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Given that the sum of the first n terms of the sequence {a_n} is S_n, and a_{n+1}+a_n=2^n, find the value of S_{10}. | 682 | 7/8 |
Sandhya must save 35 files onto disks, each with 1.44 MB space. 5 of the files take up 0.6 MB, 18 of the files take up 0.5 MB, and the rest take up 0.3 MB. Files cannot be split across disks. Calculate the smallest number of disks needed to store all 35 files. | 12 | 1/8 |
The different ways to obtain the number of combinations of dice, as discussed in Example 4-15 of Section 4.6, can also be understood using the generating function form of Pólya’s enumeration theorem as follows:
$$
\begin{aligned}
P= & \frac{1}{24} \times\left[\left(x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}\right)^{6}\right. \\
& +6\left(x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}\right)^{2}\left(x_{1}^{4}+x_{2}^{4}+x_{3}^{4}+x_{4}^{4}+x_{5}^{4}+x_{6}^{4}\right) \\
& +3\left(x_{1}+x+x_{3}+x_{4}+x_{5}+x_{6}\right)^{2}\left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}\right)^{2} \\
& \left.+6\left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}\right)^{3}+8\left(x_{1}^{3}+x_{2}^{3}+x_{3}^{3}+x_{4}^{3}+x_{5}^{3}+x_{6}^{3}\right)^{2}\right],
\end{aligned}
$$
where $x_{i}$ represents the $i^{th}$ color for $i=1,2,\cdots,6$. | 30 | 1/8 |
Let \[P(x) = (3x^4 - 39x^3 + ax^2 + bx + c)(4x^4 - 96x^3 + dx^2 + ex + f),\] where $a, b, c, d, e, f$ are real numbers. Suppose that the set of all complex roots of $P(x)$ is $\{1, 2, 2, 3, 3, 4, 6\}.$ Find $P(7).$ | 86400 | 1/8 |
The lateral edge of a pyramid is divided into 100 equal parts, and through the points of division, planes parallel to the base are drawn. Find the ratio of the areas of the largest and smallest of the resulting cross-sections. | 9801 | 5/8 |
The diagonal of an isosceles trapezoid bisects its obtuse angle. The smaller base of the trapezoid is 3, and the perimeter is 42.
Find the area of the trapezoid. | 96 | 3/8 |
A parabola has focus $(3,3)$ and directrix $3x + 7y = 21.$ Express the equation of the parabola in the form
\[ax^2 + bxy + cy^2 + dx + ey + f = 0,\]where $a,$ $b,$ $c,$ $d,$ $e,$ $f$ are integers, $a$ is a positive integer, and $\gcd(|a|,|b|,|c|,|d|,|e|,|f|) = 1.$ | 49x^2 - 42xy + 9y^2 - 222x - 54y + 603 = 0 | 7/8 |
For given natural numbers \( k_{0}<k_{1}<k_{2} \), determine the minimum number of roots in the interval \([0, 2\pi)\) that the equation
$$
\sin \left(k_{0} x\right) + A_{1} \cdot \sin \left(k_{1} x\right) + A_{2} \cdot \sin \left(k_{2} x\right) = 0
$$
can have, where \( A_{1}, A_{2} \) are real numbers. | 2k_0 | 1/8 |
A bag of popping corn contains $\frac{2}{3}$ white kernels and $\frac{1}{3}$ yellow kernels. Only $\frac{1}{2}$ of the white kernels will pop, whereas $\frac{2}{3}$ of the yellow ones will pop. A kernel is selected at random from the bag, and pops when placed in the popper. What is the probability that the kernel selected was white?
