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For some real number $a$ , define two parabolas on the coordinate plane with equations $x = y^2 + a$ and $y = x^2 + a$ . Suppose there are $3$ lines, each tangent to both parabolas, that form an equilateral triangle with positive area $s$ . If $s^2 = \tfrac pq$ for coprime positive integers $p$ , $q$ , find $p + q$ .
*Proposed by Justin Lee* | 91 | 1/8 |
A rigid board with a mass \( m \) and a length \( l = 20 \) meters partially lies on the edge of a horizontal surface, overhanging it by three quarters of its length. To prevent the board from falling, a stone with a mass of \( 2m \) was placed at its very edge. How far from the stone can a person with a mass of \( m / 2 \) walk on the board? Neglect the sizes of the stone and the person compared to the size of the board. | 15 | 1/8 |
Angle bisectors $AA', BB'$ and $CC'$ are drawn in triangle $ABC$ with angle $\angle B= 120^o$ . Find $\angle A'B'C'$ . | 90 | 3/8 |
There are two rows of seats, with 4 seats in the front row and 5 seats in the back row. Now, we need to arrange seating for 2 people, and these 2 people cannot sit next to each other (sitting one in front and one behind is also considered as not adjacent). How many different seating arrangements are there? | 58 | 5/8 |
In a circle, a chord of length 10 cm is drawn. A tangent to the circle is drawn through one end of the chord, and a secant, parallel to the tangent, is drawn through the other end of the chord. The internal segment of the secant is equal to 12 cm. Find the radius of the circle. | \frac{25}{4} | 6/8 |
The distance from $A$ to $B$ is 999 km. Along the road, there are kilometer markers that indicate the distance to $A$ and to $B$ as follows:
$0|999, 1|998, \ldots, 999|0$.
How many of these markers have signs that contain only two different digits? | 40 | 5/8 |
On a 240 kilometre trip, Corey's father drove $\frac{1}{2}$ of the distance. His mother drove $\frac{3}{8}$ of the total distance and Corey drove the remaining distance. How many kilometres did Corey drive?
(A) 80
(B) 40
(C) 210
(D) 30
(E) 55 | 30 | 1/8 |
In a castle, there are 9 identical square rooms forming a $3 \times 3$ grid. Each of these rooms is occupied by one of 9 people: either liars or knights (liars always lie, knights always tell the truth). Each of these 9 people said: "At least one of my neighboring rooms has a liar." Rooms are considered neighbors if they share a wall. What is the maximum number of knights that could be among these 9 people? | 6 | 1/8 |
A city uses a lottery system for assigning car permits, with 300,000 people participating in the lottery and 30,000 permits available each month.
1. If those who win the lottery each month exit the lottery, and those who do not win continue in the following month's lottery, with an additional 30,000 new participants added each month, how long on average does it take for each person to win a permit?
2. Under the conditions of part (1), if the lottery authority can control the proportion of winners such that in the first month of each quarter the probability of winning is $\frac{1}{11}$, in the second month $\frac{1}{10}$, and in the third month $\frac{1}{9}$, how long on average does it take for each person to win a permit? | 10 | 3/8 |
The function \( f(x) \) defined on the set of real numbers \( \mathbf{R} \) satisfies \( f(x+1) = \frac{1+f(x+3)}{1-f(x+3)} \). Determine the value of \( f(1) \cdot f(2) \cdots f(2008) + 2009 \). | 2010 | 5/8 |
Assume the number of passengers traveling from location A to location B per day, $X$, follows a normal distribution $N(800, 50^2)$. Let $p_0$ denote the probability that the number of passengers traveling from A to B in a day does not exceed 900.
