problem
stringlengths 10
7.44k
| answer
stringlengths 1
270
| difficulty
stringclasses 8
values |
|---|---|---|
Starting with two real numbers \( a \) and \( b \), we define \( x_{0} = a \), \( y_{0} = b \), and at each step:
\[
\begin{aligned}
x_{n+1} & = \frac{x_{n} + y_{n}}{2}, \\
y_{n+1} & = \frac{2 x_{n} y_{n}}{x_{n} + y_{n}}
\end{aligned}
\]
Show that the sequences \(\left(x_{n}\right)\) and \(\left(y_{n}\right)\) converge to a common limit \(\ell\) (i.e., they get arbitrarily close to \(\ell\) as \(n\) becomes large), and express \(\ell\) as a function of \(a\) and \(b\). | \sqrt{} | 7/8 |
Given that $F$ is the focus of the parabola $C_{1}$: $y^{2}=2ρx (ρ > 0)$, and point $A$ is a common point of one of the asymptotes of the hyperbola $C_{2}$: $\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1 (a > 0, b > 0)$ and $AF \perp x$-axis, find the eccentricity of the hyperbola. | \sqrt{5} | 6/8 |
Prove that $x = y = z = 1$ is the only positive solution of the system \[\left\{ \begin{array}{l}
x+y^2 +z^3 = 3
y+z^2 +x^3 = 3
z+x^2 +y^3 = 3
\end{array} \right.
\] | x=y=z=1 | 5/8 |
Select 5 different letters from the word "equation" to arrange in a row, including the condition that the letters "qu" are together and in the same order. | 480 | 7/8 |
A baseball league consists of two four-team divisions. Each team plays every other team in its division $N$ games. Each team plays every team in the other division $M$ games with $N>2M$ and $M>4$. Each team plays a $76$ game schedule. How many games does a team play within its own division? | 48 | 7/8 |
Vasya and Petya started running simultaneously from the starting point of a circular track and ran in opposite directions. While running, they met at some point on the track. Vasya ran a full lap and, continuing to run in the same direction, reached the point where they had previously met at the moment when Petya completed a full lap. How many times faster was Vasya running compared to Petya? | \frac{\sqrt{5}+1}{2} | 1/8 |
A number is composed of 10 ones, 9 tenths (0.1), and 6 hundredths (0.01). This number is written as ____, and when rounded to one decimal place, it is approximately ____. | 11.0 | 6/8 |
$\_$\_$\_$:20=24÷$\_$\_$\_$=80%=$\_$\_$\_$(fill in the blank with a fraction)=$\_$\_$\_$(fill in the blank with a decimal) | 0.8 | 7/8 |
The diagonals of a quadrilateral $ABCD$ inscribed in a circle intersect at point $E$. Given that $\angle ADB = \frac{\pi}{8}$, $BD = 6$, and $AD \cdot CE = DC \cdot AE$, find the area of quadrilateral $ABCD$. | 9\sqrt{2} | 2/8 |
Triangle $ABC$ with $AB=50$ and $AC=10$ has area $120$. Let $D$ be the midpoint of $\overline{AB}$, and let $E$ be the midpoint of $\overline{AC}$. The angle bisector of $\angle BAC$ intersects $\overline{DE}$ and $\overline{BC}$ at $F$ and $G$, respectively. What is the area of quadrilateral $FDBG$? | 75 | 5/8 |
Given a non-empty subset family \( U \) of \( S = \{a_1, a_2, \ldots, a_n\} \) that satisfies the property: if \( A \in U \) and \( A \subseteq B \), then \( B \in U \); and a non-empty subset family \( V \) of \( S \) that satisfies the property: if \( A \in V \) and \( A \supseteq B \), then \( B \in V \). Find the maximum possible value of \( \frac{|U \cap V|}{|U| \cdot |V|} \). | \frac{1}{2^n} | 1/8 |