$\textbf{(A)}\ \frac{1}{2} \qquad\textbf{(B)}\ \frac{5}{9} \qquad\textbf{(C)}\ \frac{4}{7} \qquad\textbf{(D)}\ \frac{3}{5} \qquad\textbf{(E)}\ \frac{2}{3}$ | \textbf{(D)}\\frac{3}{5} | 1/8 |
A storybook contains 30 stories, each with a length of $1, 2, \cdots, 30$ pages respectively. Every story starts on a new page starting from the first page of the book. What is the maximum number of stories that can start on an odd-numbered page? | 23 | 1/8 |
Given that point $P$ is a moving point on the parabola $y^{2}=2x$, find the minimum value of the sum of the distance from point $P$ to point $D(2, \frac{3}{2} \sqrt{3})$ and the distance from point $P$ to the $y$-axis. | \frac{5}{2} | 4/8 |
Given a cone with vertex \( P \), a base radius of 2, and a height of 1. On the base of the cone, a point \( Q \) is chosen such that the angle between the line \( PQ \) and the base is no greater than \( 45^{\circ} \). Find the area of the region that point \( Q \) satisfies. | 3\pi | 7/8 |
In the year 2009, there is a property that rearranging the digits of the number 2009 cannot yield a smaller four-digit number (numbers do not start with zero). In what subsequent year does this property first repeat again? | 2022 | 7/8 |
Find all pairs of positive integers \((a, b)\) such that \(a - b\) is a prime and \(ab\) is a perfect square. | ((\frac{p+1}{2})^2,(\frac{p-1}{2})^2) | 7/8 |
In triangle $XYZ$, $XY = 12$, $YZ = 16$, and $XZ = 20$, with $ZD$ as the angle bisector. Find the length of $ZD$. | \frac{16\sqrt{10}}{3} | 7/8 |
For some particular value of $N$, when $(a+b+c+d+1)^N$ is expanded and like terms are combined, the resulting expression contains exactly $1001$ terms that include all four variables $a, b,c,$ and $d$, each to some positive power. What is $N$?
$\textbf{(A) }9 \qquad \textbf{(B) } 14 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 17 \qquad \textbf{(E) } 19$ | \textbf{(B)}14 | 1/8 |
Misha thought of a number not less than 1 and not greater than 1000. Vasya is allowed to ask only questions to which Misha can answer "yes" or "no" (Misha always tells the truth). Can Vasya determine the number Misha thought of in 10 questions? | Yes | 4/8 |
Given the sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1,...,$ find $n$ such that the sum of the first $n$ terms is $2008$ or $2009$ . | 1026 | 4/8 |
Let set $\mathcal{C}$ be a 70-element subset of $\{1,2,3,\ldots,120\},$ and let $P$ be the sum of the elements of $\mathcal{C}.$ Find the number of possible values of $P.$ | 3501 | 7/8 |
Let $c$ be a real number randomly selected from the interval $[-20,20]$. Then, $p$ and $q$ are two relatively prime positive integers such that $\frac{p}{q}$ is the probability that the equation $x^4 + 36c^2 = (9c^2 - 15c)x^2$ has at least two distinct real solutions. Find the value of $p + q$. | 29 | 1/8 |
Given the sequence
$$
a_{n}=\frac{(n+3)^{2}+3}{n(n+1)(n+2)} \cdot \frac{1}{2^{n+1}}
$$
defined by its general term, form the sequence
$$
b_{n}=\sum_{k=1}^{n} a_{k}
$$
Determine the limit of the sequence $b_{n}$ as $n$ approaches $+\infty$. | 1 | 2/8 |
A company has five directors. The regulations of the company require that any majority (three or more) of the directors should be able to open its strongroom, but any minority (two or less) should not be able to do so. The strongroom is equipped with ten locks, so that it can only be opened when keys to all ten locks are available. Find all positive integers $n$ such that it is possible to give each of the directors a set of keys to $n$ different locks, according to the requirements and regulations of the company. | 6 | 5/8 |
Consider triangle \(ABC\) where \(BC = 7\), \(CA = 8\), and \(AB = 9\). \(D\) and \(E\) are the midpoints of \(BC\) and \(CA\), respectively, and \(AD\) and \(BE\) meet at \(G\). The reflection of \(G\) across \(D\) is \(G'\), and \(G'E\) meets \(CG\) at \(P\). Find the length \(PG\). | \frac{\sqrt{145}}{9} | 6/8 |
Find the times between $8$ and $9$ o'clock, correct to the nearest minute, when the hands of a clock will form an angle of $120^{\circ}$. | 8:22 | 3/8 |
Function $f(x, y): \mathbb N \times \mathbb N \to \mathbb Q$ satisfies the conditions:
(i) $f(1, 1) =1$ ,
(ii) $f(p + 1, q) + f(p, q + 1) = f(p, q)$ for all $p, q \in \mathbb N$ , and
(iii) $qf(p + 1, q) = pf(p, q + 1)$ for all $p, q \in \mathbb N$ .