(1) Find the value of $p_0$. (Reference data: If $X \sim N(\mu, \sigma^2)$, then $P(\mu - \sigma < X \leq \mu + \sigma) = 0.6826$, $P(\mu - 2\sigma < X \leq \mu + 2\sigma) = 0.9544$, $P(\mu - 3\sigma < X \leq \mu + 3\sigma) = 0.9974$)
(2) A passenger transport company uses two models of vehicles, A and B, for long-distance passenger transport services between locations A and B, with each vehicle making one round trip per day. The passenger capacities of models A and B are 36 and 60, respectively, and the operating costs from A to B are 1,600 yuan per vehicle for model A and 2,400 yuan per vehicle for model B. The company plans to form a passenger transport fleet of no more than 21 vehicles, with the number of model B vehicles not exceeding the number of model A vehicles by more than 7. If the company needs to transport all passengers from A to B each day with a probability of at least $p_0$ and aims to minimize the operating cost from A to B, how many vehicles of models A and B should be equipped? | 12 | 7/8 |
Consider all possible broken lines that travel along the sides of the cells and connect two opposite corners of a square sheet of grid paper with dimensions $100 \times 100$ by the shortest path. What is the minimum number of such broken lines that need to be taken so that their union contains all the vertices of the cells? | 101 | 4/8 |
Given real numbers \( x \) and \( y \) satisfy
\[
\left\{
\begin{array}{l}
x - y \leq 0, \\
x + y - 5 \geq 0, \\
y - 3 \leq 0
\end{array}
\right.
\]
If the inequality \( a(x^2 + y^2) \leq (x + y)^2 \) always holds, then the maximum value of the real number \( a \) is $\qquad$. | 25/13 | 6/8 |
In the base of the pyramid \( S A B C D \), there is a trapezoid \( A B C D \) with bases \( B C \) and \( A D \), where \( B C = 2 A D \). Points \( K \) and \( L \) are taken on the edges \( S A \) and \( S B \) such that \( 2 S K = K A \) and \( 3 S L = L B \). In what ratio does the plane \( K L C \) divide the edge \( S D \)? | 2:1 | 5/8 |
Let \( a_{1}, a_{2}, \cdots, a_{2018} \) be the roots of the polynomial
\[ x^{2018}+x^{2017}+\cdots+x^{2}+x-1345=0. \]
Calculate \(\sum_{n=1}^{2018} \frac{1}{1-a_{n}}\). | 3027 | 7/8 |
What is the correct order of the fractions $\frac{15}{11}, \frac{19}{15},$ and $\frac{17}{13},$ from least to greatest? | \frac{19}{15}<\frac{17}{13}<\frac{15}{11} | 4/8 |
Barbara, Edward, Abhinav, and Alex took turns writing this test. Working alone, they could finish it in $10$ , $9$ , $11$ , and $12$ days, respectively. If only one person works on the test per day, and nobody works on it unless everyone else has spent at least as many days working on it, how many days (an integer) did it take to write this test? | 12 | 1/8 |
Given a pair of standard $8$-sided dice is rolled once. The sum of the numbers rolled, if it is a prime number, determines the diameter of a circle. Find the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference. | \frac{3}{64} | 6/8 |
Two heaters are alternately connected to the same direct current source. Water in a pot boiled in $t_{1}=120$ seconds with the first heater. With the second heater, the same water taken at the same initial temperature boiled in $t_{2}=180$ seconds. How much time would it take for this water to boil if the heaters are connected in parallel to each other? Ignore heat dissipation to the surrounding space. | 72\, | 1/8 |
For a real number \( x \), let \( \lfloor x \rfloor \) denote the greatest integer not exceeding \( x \). Consider the function
\[ f(x, y) = \sqrt{M(M+1)}(|x-m| + |y-m|), \]
where \( M = \max(\lfloor x \rfloor, \lfloor y \rfloor) \) and \( m = \min(\lfloor x \rfloor, \lfloor y \rfloor) \). The set of all real numbers \( (x, y) \) such that \( 2 \leq x, y \leq 2022 \) and \( f(x, y) \leq 2 \) can be expressed as a finite union of disjoint regions in the plane. The sum of the areas of these regions can be expressed as a fraction \( \frac{a}{b} \) in lowest terms. What is the value of \( a + b \)? | 2021 | 1/8 |
