your answer as a common fraction. | \frac{2017}{36324} | 7/8 |
In quadrilateral $ABCD$, side $AB$ is equal to side $BC$, diagonal $AC$ is equal to side $CD$, and $\angle ACB = \angle ACD$. The radii of the circles inscribed in triangles $ACB$ and $ACD$ are in the ratio $3:4$. Find the ratio of the areas of these triangles. | \frac{9}{14} | 2/8 |
In a football tournament, six teams played: each team played once with every other team. Teams that scored twelve or more points advance to the next round. For a victory, 3 points are awarded, for a draw - 1 point, for a loss - 0 points. What is the maximum number of teams that can advance to the next round? | 3 | 5/8 |
In the rectangular coordinate system xOy, the parametric equations of the curve C1 are given by $$\begin{cases} x=t\cos\alpha \\ y=1+t\sin\alpha \end{cases}$$, and the polar coordinate equation of the curve C2 with the origin O as the pole and the positive semi-axis of the x-axis as the polar axis is ρ=2cosθ.
1. If the parameter of curve C1 is α, and C1 intersects C2 at exactly one point, find the Cartesian equation of C1.
2. Given point A(0, 1), if the parameter of curve C1 is t, 0<α<π, and C1 intersects C2 at two distinct points P and Q, find the maximum value of $$\frac {1}{|AP|}+\frac {1}{|AQ|}$$. | 2\sqrt{2} | 7/8 |
Let \( T \) be the set of numbers of the form \( 2^{a} 3^{b} \) where \( a \) and \( b \) are integers satisfying \( 0 \leq a, b \leq 5 \). How many subsets \( S \) of \( T \) have the property that if \( n \) is in \( S \) then all positive integer divisors of \( n \) are in \( S \)? | 924 | 5/8 |
There are five students: A, B, C, D, and E;
(1) If these five students line up in a row, in how many ways can A not stand in the first position?
(2) If these five students line up in a row, and A and B must be next to each other while C and D must not be next to each other, in how many ways can they line up?
(3) If these five students participate in singing, dancing, chess, and drawing competitions, with at least one person in each competition, and each student must participate in exactly one competition, and A cannot participate in the dancing competition, how many participating arrangements are there? | 180 | 1/8 |
There are 20 chairs in a room of two colors: blue and red. Seated on each chair is either a knight or a liar. Knights always tell the truth, and liars always lie. Each of the seated individuals initially declared that they were sitting on a blue chair. After that, they somehow changed seats, and now half of the seated individuals claim to be sitting on blue chairs while the other half claim to be sitting on red chairs. How many knights are now sitting on red chairs? | 5 | 7/8 |
Given a rectangle \(ABCD\). A circle intersects the side \(AB\) at the points \(K\) and \(L\) and the side \(CD\) at the points \(M\) and \(N\). Find the length of segment \(MN\) if \(AK=10\), \(KL=17\), and \(DN=7\). | 23 | 5/8 |
Define a $\it{good\ word}$ as a sequence of letters that consists only of the letters $A$, $B$, $C$, and $D$ --- some of these letters may not appear in the sequence --- and in which $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, $C$ is never immediately followed by $D$, and $D$ is never immediately followed by $A$. How many eight-letter good words are there? | 8748 | 7/8 |
There are 4 different colors of light bulbs, with each color representing a different signal. Assuming there is an ample supply of each color, we need to install one light bulb at each vertex $P, A, B, C, A_{1}, B_{1}, C_{1}$ of an airport signal tower (as shown in the diagram), with the condition that the two ends of the same line segment must have light bulbs of different colors. How many different installation methods are there? | 2916 | 1/8 |
Let $A\in\mathcal{M}_n(\mathbb{C})$ be an antisymmetric matrix, i.e. $A=-A^t.$ [list=a]
[*]Prove that if $A\in\mathcal{M}_n(\mathbb{R})$ and $A^2=O_n$ then $A=O_n.$ [*]Assume that $n{}$ is odd. Prove that if $A{}$ is the adjoint of another matrix $B\in\mathcal{M}_n(\mathbb{C})$ then $A^2=O_n.$ [/list] | O_n | 1/8 |
Let \( ABC \) be a right triangle with hypotenuse \( AC \). Let \( B' \) be the reflection of point \( B \) across \( AC \), and let \( C' \) be the reflection of \( C \) across \( AB' \). Find the ratio of \([BCB']\) to \([BC'B']\). | 1 | 4/8 |
Let $ABCD$ be a nondegenerate isosceles trapezoid with integer side lengths such that $BC \parallel AD$ and $AB=BC=CD$ . Given that the distance between the incenters of triangles $ABD$ and $ACD$ is $8!$ , determine the number of possible lengths of segment $AD$ .