Find $f(1990, 31).$ | \frac{30! \cdot 1989!}{2020!} | 1/8 |
Let \( x \in \mathbf{R} \). Find the minimum value of the algebraic expression \( (x+1)(x+2)(x+3)(x+4) + 2019 \). | 2018 | 7/8 |
In the diagram, $ABCD$ is a trapezoid with an area of $18.$ $CD$ is three times the length of $AB.$ What is the area of $\triangle ABD?$
[asy]
draw((0,0)--(1,4)--(9,4)--(18,0)--cycle);
draw((9,4)--(0,0));
label("$D$",(0,0),W);
label("$A$",(1,4),NW);
label("$B$",(9,4),NE);
label("$C$",(18,0),E);
[/asy] | 4.5 | 1/8 |
Consider the sequence \( 5, 55, 555, 5555, 55555, \ldots \). Are any of the numbers in this sequence divisible by 495; if so, what is the smallest such number? | 555555555555555555 | 7/8 |
How many ordered triples $(a,b,c)$ of integers with $1\le a\le b\le c\le 60$ satisfy $a\cdot b=c$ ? | 134 | 7/8 |
Let $P$ be a point inside the equilateral triangle $ABC$ such that $6\angle PBC = 3\angle PAC = 2\angle PCA$ . Find the measure of the angle $\angle PBC$ . | 15 | 6/8 |
In an election for the Peer Pressure High School student council president, there are 2019 voters and two candidates Alice and Celia (who are voters themselves). At the beginning, Alice and Celia both vote for themselves, and Alice's boyfriend Bob votes for Alice as well. Then one by one, each of the remaining 2016 voters votes for a candidate randomly, with probabilities proportional to the current number of the respective candidate's votes. For example, the first undecided voter David has a $\frac{2}{3}$ probability of voting for Alice and a $\frac{1}{3}$ probability of voting for Celia. What is the probability that Alice wins the election (by having more votes than Celia)? | \frac{1513}{2017} | 1/8 |
Let a positive integer \( k \) be called interesting if the product of the first \( k \) prime numbers is divisible by \( k \) (for example, the product of the first two prime numbers is \(2 \cdot 3 = 6\), and 2 is an interesting number).