Points \( M \) and \( N \) are located on the lateral sides \( AB \) and \( CD \) of trapezoid \( ABCD \), respectively. Line \( MN \) is parallel to \( AD \), and segment \( MN \) is divided by the diagonals of the trapezoid into three equal parts. Find the length of segment \( MN \), if \( AD = a \), \( BC = b \), and the intersection point of the diagonals of the trapezoid lies inside quadrilateral \( MBCN \). | \frac{3ab}{2b} | 4/8 |
Let quadrilateral \( A B C D \) be inscribed in a circle. The rays \( A B \) and \( D C \) intersect at point \( K \). It turns out that points \( B, D \), and the midpoints \( M \) and \( N \) of segments \( A C \) and \( K C \) lie on the same circle. What values can the angle \( A D C \) take? | 90 | 1/8 |
The shortest distances between an interior [diagonal](https://artofproblemsolving.com/wiki/index.php/Diagonal) of a rectangular [parallelepiped](https://artofproblemsolving.com/wiki/index.php/Parallelepiped), $P$, and the edges it does not meet are $2\sqrt{5}$, $\frac{30}{\sqrt{13}}$, and $\frac{15}{\sqrt{10}}$. Determine the [volume](https://artofproblemsolving.com/wiki/index.php/Volume) of $P$. | 750 | 4/8 |
Given that \(A, B, C,\) and \(D\) are points on a circle with radius 1, \(\overrightarrow{AB} + 2 \overrightarrow{AC} = \overrightarrow{AD}\), and \(|AC| = 1\). Find the area of the quadrilateral \(ABDC\). | \frac{3 \sqrt{3}}{4} | 1/8 |
A segment \( AB \) of unit length, which is a chord of a sphere with radius 1, is positioned at an angle of \( \pi / 3 \) to the diameter \( CD \) of this sphere. The distance from the end \( C \) of the diameter to the nearest end \( A \) of the chord \( AB \) is \( \sqrt{2} \). Determine the length of segment \( BD \). | \sqrt{3} | 3/8 |
In the $4 \times 5$ grid shown, six of the $1 \times 1$ squares are not intersected by either diagonal. When the two diagonals of an $8 \times 10$ grid are drawn, how many of the $1 \times 1$ squares are not intersected by either diagonal? | 48 | 1/8 |
Write $-\left(-3\right)-4+\left(-5\right)$ in the form of algebraic sum without parentheses. | 3-4-5 | 6/8 |
Let $C=\bigcup_{N=1}^{\infty}C_N,$ where $C_N$ denotes the set of 'cosine polynomials' of the form \[f(x)=1+\sum_{n=1}^Na_n\cos(2\pi nx)\] for which:
(i) $f(x)\ge 0$ for all real $x,$ and
(ii) $a_n=0$ whenever $n$ is a multiple of $3.$ Determine the maximum value of $f(0)$ as $f$ ranges through $C,$ and prove that this maximum is attained. | 3 | 1/8 |
Within a triangular piece of paper, there are 100 points, along with the 3 vertices of the triangle, making it a total of 103 points, and no three of these points are collinear. If these points are used as vertices to create triangles, and the paper is cut into small triangles, then the number of such small triangles is ____. | 201 | 7/8 |
Prove that the sum of all fractions \( \frac{1}{rs} \), where \( r \) and \( s \) are relatively prime integers satisfying \( 0 < r < s \leq n \) and \( r + s > n \), is \( \frac{1}{2} \). | \frac{1}{2} | 1/8 |
Let \( A \) be a positive integer, and let \( B \) denote the number obtained by writing the digits of \( A \) in reverse order. Show that at least one of the numbers \( A+B \) or \( A-B \) is divisible by 11. | 1 | 1/8 |
Given the function \( f(x) = \frac{1}{\sqrt[3]{1 - x^3}} \). Find \( f(...f(f(19))...) \) applied 95 times. | \sqrt[3]{1-\frac{1}{19^3}} | 3/8 |
The diagonals of a trapezoid are 3 and 5, and the segment connecting the midpoints of the bases is 2. Find the area of the trapezoid. | 6 | 3/8 |
Let $r$ be the number that results when both the base and the exponent of $a^b$ are tripled, where $a,b>0$. If $r$ equals the product of $a^b$ and $x^b$ where $x>0$, then $x=$
$\text{(A) } 3\quad \text{(B) } 3a^2\quad \text{(C) } 27a^2\quad \text{(D) } 2a^{3b}\quad \text{(E) } 3a^{2b}$ | (C)27a^2 | 1/8 |