Let \( S = \{1, 2, \cdots, 98\} \). Find the smallest natural number \( n \) such that in any subset of \( S \) with \( n \) elements, it is possible to select 10 numbers and no matter how these 10 numbers are divided into two groups, there will always be one group containing one number that is coprime (mutually prime) with four other numbers in that group, and the other group will contain one number that is not coprime with four other numbers in that group. | 50 | 1/8 |
Hexagon $ABCDEF$ is divided into five rhombuses, $P, Q, R, S,$ and $T$ , as shown. Rhombuses $P, Q, R,$ and $S$ are congruent, and each has area $\sqrt{2006}.$ Let $K$ be the area of rhombus $T$ . Given that $K$ is a positive integer, find the number of possible values for $K.$ [asy] // TheMathGuyd size(8cm); pair A=(0,0), B=(4.2,0), C=(5.85,-1.6), D=(4.2,-3.2), EE=(0,-3.2), F=(-1.65,-1.6), G=(0.45,-1.6), H=(3.75,-1.6), I=(2.1,0), J=(2.1,-3.2), K=(2.1,-1.6); draw(A--B--C--D--EE--F--cycle); draw(F--G--(2.1,0)); draw(C--H--(2.1,0)); draw(G--(2.1,-3.2)); draw(H--(2.1,-3.2)); label("$\mathcal{T}$",(2.1,-1.6)); label("$\mathcal{P}$",(0,-1),NE); label("$\mathcal{Q}$",(4.2,-1),NW); label("$\mathcal{R}$",(0,-2.2),SE); label("$\mathcal{S}$",(4.2,-2.2),SW); [/asy] | 89 | 1/8 |
At the World Meteorologist Conference, each participant took turns announcing the average monthly temperature in their hometown. All other participants recorded the product of the temperatures in their own city and the announced temperature. A total of 54 positive and 56 negative numbers were recorded. What is the minimum number of times a positive temperature could have been announced? | 4 | 6/8 |
Two parabolas are the graphs of the equations $y=2x^2-10x-10$ and $y=x^2-4x+6$. Find all points where they intersect. List the points in order of increasing $x$-coordinate, separated by semicolons. | (8,38) | 2/8 |
If $k, l, m$ are positive integers with $\frac{1}{k}+\frac{1}{l}+\frac{1}{m}<1$ , find the maximum possible value of $\frac{1}{k}+\frac{1}{l}+\frac{1}{m}$ . | \frac{41}{42} | 7/8 |
Let \( n \) be the second smallest integer that can be written as the sum of two positive cubes in two different ways. Compute \( n \). | 4104 | 7/8 |
11 people were standing in line under the rain, each holding an umbrella. They stood so close together that the umbrellas touched each other. Once the rain stopped, people closed their umbrellas and maintained a distance of 50 cm between each other. By how many times did the length of the queue decrease? Assume people can be considered as points and umbrellas as circles with a radius of 50 cm. | 2.2 | 1/8 |
The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to
[asy] unitsize(3mm); defaultpen(linewidth(0.8pt)); path p1=(0,0)--(3,0)--(3,3)--(0,3)--(0,0); path p2=(0,1)--(1,1)--(1,0); path p3=(2,0)--(2,1)--(3,1); path p4=(3,2)--(2,2)--(2,3); path p5=(1,3)--(1,2)--(0,2); path p6=(1,1)--(2,2); path p7=(2,1)--(1,2); path[] p=p1^^p2^^p3^^p4^^p5^^p6^^p7; for(int i=0; i<3; ++i) { for(int j=0; j<3; ++j) { draw(shift(3*i,3*j)*p); } } [/asy] | 56 | 5/8 |
If 600 were expressed as a sum of at least two distinct powers of 2, what would be the least possible sum of the exponents of these powers? | 22 | 2/8 |
We are given $n$ distinct rectangles in the plane. Prove that between the $4n$ interior angles formed by these rectangles at least $4\sqrt n$ are distinct. | 4\sqrt{n} | 1/8 |