What is the largest number of consecutive interesting numbers that can occur? | 3 | 5/8 |
Which are more: three-digit numbers where all digits have the same parity (all even or all odd), or three-digit numbers where adjacent digits have different parity? | 225 | 5/8 |
Can each of the means in a proportion be less than each of the extremes? | yes | 3/8 |
A malfunctioning thermometer shows a temperature of $+1^{\circ}$ in freezing water and $+105^{\circ}$ in the steam of boiling water. Currently, this thermometer shows $+17^{\circ}$; what is the true temperature? | 15.38 | 2/8 |
A Ferris wheel rotates at a constant speed, completing one revolution every 12 minutes. The lowest point of the Ferris wheel is 2 meters above the ground, and the highest point is 18 meters above the ground. Let P be a fixed point on the circumference of the Ferris wheel. Starting the timing when P is at the lowest point, the height of point P above the ground 16 minutes later is _______. | 14 | 7/8 |
Triangle $ABC$ is scalene. Points $P$ and $Q$ are on segment $BC$ with $P$ between $B$ and $Q$ such that $BP=21$ , $PQ=35$ , and $QC=100$ . If $AP$ and $AQ$ trisect $\angle A$ , then $\tfrac{AB}{AC}$ can be written uniquely as $\tfrac{p\sqrt q}r$ , where $p$ and $r$ are relatively prime positive integers and $q$ is a positive integer not divisible by the square of any prime. Determine $p+q+r$ . | 92 | 6/8 |
There are three piles with 40 stones each. Petya and Vasya take turns, starting with Petya. Each turn, a player must combine two piles, then divide the combined stones into four piles. The player who cannot make a move loses. Which player (Petya or Vasya) can guarantee a win, regardless of the opponent's moves? | Vasya | 3/8 |
Triangle $ABC$ has side lengths $AB=19, BC=20$, and $CA=21$. Points $X$ and $Y$ are selected on sides $AB$ and $AC$, respectively, such that $AY=XY$ and $XY$ is tangent to the incircle of $\triangle ABC$. If the length of segment $AX$ can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers, compute $100 a+b$. | 6710 | 4/8 |
In a row, there are $n$ integers such that the sum of any three consecutive numbers is positive, and the sum of any five consecutive numbers is negative. What is the maximum possible value of $n$? | 6 | 2/8 |
Given the hyperbola $\frac{x^{2}}{4} - \frac{y^{2}}{2} = 1$ with three non-collinear points $A$, $B$, $C$ on it. The midpoints of $AB$, $BC$, $AC$ are $D$, $E$, $F$ respectively. If the sum of the slopes of $OD$, $OE$, $OF$ is $-1$, find the value of $\frac{1}{k_{AB}} + \frac{1}{k_{BC}} + \frac{1}{k_{AC}}$. | -2 | 7/8 |
Show that for all \( a, b, c, d > 0 \),
\[
\sum_{\text{cyc}} \frac{a}{b + 2c + 3d} \geq \frac{2}{3}
\] | \frac{2}{3} | 7/8 |
Three three-digit numbers, with all digits except zero being used in their digits, sum up to 1665. In each number, the first digit was swapped with the last digit. What is the sum of the new numbers? | 1665 | 3/8 |
P1, P2, ..., Pn are distinct subsets of {1, 2, ..., n} with two elements each. Distinct subsets Pi and Pj have an element in common if and only if {i, j} is one of the Pk. Show that each member of {1, 2, ..., n} belongs to just two of the subsets. | 2 | 7/8 |
Given non-zero vectors \\(a\\) and \\(b\\) satisfying \\(|b|=2|a|\\) and \\(a \perp (\sqrt{3}a+b)\\), find the angle between \\(a\\) and \\(b\\). | \dfrac{5\pi}{6} | 3/8 |
In Perfectville, the streets are all $30$ feet wide and the blocks they enclose are all squares of side length $500$ feet. Jane runs around the block on the $500$-foot side of the street, while John runs on the opposite side of the street. How many more feet than Jane does John run for every lap around the block? | 240 | 7/8 |
Compute
$$
\int_{L} \frac{60 e^{z}}{z(z+3)(z+4)(z+5)} d z
$$