Given five distinct positive numbers, in how many ways can we split them into two groups such that the sums of the numbers in each group are equal? | 1 | 1/8 |
Let the sequence $\left\{x_{n}\right\}$ be defined such that $x_{1}=5$, and
$$
x_{n+1}=x_{n}^{2}-2 \quad \text{for} \quad n=1,2, \ldots
$$
Find:
$$
\lim _{n \rightarrow \infty} \frac{x_{n+1}}{x_{1} x_{2} \cdots x_{n}}
$$ | \sqrt{21} | 7/8 |
In a mathematics competition consisting of three problems, A, B, and C, among the 39 participants, each person solved at least one problem. Among those who solved problem A, there are 5 more people who only solved A than those who solved A and any other problems. Among those who did not solve problem A, the number of people who solved problem B is twice the number of people who solved problem C. Additionally, the number of people who only solved problem A is equal to the combined number of people who only solved problem B and those who only solved problem C. What is the maximum number of people who solved problem A? | 23 | 7/8 |
Compute the sum of all integers \( n \) such that \( n^2 - 3000 \) is a perfect square. | 0 | 7/8 |
Let \( P_{1} \) and \( P_{2} \) be any two different points on the ellipse \(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1\), and let \( P \) be a variable point on the circle with diameter \( P_{1} P_{2} \). Find the maximum area of the circle with radius \( OP \). | 13 \pi | 1/8 |
Determine the volume of a regular quadrangular prism if its diagonal forms an angle of $30^{\circ}$ with the plane of one of its lateral faces, and the side of its base is equal to $a$. | ^3\sqrt{2} | 3/8 |
Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[f(x(x + f(y))) = (x + y)f(x),\]
for all $x, y \in\mathbb{R}$. | f(x) = 0 \text{ and } f(x) = x. | 1/8 |
Let $u$ be a positive rational number and $m$ be a positive integer. Define a sequence $q_1,q_2,q_3,\dotsc$ such that $q_1=u$ and for $n\geqslant 2$ : $$ \text{if }q_{n-1}=\frac{a}{b}\text{ for some relatively prime positive integers }a\text{ and }b, \text{ then }q_n=\frac{a+mb}{b+1}. $$
Determine all positive integers $m$ such that the sequence $q_1,q_2,q_3,\dotsc$ is eventually periodic for any positive rational number $u$ .
*Remark:* A sequence $x_1,x_2,x_3,\dotsc $ is *eventually periodic* if there are positive integers $c$ and $t$ such that $x_n=x_{n+t}$ for all $n\geqslant c$ .
*Proposed by Petar Nizié-Nikolac* | m | 1/8 |
Given a four-digit number that satisfies the following conditions: (1) If the units digit and the hundreds digit are swapped, and the tens digit and the thousands digit are swapped, the number increases by 5940; (2) When divided by 9, the remainder is 8. Find the smallest odd number that meets these conditions. | 1979 | 2/8 |
Evaluate $|5 - e|$ where $e$ is the base of the natural logarithm. | 2.28172 | 1/8 |
Define $f(x) = \frac{3}{27^x + 3}.$ Calculate the sum
\[ f\left(\frac{1}{2001}\right) + f\left(\frac{2}{2001}\right) + f\left(\frac{3}{2001}\right) + \dots + f\left(\frac{2000}{2001}\right). \] | 1000 | 1/8 |
Calculate the lengths of the arcs of the curves given by the equations in the rectangular coordinate system.
\[ y = \ln \frac{5}{2 x}, \quad \sqrt{3} \leq x \leq \sqrt{8} \] | 1 + \frac{1}{2} \ln \frac{3}{2} | 7/8 |
The number of integers between 1 and 100 that are not divisible by 2, 3, or 5 is \( (\ \ \ ). | 26 | 7/8 |
The midsegment of a trapezoid divides it into two quadrilaterals. The difference in the perimeters of these two quadrilaterals is 24, and the ratio of their areas is $\frac{20}{17}$. Given that the height of the trapezoid is 2, what is the area of this trapezoid? | 148 | 7/8 |
Two circles $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ intersect at two distinct points $A$ and $B$. Let $I$ be the midpoint of the segment $[A B]$.
a) Is it possible to draw distinct lines concurrent at $I$ such that these lines and the circles $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ form exactly 2017 points of intersection? What about 2018?