Two cards are chosen at random from a standard 52-card deck. What is the probability that both cards are numbers (2 through 10) totaling to 12? | \frac{35}{663} | 7/8 |
In acute triangle $ABC$, let $H$ be the orthocenter and $D$ the foot of the altitude from $A$. The circumcircle of triangle $BHC$ intersects $AC$ at $E \neq C$, and $AB$ at $F \neq B$. If $BD=3, CD=7$, and $\frac{AH}{HD}=\frac{5}{7}$, the area of triangle $AEF$ can be expressed as $\frac{a}{b}$, where $a, b$ are relatively prime positive integers. Compute $100a+b$. | 12017 | 7/8 |
Two different points, $C$ and $D$, lie on the same side of line $AB$ so that $\triangle ABC$ and $\triangle BAD$ are congruent with $AB = 9$, $BC=AD=10$, and $CA=DB=17$. The intersection of these two triangular regions has area $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 59 | 7/8 |
Compute $1011_2 + 101_2 - 1100_2 + 1101_2$. Express your answer in base 2. | 10001_2 | 6/8 |
Given a geometric sequence $\{a_{n}\}$ that satisfies $\lim_{n \rightarrow +\infty} \sum_{i=1}^{n} a_{4} = 4$ and $\lim_{n \rightarrow +\infty} \sum_{i=1}^{n} a_{i}^{2} = 8$, determine the common ratio $q$. | \frac{1}{3} | 5/8 |
Vasya and Petya are playing "Take or Divide." In this game, there is initially one large pile of stones. On each turn, the current player either divides one of the existing piles into two smaller piles or takes one of the existing piles. The player who removes the last pile of stones from the field wins. Players take turns, with Vasya going first, but on this initial turn, he is not allowed to take the entire pile immediately. Who will win this game? Note: the pile can contain just one stone. | Vasya | 1/8 |
Given that \( f(x) \) is a function defined on \(\mathbf{R}\) such that \( f(1) = 1 \) and for any \( x \in \mathbf{R} \), \( f(x+5) \geq f(x) + 5 \) and \( f(x+1) \leq f(x) + 1 \). Find \( g(2002) \) given \( g(x) = f(x) + 1 - x \). | 1 | 7/8 |
A convex quadrilateral \(EFGH\) has vertices \(E, F, G, H\) lying respectively on the sides \(AB, BC, CD,\) and \(DA\) of another quadrilateral \(ABCD\). It satisfies the equation \(\frac{AE}{EB} \cdot \frac{BF}{FC} \cdot \frac{CG}{GD} \cdot \frac{DH}{HA} = 1\). Given that points \(E, F, G,\) and \(H\) lie on the sides of quadrilateral \(E_1F_1G_1H_1\) respectively with \(\frac{E_1A}{AH_1} = \lambda\), find the value of \(\frac{F_1C}{CG_1}\). | \lambda | 4/8 |
Kacey is handing out candy for Halloween. She has only $15$ candies left when a ghost, a goblin, and a vampire arrive at her door. She wants to give each trick-or-treater at least one candy, but she does not want to give any two the same number of candies. How many ways can she distribute all $15$ identical candies to the three trick-or-treaters given these restrictions? | 72 | 7/8 |
For positive integer $k>1$, let $f(k)$ be the number of ways of factoring $k$ into product of positive integers greater than $1$ (The order of factors are not countered, for example $f(12)=4$, as $12$ can be factored in these $4$ ways: $12,2\cdot 6,3\cdot 4, 2\cdot 2\cdot 3$.
Prove: If $n$ is a positive integer greater than $1$, $p$ is a prime factor of $n$, then $f(n)\leq \frac{n}{p}$ | \frac{n}{p} | 1/8 |
Define sequences $\{a_n\},\ \{b_n\}$ by
\[a_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}d\theta,\ b_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}\cos \theta d\theta\ (n=1,\ 2,\ 3,\ \cdots).\]
(1) Find $b_n$ .