where \( L \) is a unit circle centered at the origin. | 2\pii | 7/8 |
Let \(ABC\) be a triangle with \(AB=13\), \(BC=14\), \(CA=15\). Let \(H\) be the orthocenter of \(ABC\). Find the radius of the circle with nonzero radius tangent to the circumcircles of \(AHB\), \(BHC\), and \(CHA\). | \frac{65}{4} | 1/8 |
Points A and B are 999 km apart, with 1000 milestones along the way. Each milestone indicates the distance to points A and B in the format (distance to A, distance to B), such as (0, 999), (1, 998), (2, 997), ..., (997, 2), (998, 1), (999, 0). How many of these milestones display exactly two different digits on the distances to A and B? | 40 | 1/8 |
In an acute triangle $\triangle ABC$, altitudes $\overline{AD}$ and $\overline{BE}$ intersect at point $H$. Given that $HD=6$ and $HE=3$, calculate $(BD)(DC)-(AE)(EC)$. | 27 | 1/8 |
M and N are real unequal \( n \times n \) matrices satisfying \( M^3 = N^3 \) and \( M^2N = N^2M \). Can we choose M and N so that \( M^2 + N^2 \) is invertible? | No | 1/8 |
A square with an area of one square unit is inscribed in an isosceles triangle such that one side of the square lies on the base of the triangle. Find the area of the triangle, given that the centers of mass of the triangle and the square coincide (the center of mass of the triangle lies at the intersection of its medians). | 9/4 | 7/8 |
Given that Alice's car averages 30 miles per gallon of gasoline, and Bob's car averages 20 miles per gallon of gasoline, and Alice drives 120 miles and Bob drives 180 miles, calculate the combined rate of miles per gallon of gasoline for both cars. | \frac{300}{13} | 4/8 |
Suppose that a sequence $(a_n)_{n=1}^{\infty}$ of integers has the following property: For all $n$ large enough (i.e. $n \ge N$ for some $N$ ), $a_n$ equals the number of indices $i$ , $1 \le i < n$ , such that $a_i + i \ge n$ . Find the maximum possible number of integers which occur infinitely many times in the sequence. | 2 | 2/8 |
In the plane, 100 points are given, no three of which are collinear. Consider all possible triangles formed using these points as vertices. Prove that at most 70% of these triangles are acute-angled triangles. | 70 | 1/8 |
Arrange the numbers $1, 2, \cdots, n^2$ in an $n$ by $n$ grid $T_{n}$ in a clockwise spiral order, starting with $1, 2, \cdots, n$ in the first row. For example, $T_{3}=\left[\begin{array}{lll}1 & 2 & 3 \\ 8 & 9 & 4 \\ 7 & 6 & 5\end{array}\right]$. If 2018 is located at the $(i, j)$ position in $T_{100}$, what is $(i, j)$? | (34,95) | 3/8 |
Into how many parts do the face planes divide the space of a) a cube; b) a tetrahedron? | 15 | 5/8 |
Point \( A \) lies on the line \( y = \frac{15}{8} x - 4 \), and point \( B \) on the parabola \( y = x^{2} \). What is the minimum length of segment \( AB \)? | 47/32 | 3/8 |
The energy stored by a pair of positive charges is inversely proportional to the distance between them, and directly proportional to their charges. Four identical point charges are initially placed at the corners of a square with each side length $d$. This configuration stores a total of $20$ Joules of energy. How much energy, in Joules, would be stored if two of these charges are moved such that they form a new square with each side doubled (i.e., side length $2d$)? | 10 | 7/8 |
Given that there are three mathematics teachers: Mrs. Germain with 13 students, Mr. Newton with 10 students, and Mrs. Young with 12 students, and 2 students are taking classes from both Mrs. Germain and Mr. Newton and 1 additional student is taking classes from both Mrs. Germain and Mrs. Young. Determine the number of distinct students participating in the competition from all three classes. | 32 | 7/8 |
Among the four statements on real numbers below, how many of them are correct?