b) What if $I$ is instead located anywhere within the area bounded by the intersection of $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ (but not on $[A B]$)? | 2018 | 5/8 |
Let \( x \) and \( y \) be two non-zero numbers such that \( x^{2} + xy + y^{2} = 0 \) ( \( x \) and \( y \) are complex numbers, but that is not very important). Find the value of
$$
\left(\frac{x}{x+y}\right)^{2013}+\left(\frac{y}{x+y}\right)^{2013}
$$ | -2 | 7/8 |
Let $a_1$ , $a_2$ , $\ldots$ be an infinite sequence of (positive) integers such that $k$ divides $\gcd(a_{k-1},a_k)$ for all $k\geq 2$ . Compute the smallest possible value of $a_1+a_2+\cdots+a_{10}$ . | 440 | 6/8 |
Let \( D = \{ 1, 2, \ldots, 10 \} \). The function \( f(x) \) is a bijection from \( D \) to \( D \). Define \( f_{n+1}(x) = f(f_n(x)) \) for \( n \in \mathbf{N}_+ \) and \( f_1(x) = f(x) \). Find a permutation \( x_1, x_2, \ldots, x_{10} \) of \( D \) such that \( \sum_{i=1}^{10} x_i f_{2520}(i) = 220 \). | 10,9,8,7,6,5,4,3,2,1 | 3/8 |
Given the function \(f(x) = \frac{x^{2} + a x + b}{x^{2} + c x + d}\), where the polynomials \(x^{2} + a x + b\) and \(x^{2} + c x + d\) have no common roots, prove that the following two statements are equivalent:
1. There exists an interval of numbers free from the values of the function.
2. \(f(x)\) can be represented in the form: \(f(x) = f_{1}\left(f_{2}\left(\ldots f_{n-1}\left(f_{n}(x)\right) \ldots\right)\right)\), where each function \(f_{i}(x)\) is of one of the following types: \(k_{i} x + b\), \(x^{-1}\), \(x^{2}\). | 2 | 2/8 |
Determine all real numbers $a$ such that the inequality $|x^{2}+2 a x+3 a| \leq 2$ has exactly one solution in $x$. | 1,2 | 1/8 |
From point \( A \), a secant and a tangent are drawn to a circle with radius \( R \). Let \( B \) be the point of tangency, and \( D \) and \( C \) be the points where the secant intersects the circle, with point \( D \) lying between \( A \) and \( C \). It is known that \( BD \) is the angle bisector of angle \( B \) in triangle \( ABC \) and its length is equal to \( R \). Find the distance from point \( A \) to the center of the circle. | \frac{R\sqrt{7}}{2} | 1/8 |
Let $A(n)$ denote the largest odd divisor of the number $n$. For example, $A(21)=21$, $A(72)=9, A(64)=1$. Find the sum $A(111)+A(112)+\ldots+A(218)+A(219)$. | 12045 | 6/8 |
A taxi has a starting fare of 10 yuan. After exceeding 10 kilometers, for every additional kilometer, the fare increases by 1.50 yuan (if the increase is less than 1 kilometer, it is rounded up to 1 kilometer; if the increase is more than 1 kilometer but less than 2 kilometers, it is rounded up to 2 kilometers, etc.). Now, traveling from A to B costs 28 yuan. If one walks 600 meters from A before taking a taxi to B, the fare is still 28 yuan. If one takes a taxi from A, passes B, and goes to C, with the distance from A to B equal to the distance from B to C, how much is the taxi fare? | 61 | 7/8 |
From $A$ to $B$ it is 999 km. Along the road, there are kilometer markers with distances written to $A$ and to $B$:
$0|999,1|998, \ldots, 999|0$.
How many of these markers have only two different digits? | 40 | 2/8 |
Let a moving line $l$ intersect a fixed ellipse $\Gamma: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ at two points $A$ and $B$. If $OA \perp OB$, prove that the line $l$ is always tangent to a fixed circle $L$, and find the equation of the circle $L$. | x^2+y^2=\frac{^2b^2}{^2+b^2} | 6/8 |
The circle is tangent to the sides \(AC\) and \(BC\) of triangle \(ABC\) at points \(A\) and \(B\) respectively. On the arc of this circle, lying inside the triangle, there is a point \(K\) such that the distances from it to the sides \(AC\) and \(BC\) are 6 and 24 respectively. Find the distance from point \(K\) to the side \(AB\). | 12 | 5/8 |
Consider the polynomial determined by the identity
$$
a_{0}+a_{1} x+a_{2} x^{2}+\cdots+a_{2 n} x^{2 n} \equiv\left(x+2 x^{2}+\cdots+n x^{n}\right)^{2}.