*Ray Li* | 337 | 1/8 |
Six students sign up for three different intellectual competition events. How many different registration methods are there under the following conditions? (Not all six students must participate)
(1) Each person participates in exactly one event, with no limit on the number of people per event;
(2) Each event is limited to one person, and each person can participate in at most one event;
(3) Each event is limited to one person, but there is no limit on the number of events a person can participate in. | 216 | 7/8 |
In a convex $n$ -gon, several diagonals are drawn. Among these diagonals, a diagonal is called *good* if it intersects exactly one other diagonal drawn (in the interior of the $n$ -gon). Find the maximum number of good diagonals. | 2\lfloor\frac{n}{2}\rfloor-2 | 1/8 |
Riquinho distributed 1000,00 reais among his friends Antônio, Bernardo, and Carlos in the following manner: he gave, successively, 1 real to Antônio, 2 reais to Bernardo, 3 reais to Carlos, 4 reais to Antônio, 5 reais to Bernardo, etc. How much money did Bernardo receive? | 345 | 6/8 |
Let " $\sum$ " denote the cyclic sum. If $a, b, c$ are given distinct real numbers, then
$$
f(x)=\sum \frac{a^{2}(x-b)(x-c)}{(a-b)(a-c)}
$$
Simplify the expression to obtain... | x^2 | 5/8 |
The area of this region formed by six congruent squares is 294 square centimeters. What is the perimeter of the region, in centimeters?
[asy]
draw((0,0)--(-10,0)--(-10,10)--(0,10)--cycle);
draw((0,10)--(0,20)--(-30,20)--(-30,10)--cycle);
draw((-10,10)--(-10,20));
draw((-20,10)--(-20,20));
draw((-20,20)--(-20,30)--(-40,30)--(-40,20)--cycle);
draw((-30,20)--(-30,30));
[/asy] | 98 | 2/8 |
Point \(P\) is located inside a square \(ABCD\) with side length 10. Let \(O_1, O_2, O_3\), and \(O_4\) be the circumcenters of \(\triangle PAB, \triangle PBC, \triangle PCD\), and \(\triangle PDA\), respectively. Given that \(PA + PB + PC + PD = 23\sqrt{2}\) and the area of quadrilateral \(O_1O_2O_3O_4\) is 50, the second largest of the lengths \(O_1O_2, O_2O_3, O_3O_4,\) and \(O_4O_1\) can be written as \(\sqrt{\frac{a}{b}}\), where \(a\) and \(b\) are relatively prime positive integers. Compute \(100a + b\). | 16902 | 1/8 |
On the edge $A A^{\prime}$ of a cube $A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ with edge length 2, a point $K$ is marked. A point $T$ is marked in space such that $T B = \sqrt{11}$ and $T C = \sqrt{15}$. Find the length of the height of the tetrahedron $T B C K$ dropped from the vertex $C$. | 2 | 4/8 |
For how many positive integers $n$ less than $2013$, does $p^2+p+1$ divide $n$ where $p$ is the least prime divisor of $n$? | 212 | 1/8 |
Let $N$ be the number of complex numbers $z$ with the properties that $|z|=1$ and $z^{6!}-z^{5!}$ is a real number. Find the remainder when $N$ is divided by $1000$. | 440 | 7/8 |
Let $\omega = -\frac{1}{2} + \frac{i\sqrt{3}}{2}$ and $\omega^2 = -\frac{1}{2} - \frac{i\sqrt{3}}{2}$. Define $T$ as the set of all points in the complex plane of the form $a + b\omega + c\omega^2 + d$, where $0 \leq a, b, c \leq 1$ and $d \in \{0, 1\}$. Find the area of $T$. | 3\sqrt{3} | 1/8 |
Given that \( x, y, z \) are positive real numbers such that \( x + y + z = 1 \), find the minimum value of the function \( f(x, y, z) = \frac{3x^{2} - x}{1 + x^{2}} + \frac{3y^{2} - y}{1 + y^{2}} + \frac{3z^{2} - z}{1 + z^{2}} \), and provide a proof. | 0 | 2/8 |
Given a regular tetrahedron of volume 1, obtain a second regular tetrahedron by reflecting the given one through its center. What is the volume of their intersection? | 1/2 | 2/8 |
Find the sum of all positive integers $b < 1000$ such that the base-$b$ integer $36_{b}$ is a perfect square and the base-$b$ integer $27_{b}$ is a perfect cube. | 371 | 6/8 |
Jar C initially contains 6 red buttons and 12 green buttons. Michelle removes the same number of red buttons as green buttons from Jar C and places them into an empty Jar D. After the removal, Jar C is left with $\frac{3}{4}$ of its initial button count. If Michelle were to randomly choose a button from Jar C and a button from Jar D, what is the probability that both chosen buttons are green? Express your answer as a common fraction. | \frac{5}{14} | 1/8 |