1. If \( a < b \) and \( a, b \neq 0 \) then \( \frac{1}{b} < \frac{1}{a} \)
2. If \( a < b \) then \( ac < bc \)
3. If \( a < b \) then \( a + c < b + c \)
4. If \( a^2 < b^2 \) then \( a < b \)
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4 | 1 | 1/8 |
The graph of $x^2-4y^2=0$ is:
$\textbf{(A)}\ \text{a parabola} \qquad \textbf{(B)}\ \text{an ellipse} \qquad \textbf{(C)}\ \text{a pair of straight lines}\qquad \\ \textbf{(D)}\ \text{a point}\qquad \textbf{(E)}\ \text{None of these}$ | \textbf{(C)}\ | 1/8 |
Let $A,$ $B,$ $C$ be the angles of a triangle. Evaluate
\[\begin{vmatrix} \sin^2 A & \cot A & 1 \\ \sin^2 B & \cot B & 1 \\ \sin^2 C & \cot C & 1 \end{vmatrix}.\] | 0 | 6/8 |
Eight paper squares of size 2 by 2 were sequentially placed on a table to form a large 4 by 4 square. The last square placed on the table was square \(E\). The picture shows that square \(E\) is fully visible, while the rest of the squares are partially visible. Which square was placed on the table third in order? | G | 1/8 |
Given four points O, A, B, C on a plane satisfying OA=4, OB=3, OC=2, and $\overrightarrow{OB} \cdot \overrightarrow{OC} = 3$, find the maximum area of $\triangle ABC$. | 2\sqrt{7} + \frac{3\sqrt{3}}{2} | 1/8 |
For the set $\{1,2,\cdots,n\}$ and each of its non-empty subsets, define a unique "alternating sum" as follows: Arrange the numbers in each subset in descending order, then start from the largest number and alternately subtract and add subsequent numbers to obtain the alternating sum (for example, the alternating sum of the set $\{1, 3, 8\}$ is $8 - 3 + 1 = 6$). For $n=8$, find the total sum of the alternating sums of all subsets. | 1024 | 7/8 |
A right triangle has legs of lengths 3 and 4. Find the volume of the solid formed by revolving the triangle about its hypotenuse. | \frac{48\pi}{5} | 6/8 |
Shift the graph of the function $y=3\sin (2x+ \frac {\pi}{6})$ to the graph of the function $y=3\cos 2x$ and determine the horizontal shift units. | \frac {\pi}{6} | 6/8 |
Two painters are painting a fence that surrounds garden plots. They come every other day and paint one plot (there are 100 plots) in either red or green. The first painter is colorblind and mixes up the colors; he remembers which plots he painted, but cannot distinguish the color painted by the second painter. The first painter aims to maximize the number of places where a green plot borders a red plot. What is the maximum number of such transitions he can achieve (regardless of how the second painter acts)?
Note: The garden plots are arranged in a single line. | 49 | 1/8 |
Find all prime numbers which can be presented as a sum of two primes and difference of two primes at the same time. | 5 | 7/8 |
Determine how many regions of space are divided by:
a) The six planes of a cube's faces.
b) The four planes of a tetrahedron's faces. | 15 | 5/8 |
The range of the function \( y = \arcsin[\sin x] + \arccos[\cos x] \) (for \( x \in [0, 2\pi) \) and where \([x]\) denotes the greatest integer less than or equal to \( x \)) is ____. | {0,\frac{\pi}{2},\pi} | 4/8 |
Let $N = 99999$. Then $N^3 = \ $ | 999970000299999 | 1/8 |
Find the smallest constant $M$, such that for any positive real numbers $x$, $y$, $z$, and $w$,
\[\sqrt{\frac{x}{y + z + w}} + \sqrt{\frac{y}{x + z + w}} + \sqrt{\frac{z}{x + y + w}} + \sqrt{\frac{w}{x + y + z}} < M.\] | \frac{4}{\sqrt{3}} | 1/8 |
If the final 5 contestants of "The Voice" season 4 must sign with one of the three companies A, B, and C, with each company signing at least 1 person and at most 2 people, calculate the total number of possible different signing schemes. | 90 | 7/8 |
As shown in the diagram, squares \( a \), \( b \), \( c \), \( d \), and \( e \) are used to form a rectangle that is 30 cm long and 22 cm wide. Find the area of square \( e \). | 36 | 1/8 |
Given a triangle \(ABC\), let \(D\), \(E\), and \(F\) be the midpoints of sides \(BC\), \(AC\), and \(AB\) respectively.
The two medians \(AD\) and \(BE\) are perpendicular to each other and have lengths \(\overline{AD}=18\) and \(\overline{BE}=13.5\).