$$
Prove that $\sum_{k=n+1}^{2 n} \cdot a_{k}=\frac{1}{24} n(n+1)\left(5 n^{2}+5 n+2\right)$. | \frac{1}{24}n(n+1)(5n^2+5n+2) | 4/8 |
If the graph of the function $f(x)=(x^{2}-4)(x^{2}+ax+b)$ is symmetric about the line $x=-1$, find the values of $a$ and $b$, and the minimum value of $f(x)$. | -16 | 7/8 |
Mena listed the numbers from 1 to 30 one by one. Emily copied these numbers and substituted every digit 2 with digit 1. Both calculated the sum of the numbers they wrote. By how much is the sum that Mena calculated greater than the sum that Emily calculated? | 103 | 4/8 |
The diagram shows a square and a regular decagon that share an edge. One side of the square is extended to meet an extended edge of the decagon. What is the value of \( x \)? | 18 | 1/8 |
The function \( y = f(t) \) is such that the sum of the roots of the equation \( f(\sin x) = 0 \) in the interval \([3 \pi / 2, 2 \pi]\) is \( 33 \pi \), and the sum of the roots of the equation \( f(\cos x) = 0 \) in the interval \([\pi, 3 \pi / 2]\) is \( 23 \pi \). What is the sum of the roots of the second equation in the interval \([\pi / 2, \pi]?\) | 17\pi | 3/8 |
Given that $a, b, c, d, e, f, p, q$ are Arabic numerals and $b > c > d > a$, the difference between the four-digit numbers $\overline{c d a b}$ and $\overline{a b c d}$ is a four-digit number of the form $\overline{p q e f}$. If $\overline{e f}$ is a perfect square and $\overline{p q}$ is not divisible by 5, determine the four-digit number $\overline{a b c d}$ and explain the reasoning. | 1983 | 1/8 |
Let $T$ be the set of points $(x, y)$ in the Cartesian plane that satisfy
\[\big|\big| |x|-3\big|-1\big|+\big|\big| |y|-3\big|-1\big|=2.\]
What is the total length of all the lines that make up $T$? | 32\sqrt{2} | 1/8 |
Regular polygons $I C A O, V E N T I$, and $A L B E D O$ lie on a plane. Given that $I N=1$, compute the number of possible values of $O N$. | 2 | 1/8 |
Given the vertex of the parabola \(C_1\) is \((\sqrt{2}-1,1)\), and the focus is \(\left(\sqrt{2}-\frac{3}{4}, 1\right)\). The equation of another parabola \(C_2\) is \(y^2 - a y + x + 2 b = 0\). It is known that at one intersection point of \(C_1\) and \(C_2\), their tangents are perpendicular. Prove that \(C_2\) must pass through a fixed point, and find the coordinates of that point. | (\sqrt{2}-\frac{1}{2},1) | 7/8 |
Calculate the limit of the numerical sequence:
$$\lim _{n \rightarrow \infty} \frac{\sqrt{n^{5}-8}-n \sqrt{n\left(n^{2}+5\right)}}{\sqrt{n}}$$ | -\frac{5}{2} | 7/8 |
The students of class 4(1) went to the park to plant trees. Along a 120-meter long road, they initially dug a pit every 3 meters on both sides of the road. Later, because the spacing was too small, they changed to digging a pit every 5 meters. How many pits can be retained at most? | 18 | 5/8 |
The sum \( b_{6} + b_{7} + \ldots + b_{2018} \) of the terms of the geometric progression \( \left\{b_{n}\right\} \) with \( b_{n}>0 \) is equal to 6. The sum of the same terms taken with alternating signs \( b_{6} - b_{7} + b_{8} - \ldots - b_{2017} + b_{2018} \) is equal to 3. Find the sum of the squares of these terms \( b_{6}^{2} + b_{7}^{2} + \ldots + b_{2018}^{2} \). | 18 | 3/8 |
Find the maximum number $N$ for which there exist $N$ consecutive natural numbers such that the sum of the digits of the first number is divisible by 1, the sum of the digits of the second number is divisible by 2, the sum of the digits of the third number is divisible by 3, and so on, up to the sum of the digits of the $N$-th number being divisible by $N$. | 21 | 1/8 |
Two chords, measuring 9 and 17, are drawn from a single point on a circle. Find the radius of the circle if the distance between the midpoints of these chords is 5. | \frac{85}{8} | 2/8 |
Suppose $a$ and $b$ be positive integers not exceeding 100 such that $$a b=\left(\frac{\operatorname{lcm}(a, b)}{\operatorname{gcd}(a, b)}\right)^{2}$$ Compute the largest possible value of $a+b$. | 78 | 2/8 |
Given that point $P$ is the circumcenter of $\triangle ABC$, and $\overrightarrow{PA} + \overrightarrow{PB} + \lambda \overrightarrow{PC} = 0$, $\angle C = 120^{\circ}$, determine the value of the real number $\lambda$. | -1 | 3/8 |
A chessboard’s squares are labeled with numbers as follows:
[asy]
unitsize(0.8 cm);
int i, j;
for (i = 0; i <= 8; ++i) {
draw((i,0)--(i,8));
draw((0,i)--(8,i));
}
for (i = 0; i <= 7; ++i) {
for (j = 0; j <= 7; ++j) {
label("$\frac{1}{" + string(9 - i + j) + "}$", (i + 0.5, j + 0.5));
}}
[/asy]
Eight of the squares are chosen such that each row and each column has exactly one selected square. Find the maximum sum of the labels of these eight chosen squares. | \frac{8}{9} | 1/8 |
A five-digit number is called irreducible if it cannot be expressed as a product of two three-digit numbers.