Given that $F_{2}$ is the right focus of the ellipse $mx^{2}+y^{2}=4m\left(0 \lt m \lt 1\right)$, point $A\left(0,2\right)$, and point $P$ is any point on the ellipse, and the minimum value of $|PA|-|PF_{2}|$ is $-\frac{4}{3}$, then $m=$____. | \frac{2}{9} | 2/8 |
Write the smallest number whose digits sum to 62 and in which at least three different digits are used. | 17999999 | 1/8 |
Let $\mathcal{S}$ be the set $\{1, 2, 3, \dots, 12\}$. Let $n$ be the number of sets of two non-empty disjoint subsets of $\mathcal{S}$. Calculate the remainder when $n$ is divided by 500. | 125 | 7/8 |
Given four points \( A, B, C, D \) on the surface of a sphere with a radius of 2, and that \( AB = CD = 2 \), determine the maximum volume of the tetrahedron \( ABCD \). | \frac{4\sqrt{3}}{3} | 1/8 |
How many triangles exist in which the measures of the angles, measured in degrees, are whole numbers? | 2700 | 2/8 |
Given a rhombus \(ABCD\), \(\Gamma_{B}\) and \(\Gamma_{C}\) are circles centered at \(B\) and \(C\) passing through \(C\) and \(B\) respectively. \(E\) is an intersection point of circles \(\Gamma_{B}\) and \(\Gamma_{C}\). The line \(ED\) intersects circle \(\Gamma_{B}\) at a second point \(F\). Find the measure of \(\angle AFB\). | 60 | 2/8 |
\(a_{1}, a_{2}, a_{3}, \ldots\) is an increasing sequence of natural numbers. It is known that \(a_{a_{k}} = 3k\) for any \(k\).
Find
a) \(a_{100}\)
b) \(a_{1983}\). | 3762 | 1/8 |
The sequence $\left\{a_{n}\right\}$ is defined as follows:
$a_{1}=1$, and $a_{n+1}=a_{n}+\frac{1}{a_{n}}$ for $n \geq 1$.
Find the integer part of $a_{100}$.
| 14 | 1/8 |
A family of 4 people, consisting of a mom, dad, and two children, arrived in city $N$ for 5 days. They plan to make 10 trips on the subway each day. What is the minimum amount they will have to spend on tickets if the following rates are available in city $N$?
| Adult ticket for one trip | 40 rubles |
| --- | --- |
| Child ticket for one trip | 20 rubles |
| Unlimited day pass for one person | 350 rubles |
| Unlimited day pass for a group of up to 5 people | 1500 rubles |
| Unlimited three-day pass for one person | 900 rubles |
| Unlimited three-day pass for a group of up to 5 people | 3500 rubles | | 5200 | 1/8 |
Show that the infinite decimal
$$
T=0, a_{1} a_{2} a_{3} \ldots
$$
is a rational number, where $a_{n}$ is the remainder when $n^{2}$ is divided by 10. Express this number as a fraction of two integers. | \frac{166285490}{1111111111} | 2/8 |
For each positive integer $n$, let $S(n)$ denote the sum of the digits of $n$. For how many values of $n$ is $n+S(n)+S(S(n))=2007$? | 4 | 3/8 |
A 9x9 chessboard has its squares labeled such that the label of the square in the ith row and jth column is given by $\frac{1}{2 \times (i + j - 1)}$. We need to select one square from each row and each column. Find the minimum sum of the labels of the nine chosen squares. | \frac{1}{2} | 3/8 |
S is a subset of {1, 2, 3, ... , 16} which does not contain three integers which are relatively prime in pairs. How many elements can S have? | 11 | 1/8 |
Show that
$$
\sqrt{x+2+2 \sqrt{x+1}}-\sqrt{x+2-2 \sqrt{x+1}}
$$
is a rational number. | 2 | 4/8 |
Consider a wardrobe that consists of $6$ red shirts, $7$ green shirts, $8$ blue shirts, $9$ pairs of pants, $10$ green hats, $10$ red hats, and $10$ blue hats. Additionally, you have $5$ ties in each color: green, red, and blue. Every item is distinct. How many outfits can you make consisting of one shirt, one pair of pants, one hat, and one tie such that the shirt and hat are never of the same color, and the tie must match the color of the hat? | 18900 | 7/8 |
Person A arrives between 7:00 and 8:00, while person B arrives between 7:20 and 7:50. The one who arrives first waits for the other for 10 minutes, after which they leave. Calculate the probability that the two people will meet. | \frac{1}{3} | 2/8 |
Let $m$ be a real number where $m > 0$. If for any $x \in (1, +\infty)$, the inequality $2e^{2mx} - \frac{ln x}{m} ≥ 0$ always holds, then find the minimum value of the real number $m$. | \frac{1}{2e} | 5/8 |
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