Calculate the length of the third median \(CF\) of this triangle. | 22.5 | 3/8 |
What is the smallest positive integer $a$ such that $a^{-1}$ is undefined $\pmod{55}$ and $a^{-1}$ is also undefined $\pmod{66}$? | 10 | 7/8 |
If $\tan{\alpha}$ and $\tan{\beta}$ are the roots of $x^2 - px + q = 0$, and $\cot{\alpha}$ and $\cot{\beta}$
are the roots of $x^2 - rx + s = 0$, then $rs$ is necessarily
$\textbf{(A)} \ pq \qquad \textbf{(B)} \ \frac{1}{pq} \qquad \textbf{(C)} \ \frac{p}{q^2} \qquad \textbf{(D)}\ \frac{q}{p^2}\qquad \textbf{(E)}\ \frac{p}{q}$ | \textbf{(C)}\\frac{p}{q^2} | 1/8 |
Given \(a, b, c\) are distinct positive integers such that
$$
\{a+b, b+c, c+a\}=\{n^{2},(n+1)^{2},(n+2)^{2}\},
$$
where \(n\) is a positive integer. Find the minimum value of \(a^{2} + b^{2} + c^{2}\). | 1297 | 5/8 |
How many pairs of positive integers have greatest common divisor 5! and least common multiple 50! ? | 16384 | 4/8 |
Square the numbers \( a = 101 \) and \( b = 10101 \). Find the square root of the number \( c = 102030405060504030201 \). | 10101010101 | 6/8 |
Compute
\[
\begin{vmatrix} \cos 1 & \cos 2 & \cos 3 \\ \cos 4 & \cos 5 & \cos 6 \\ \cos 7 & \cos 8 & \cos 9 \end{vmatrix}
.\]All the angles are in radians. | 0 | 6/8 |
Given that 216 sprinters enter a 100-meter dash competition, and the track has 6 lanes, determine the minimum number of races needed to find the champion sprinter. | 43 | 7/8 |
A cube has six faces. Each face has some dots on it. The numbers of dots on the six faces are 2, 3, 4, 5, 6, and 7. Harry removes one of the dots at random, with each dot equally likely to be removed. When the cube is rolled, each face is equally likely to be the top face. What is the probability that the top face has an odd number of dots on it? | \frac{13}{27} | 7/8 |
**p1.** For what integers $n$ is $2^6 + 2^9 + 2^n$ the square of an integer?**p2.** Two integers are chosen at random (independently, with repetition allowed) from the set $\{1,2,3,...,N\}$ . Show that the probability that the sum of the two integers is even is not less than the probability that the sum is odd.**p3.** Let $X$ be a point in the second quadrant of the plane and let $Y$ be a point in the first quadrant. Locate the point $M$ on the $x$ -axis such that the angle $XM$ makes with the negative end of the $x$ -axis is twice the angle $YM$ makes with the positive end of the $x$ -axis.**p4.** Let $a,b$ be positive integers such that $a \ge b \sqrt3$ . Let $\alpha^n = (a + b\sqrt3)^n = a_n + b_n\sqrt3$ for $n = 1,2,3,...$ .
i. Prove that $\lim_{n \to + \infty} \frac{a_n}{b_n}$ exists.
ii. Evaluate this limit.**p5.** Suppose $m$ and $n$ are the hypotenuses are of Pythagorean triangles, i.e,, there are positive integers $a,b,c,d$ , so that $m^2 = a^2 + b^2$ and $n^2= c^2 + d^2$ . Show than $mn$ is the hypotenuse of at least two distinct Pythagorean triangles.
Hint: you may not assume that the pair $(a,b)$ is different from the pair $(c,d)$ .