What is the greatest number of consecutive irreducible five-digit numbers? | 99 | 1/8 |
On a circular keychain, I need to arrange six keys. Two specific keys, my house key and my car key, must always be together. How many different ways can I arrange these keys considering that arrangements can be rotated or reflected? | 48 | 1/8 |
Let $N$ be a three-digit integer such that the difference between any two positive integer factors of $N$ is divisible by 3 . Let $d(N)$ denote the number of positive integers which divide $N$. Find the maximum possible value of $N \cdot d(N)$. | 5586 | 1/8 |
The slope angle of the line $x - y + 3 = 0$ is $\tan^{-1}(1)$. | \frac{\pi}{4} | 3/8 |
Let $P$ be an interior point of a triangle of area $T$ . Through the point $P$ , draw lines parallel to the three sides, partitioning the triangle into three triangles and three parallelograms. Let $a$ , $b$ and $c$ be the areas of the three triangles. Prove that $ \sqrt { T } = \sqrt { a } + \sqrt { b } + \sqrt { c } $ . | \sqrt{T}=\sqrt{}+\sqrt{b}+\sqrt{} | 2/8 |
Given an integer \( n \geq 2 \), for any pairwise coprime positive integers \( a_1, a_2, \ldots, a_n \), let \( A = a_1 + a_2 + \ldots + a_n \). Denote by \( d_i \) the greatest common divisor (gcd) of \( A \) and \( a_i \) for \( i = 1, 2, \ldots, n \). Denote by \( D_i \) the gcd of the remaining \( n-1 \) numbers after removing \( a_i \). Find the minimum value of \( \prod_{i=1}^{n} \frac{A - a_i}{d_i D_i} \). | (n-1)^n | 2/8 |
In $\triangle ABC$, point $E$ is on $AB$, point $F$ is on $AC$, and $BF$ intersects $CE$ at point $P$. If the areas of quadrilateral $AEPF$ and triangles $BEP$ and $CFP$ are all equal to 4, what is the area of $\triangle BPC$? | 12 | 2/8 |
Given \( a \in \mathbf{R} \) and the function \( f(x) = a x^{2} - 2 x - 2a \). If the solution set of \( f(x) > 0 \) is \( A \) and \( B = \{ x \mid 1 < x < 3 \} \), with \( A \cap B \neq \varnothing \), find the range of the real number \( a \). | (-\infty,-2)\cup(\frac{6}{7},+\infty) | 1/8 |
Use the bisection method to find an approximate zero of the function $f(x) = \log x + x - 3$, given that approximate solutions (accurate to 0.1) are $\log 2.5 \approx 0.398$, $\log 2.75 \approx 0.439$, and $\log 2.5625 \approx 0.409$. | 2.6 | 3/8 |
Let \( A, B, C \) be distinct points on the circle \( O \) (where \( O \) is the origin of coordinates), and \(\angle AOB = \frac{2\pi}{3}\). If \(\overrightarrow{OC} = \lambda \overrightarrow{OA} + \mu \overrightarrow{OB}\) (\(\lambda, \mu \in \mathbf{R}\)), find \(\frac{\lambda}{\mu}\) when \(\omega = \sqrt{3} \lambda + \lambda + \mu\) reaches its maximum value. | \frac{\sqrt{3}+1}{2} | 3/8 |
With inspiration drawn from the rectilinear network of streets in New York, the Manhattan distance between two points \((a, b)\) and \((c, d)\) in the plane is defined to be
\[
|a-c| + |b-d| \text{.}
\]
Suppose only two distinct Manhattan distances occur between all pairs of distinct points of some point set. What is the maximal number of points in such a set? | 9 | 1/8 |
In the expansion of the binomial ${(\sqrt{x}-\frac{1}{{2x}}})^n$, only the coefficient of the 4th term is the largest. The constant term in the expansion is ______. | \frac{15}{4} | 4/8 |
The side \( AB \) of the trapezoid \( ABCD \) is divided into five equal parts, and through the third division point, counting from point \( B \), a line is drawn parallel to the bases \( BC \) and \( AD \). Find the segment of this line that is enclosed between the sides of the trapezoid, given \( BC = a \) and \( AD = b \). | \frac{2a+3b}{5} | 5/8 |
Suppose $F$ is a polygon with lattice vertices and sides parralell to x-axis and y-axis.Suppose $S(F),P(F)$ are area and perimeter of $F$ .