(2) Prove that for each $n$ , $b_n\leq a_n\leq \frac 2{\sqrt{3}}b_n.$ (3) Find $\lim_{n\to\infty} \frac 1{n}\ln (na_n).$ | \frac{1}{2} | 5/8 |
Use the Horner's method to compute the value of the polynomial $f(x)=0.5x^{5}+4x^{4}-3x^{2}+x-1$ when $x=3$, and determine the first operation to perform. | 5.5 | 1/8 |
Let \( ABC \) be an equilateral triangle with side length 1. Let \( D \) be the reflection of point \( C \) over point \( A \). On the line \( AB \) beyond point \( B \), there is a point \( F \) such that \( EF=1 \), where \( E \) is the intersection point of the lines \( FC \) and \( DB \). Determine the lengths of the segments \( CE = x \) and \( BF = y \). | \sqrt[3]{4} | 3/8 |
In space, 4 points are given that are not on the same plane. How many planes can be drawn equidistant from these points? | 7 | 2/8 |
A woman wants freshly baked cookies delivered exactly at 18:00 for an event. Delivery trucks, upon finishing baking, travel with varying speeds due to potential traffic conditions:
- If there is moderate traffic, the trucks travel at an average speed of 60 km/h and would arrive at 17:45.
- If there are traffic jams, the trucks travel at an average speed of 20 km/h and would arrive at 18:15.
Determine the average speed the delivery truck must maintain to arrive exactly at 18:00. | 30 | 6/8 |
Find all primes $p$ and $q$ such that $3p^{q-1}+1$ divides $11^p+17^p$ | (3, 3) | 1/8 |
The complete set of $x$-values satisfying the inequality $\frac{x^2-4}{x^2-1}>0$ is the set of all $x$ such that:
$\text{(A) } x>2 \text{ or } x<-2 \text{ or} -1<x<1\quad \text{(B) } x>2 \text{ or } x<-2\quad \\ \text{(C) } x>1 \text{ or } x<-2\qquad\qquad\qquad\quad \text{(D) } x>1 \text{ or } x<-1\quad \\ \text{(E) } x \text{ is any real number except 1 or -1}$ | \textbf{(A)}\x>2orx<-2or-1<x<1 | 1/8 |
Find the number of positive integers $n$ that satisfy
\[(n - 1)(n - 3)(n - 5) \dotsm (n - 97) < 0.\] | 24 | 4/8 |
You have $8$ friends, each of whom lives at a different vertex of a cube. You want to chart a path along the cube’s edges that will visit each of your friends exactly once. You can start at any vertex, but you must end at the vertex you started at, and you cannot travel on any edge more than once. How many different paths can you take? | 96 | 7/8 |
A line parallel to leg \(AC\) of right triangle \(ABC\) intersects leg \(BC\) at point \(K\) and the hypotenuse \(AB\) at point \(N\). On leg \(AC\), a point \(M\) is chosen such that \(MK = MN\). Find the ratio \(\frac{AM}{MC}\) if \(\frac{BK}{BC} = 14\). | 27 | 3/8 |
Calculate the volumes of solids formed by rotating the regions bounded by the graphs of the functions around the y-axis.