PS. You should use hide for answers. Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309). | 10 | 5/8 |
The last three digits of \( 1978^n \) and \( 1978^m \) are the same. Find the positive integers \( m \) and \( n \) such that \( m+n \) is minimized (here \( n > m \geq 1 \)). | 106 | 4/8 |
A random variable \(X\) is given by the probability density function \(f(x) = \frac{1}{2} \sin x\) within the interval \((0, \pi)\); outside this interval, \(f(x) = 0\). Find the variance of the function \(Y = \varphi(X) = X^2\) using the probability density function \(g(y)\). | \frac{\pi^4 - 16\pi^2 + 80}{4} | 4/8 |
A finite arithmetic progression \( a_1, a_2, \ldots, a_n \) with a positive common difference has a sum of \( S \), and \( a_1 > 0 \). It is known that if the common difference of the progression is increased by 3 times while keeping the first term unchanged, the sum \( S \) doubles. By how many times will \( S \) increase if the common difference of the initial progression is increased by 4 times (keeping the first term unchanged)? | 5/2 | 7/8 |
Find the area of the region enclosed by the graph of $|x-60|+|y|=\left|\frac{x}{4}\right|.$ | 480 | 7/8 |
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ that satisfy the conditions
\[f(1+xy)-f(x+y)=f(x)f(y) \quad \text{for all } x,y \in \mathbb{R},\]
and $f(-1) \neq 0$. | f(x) = x - 1 | 1/8 |
0 < \(a_1 \le a_2 \le a_3 \le \ldots\) is an unbounded sequence of integers. Let \(b_n = m\) if \(a_m\) is the first member of the sequence to equal or exceed \(n\). Given that \(a_{19} = 85\), what is the maximum possible value of \(a_1 + a_2 + \ldots + a_{19} + b_1 + b_2 + \ldots + b_{85}\)? | 1700 | 1/8 |
A permutation $(a_1, a_2, a_3, \dots, a_{2012})$ of $(1, 2, 3, \dots, 2012)$ is selected at random. If $S$ is the expected value of
\[
\sum_{i = 1}^{2012} | a_i - i |,
\]
then compute the sum of the prime factors of $S$ .
*Proposed by Aaron Lin* | 2083 | 5/8 |
Prove the inequality \((\sqrt{x}+\sqrt{y})^{8} \geq 64xy(x+y)^{2}\) for \(x, y \geq 0\). | (\sqrt{x}+\sqrt{y})^8\ge64xy(x+y)^2 | 6/8 |
In an isosceles right triangle \( \triangle ABC \), \( \angle A = 90^\circ \), \( AB = 1 \). \( D \) is the midpoint of \( BC \), \( E \) and \( F \) are two other points on \( BC \). \( M \) is the other intersection point of the circumcircles of \( \triangle ADE \) and \( \triangle ABF \); \( N \) is the other intersection point of line \( AF \) with the circumcircle of \( \triangle ACE \); \( P \) is the other intersection point of line \( AD \) with the circumcircle of \( \triangle AMN \). Find the length of \( AP \). | \sqrt{2} | 1/8 |
Given the sets \( A = \{2, 0, 1, 7\} \) and \( B = \{ x \mid x^2 - 2 \in A, \, x - 2 \notin A \} \), the product of all elements in set \( B \) is: | 36 | 7/8 |
The hypotenuse of a right triangle whose legs are consecutive even numbers is 50 units. What is the sum of the lengths of the two legs? | 70 | 1/8 |
Find all integers $k\ge 2$ such that for all integers $n\ge 2$ , $n$ does not divide the greatest odd divisor of $k^n+1$ . | 2^-1 | 1/8 |
Which of the following describes the graph of the equation $(x+y)^2=x^2+y^2$?
$\textbf{(A) } \text{the\,empty\,set}\qquad \textbf{(B) } \textrm{one\,point}\qquad \textbf{(C) } \textrm{two\,lines} \qquad \textbf{(D) } \textrm{a\,circle} \qquad \textbf{(E) } \textrm{the\,entire\,plane}$ | \textbf{(C)}\ | 1/8 |
Let \(\alpha_{n}\) be a real root of the cubic equation \(n x^{3}+2 x-n=0\), where \(n\) is a positive integer. If \(\beta_{n}=\left\lfloor(n+1) \alpha_{n}\right\rfloor\) for \(n=234 \cdots\), find the value of \(\frac{1}{1006} \sum_{k=2}^{2013} \beta_{k}\). | 2015 | 5/8 |
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