Find the smallest k that:
$S(F) \leq k.P(F)^2$ | 1/16 | 6/8 |
Given that $O$ is the center of the circumcircle of $\triangle ABC$, $D$ is the midpoint of side $BC$, and $BC=4$, and $\overrightarrow{AO} \cdot \overrightarrow{AD} = 6$, find the maximum value of the area of $\triangle ABC$. | 4\sqrt{2} | 7/8 |
A tetrahedron has all its faces as triangles with sides 13, 14, and 15. What is its volume? | 42\sqrt{55} | 1/8 |
Inside square \(ABCD\), point \(M\) is positioned such that \(\angle DCM = \angle MAC = 25^\circ\). What is the measure of angle \(ABM\)? | 40 | 2/8 |
Define \( f_0(x) = e^x \), \( f_{n+1}(x) = x f_n '(x) \). Show that \( \sum_{n=0}^{\infty} \frac{f_n(1)}{n!} = e^e \). | e^e | 4/8 |
The lengths of two parallel sides of a rectangle are 1 cm. Additionally, it is known that the rectangle can be divided by two perpendicular lines into four smaller rectangles, three of which have an area of at least \(1 \, \mathrm{cm}^2\), and the fourth has an area of at least \(2 \, \mathrm{cm}^2\). What is the minimum possible length of the other two sides of the rectangle? | 3+2\sqrt{2} | 1/8 |
In a mathematics competition, the sum of the scores of Bill and Dick equalled the sum of the scores of Ann and Carol.
If the scores of Bill and Carol had been interchanged, then the sum of the scores of Ann and Carol would have exceeded
the sum of the scores of the other two. Also, Dick's score exceeded the sum of the scores of Bill and Carol.
Determine the order in which the four contestants finished, from highest to lowest. Assume all scores were nonnegative.
$\textbf{(A)}\ \text{Dick, Ann, Carol, Bill} \qquad \textbf{(B)}\ \text{Dick, Ann, Bill, Carol} \qquad \textbf{(C)}\ \text{Dick, Carol, Bill, Ann}\\ \qquad \textbf{(D)}\ \text{Ann, Dick, Carol, Bill}\qquad \textbf{(E)}\ \text{Ann, Dick, Bill, Carol}$ | \textbf{(E)}\ | 1/8 |
Let \([x]\) denote the greatest integer less than or equal to the real number \(x\). Define
\[ A = \left[\frac{19}{20}\right] + \left[\frac{19^2}{20}\right] + \cdots + \left[\frac{19^{2020}}{20}\right]. \]
Find the remainder when \(A\) is divided by 11. | 2 | 5/8 |
How many distinct sets of 8 positive odd integers sum to 20 ? | 11 | 5/8 |
Let \( R \) be the reals and \( R^* \) the non-negative reals. \( f: R^* \to R \) satisfies the following conditions:
1. It is differentiable and \( f'(x) = -3 f(x) + 6 f(2x) \) for \( x > 0 \).
2. \( |f(x)| \leq e^{-\sqrt{x}} \) for \( x \geq 0 \).
Define \( u_n = \int_0^\infty x^n f(x) \, dx \) for \( n \geq 0 \). Express \( u_n \) in terms of \( u_0 \), prove that the sequence \( \frac{u_n 3^n}{n!} \) converges, and show that the limit is 0 iff \( u_0 = 0 \). | 0 | 5/8 |
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