$$
y = (x-1)^{2}, x=0, x=2, y=0
$$ | \frac{4\pi}{3} | 3/8 |
Find all positive integers $ k$ with the following property: There exists an integer $ a$ so that $ (a\plus{}k)^{3}\minus{}a^{3}$ is a multiple of $ 2007$ . | 669 | 2/8 |
In a convex 1950-gon, all diagonals are drawn. They divide it into polygons. Consider the polygon with the largest number of sides. What is the greatest number of sides it can have? | 1949 | 2/8 |
A triangle and a rhombus inscribed in it share a common angle. The sides of the triangle that include this angle are in the ratio \(\frac{m}{n}\). Find the ratio of the area of the rhombus to the area of the triangle. | \frac{2mn}{(+n)^2} | 6/8 |
The TV show "The Mystery of Santa Barbara" features 20 characters. In each episode, one of the following events occurs: a certain character learns the Mystery, a certain character learns that someone knows the Mystery, or a certain character learns that someone does not know the Mystery. What is the maximum number of episodes the series can continue? | 780 | 4/8 |
Perpendiculars \( AP \) and \( AK \) are dropped from vertex \( A \) of triangle \( ABC \) onto the angle bisectors of the external angles at \( B \) and \( C \) respectively. Find the length of segment \( PK \), given that the perimeter of triangle \( ABC \) is \( P \). | \frac{P}{2} | 1/8 |
Find the volume of an inclined triangular prism, where the area of one of its lateral faces is \( S \), and the distance from the plane of this face to the opposite edge is \( d \). | \frac{1}{2}S | 4/8 |
Let $ABCD$ be a convex quadrilateral such that $AB + BC = 2021$ and $AD = CD$ . We are also given that $\angle ABC = \angle CDA = 90^o$ . Determine the length of the diagonal $BD$ . | \frac{2021 \sqrt{2}}{2} | 4/8 |
Given the function $y=4^{x}-6\times2^{x}+8$, find the minimum value of the function and the value of $x$ when the minimum value is obtained. | -1 | 2/8 |
Alice wants to write down a list of prime numbers less than 100, using each of the digits 1, 2, 3, 4, and 5 once and no other digits. Which prime number must be in her list? | 41 | 4/8 |
In the final of the giraffe beauty contest, two giraffes, Tall and Spotted, reached the finals. There are 135 voters divided into 5 districts, with each district divided into 9 precincts, and each precinct having 3 voters. The voters in each precinct choose the winner by majority vote; in a district, the giraffe that wins in the majority of precincts wins the district; finally, the giraffe that wins in the majority of the districts is declared the winner of the final. The giraffe Tall won. What is the minimum number of voters who could have voted for Tall? | 30 | 2/8 |
Let $ABCD$ be a parallelogram with area $15$. Points $P$ and $Q$ are the projections of $A$ and $C,$ respectively, onto the line $BD;$ and points $R$ and $S$ are the projections of $B$ and $D,$ respectively, onto the line $AC.$ See the figure, which also shows the relative locations of these points.
Suppose $PQ=6$ and $RS=8,$ and let $d$ denote the length of $\overline{BD},$ the longer diagonal of $ABCD.$ Then $d^2$ can be written in the form $m+n\sqrt p,$ where $m,n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. What is $m+n+p?$ | 81 | 5/8 |
Let $L$ be the intersection point of the diagonals $C E$ and $D F$ of a regular hexagon $A B C D E F$ with side length 5. Point $K$ is such that $\overrightarrow{L K}=\overrightarrow{F B}-3 \overrightarrow{A B}$. Determine whether point $K$ lies inside, on the boundary, or outside of $A B C D E F$, and also find the length of the segment $K F$. | \frac{5 \sqrt{3}}{3} | 7/8 |
The chord \( AB \) of a sphere with a radius of 1 has a length of 1 and is located at an angle of \( 60^\circ \) to the diameter \( CD \) of this sphere. It is known that \( AC = \sqrt{2} \) and \( AC < BC \). Find the length of the segment \( BD \). | 1 | 6/8 |
Given triangle \( \triangle ABC \) with \( AB = 1 \), \( AC = 2 \), and \( \cos B + \sin C = 1 \), find the length of side \( BC \). | \frac{3 + 2 \sqrt{21}}{5} | 7/8 |
Let the function \( f(x) = 4x^3 + bx + 1 \) with \( b \in \mathbb{R} \). For any \( x \in [-1, 1] \), \( f(x) \geq 0 \). Find the range of the real number \( b \). | -3 | 1/8 |
Given the integer \( n > 1 \) and the real number \( a > 0 \), determine the maximum of \(\sum_{i=1}^{n-1} x_{i} x_{i+1}\) taken over all nonnegative numbers \( x_{i} \) with sum \( a \). | \frac{^2}{4} | 5/8 |
Given the function $$f(x)=\sin(x+ \frac {\pi}{6})+2\sin^{2} \frac {x}{2}$$.
(1) Find the equation of the axis of symmetry and the coordinates of the center of symmetry for the function $f(x)$.
(2) Determine the intervals of monotonicity for the function $f(x)$.
(3) In triangle $ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$, respectively, and $a= \sqrt {3}$, $f(A)= \frac {3}{2}$, the area of triangle $ABC$ is $\frac { \sqrt {3}}{2}$. Find the value of $\sin B + \sin C$. | \frac {3}{2} | 3/8 |
Call a positive integer $n$ $k$-pretty if $n$ has exactly $k$ positive divisors and $n$ is divisible by $k$. For example, $18$ is $6$-pretty. Let $S$ be the sum of positive integers less than $2019$ that are $20$-pretty. Find $\tfrac{S}{20}$.
| 472 | 5/8 |
On the sides \( AB \) and \( BC \) of triangle \( ABC \), squares \( ABDE \) and \( BCLK \) with centers \( O_1 \) and \( O_2 \) are constructed. Let \( M_1 \) and \( M_2 \) be the midpoints of segments \( DL \) and \( AC \). Prove that \( O_1M_1O_2M_2 \) is a square. | O_1M_1O_2M_2 | 1/8 |
The sum of 20 consecutive integers is a triangular number. What is the smallest such sum? | 190 | 1/8 |
All positive integers are colored either red or blue in such a way that the sum of numbers of different colors is blue, and the product of numbers of different colors is red.
What is the color of the product of two red numbers? | Red | 7/8 |
Find the least positive integer that cannot be represented as $\frac{2^a-2^b}{2^c-2^d}$ for some positive integers $a, b, c, d$ . | 11 | 1/8 |
Find the range of the function \( f(x) = \sin^4(x) \cdot \tan(x) + \cos^4(x) \cdot \cot(x) \). | (-\infty,-\frac{1}{2}]\cup[\frac{1}{2},\infty) | 6/8 |
Given is an acute angled triangle $ ABC$ . Determine all points $ P$ inside the triangle with
\[1\leq\frac{\angle APB}{\angle ACB},\frac{\angle BPC}{\angle BAC},\frac{\angle CPA}{\angle CBA}\leq2\] | O | 3/8 |
A box contains 5 white balls and 3 black balls. What is the probability that when drawing the balls one at a time, all draws alternate in color starting with a white ball? | \frac{1}{56} | 3/8 |
In $\triangle ABC$, \(AB = 5\), \(AC = 4\), and \(\overrightarrow{AB} \cdot \overrightarrow{AC} = 12\). Let \(P\) be a point on the plane of \(\triangle ABC\). Find the minimum value of \(\overrightarrow{PA} \cdot (\overrightarrow{PB} + \overrightarrow{PC})\). | -\frac{65}{8} | 7/8 |
A positive number \(a\) is the coefficient of \(x^{2}\) in the quadratic polynomial \(f(x)\), which has no roots. Prove that for any \(x\), the inequality \(f(x) + f(x-1) - f(x+1) > -4a\) holds. | f(x)+f(x-1)-f(x+1)>-4a | 6/8 |
Three people played. $A$ had $10\mathrm{~K}$, $B$ had $57\mathrm{~K}$, and $C$ had $29\mathrm{~K}$. By the end of the game, $B$ had three times as much money as $A$, and $C$ had three times as much money as $A$ won. How much did $C$ win or lose? | 5 | 7/8 |
In triangle $ABC$, the angle bisectors are $AD$, $BE$, and $CF$, which intersect at the incenter $I$. If $\angle ACB = 38^\circ$, then find the measure of $\angle AIE$, in degrees. | 71^\circ | 2/8 |
For rational numbers $x$, $y$, $a$, $t$, if $|x-a|+|y-a|=t$, then $x$ and $y$ are said to have a "beautiful association number" of $t$ with respect to $a$. For example, $|2-1|+|3-1|=3$, then the "beautiful association number" of $2$ and $3$ with respect to $1$ is $3$. <br/> $(1)$ The "beautiful association number" of $-1$ and $5$ with respect to $2$ is ______; <br/> $(2)$ If the "beautiful association number" of $x$ and $5$ with respect to $3$ is $4$, find the value of $x$; <br/> $(3)$ If the "beautiful association number" of $x_{0}$ and $x_{1}$ with respect to $1$ is $1$, the "beautiful association number" of $x_{1}$ and $x_{2}$ with respect to $2$ is $1$, the "beautiful association number" of $x_{2}$ and $x_{3}$ with respect to $3$ is $1$, ..., the "beautiful association number" of $x_{1999}$ and $x_{2000}$ with respect to $2000$ is $1$, ... <br/> ① The minimum value of $x_{0}+x_{1}$ is ______; <br/> ② What is the minimum value of $x_{1}+x_{2}+x_{3}+x_{4}+...+x_{2000}$? | 2001000 | 3/8 |
A school arranges five classes every morning from Monday to Friday, each lasting 40 minutes. The first class starts from 7:50 to 8:30, with a 10-minute break between classes. A student returns to school after taking leave. If he arrives at the classroom randomly between 8:50 and 9:30, calculate the probability that he listens to the second class for no less than 20 minutes. | \dfrac{1}{4} | 7/8 |
In the acute triangle \( \triangle ABC \), the sides \( a, b, c \) are opposite to the angles \( \angle A, \angle B, \angle C \) respectively, and \( a, b, c \) form an arithmetic sequence. Also, \( \sin (A - C) = \frac{\sqrt{3}}{2} \). Find \( \sin (A + C) \). | \frac{\sqrt{39}}{8} | 1/8 |
3.8. Prove, using the definition of the limit of a sequence of complex numbers, that:
a) $\lim _{n \rightarrow \infty} \frac{n^{a}+i n^{b}}{n^{b}+i n^{a}}=\left\{\begin{array}{cc}1, & a=b ; \\ i, & a<b ; \\ -i, & a>b,\end{array}\right.$
where $a, b \in \mathbb{R}, n \in \mathbb{N}$
b) $\lim _{n \rightarrow x} \frac{\left(\sqrt{2} \alpha n^{3}+a n^{2}+b n+c\right)+i\left(\sqrt{2} \beta n^{3}+d n^{2}+e n+f\right)}{\left\|n \sqrt{n}-i n \sqrt{n^{4}-n}|+i| n \sqrt{n}+i n \sqrt{n^{4}-n}\right\|}=\alpha+i \beta$,
where $\alpha, \beta, a, b, c, d, e, f$ are any complex numbers. | \alpha+i\beta | 3/8 |
Chewbacca has 25 pieces of orange gum and 35 pieces of apple gum. Some of the pieces are in complete packs, while others are loose. Each complete pack has exactly $y$ pieces of gum. If Chewbacca loses two packs of orange gum, then the ratio of the number of pieces of orange gum he has to the number of pieces of apple gum will be exactly the same as if he instead finds 4 packs of apple gum. Find $y$. | \frac{15}{4} | 6/8 |
How many different divisors does the number 86,400,000 have (including 1 and the number 86,400,000 itself)? Find the sum of all these divisors. | 319823280 | 7/8 |
If \(x\) and \(y\) are positive real numbers such that \(6x^2 + 12xy + 6y^2 = x^3 + 3x^2 y + 3xy^2\), find the value of \(x\). | \frac{24}{7} | 2/8 |
Two circles with radii \( R \) and \( r \) intersect at point \( A \). Let \( BC \) be a common tangent to both circles at points \( B \) and \( C \). Find the radius of the circumcircle of triangle \( ABC \). | \sqrt{Rr} | 1/8 |
A firecracker was thrown vertically upwards with a speed of $20 \mathrm{~m/s}$. Three seconds after the start of the flight, it exploded into two fragments of equal mass. The first fragment flew horizontally immediately after the explosion with a speed of $48 \mathrm{~m/s}$. Find the speed of the second fragment (in m/s) right after the explosion. Assume the acceleration due to gravity is $10 \mathrm{~m/s}^2$. | 52 | 7/8 |
For a positive integer $p$, define the positive integer $n$ to be $p$-safe if $n$ differs in absolute value by more than $2$ from all multiples of $p$. For example, the set of $10$-safe numbers is $\{ 3, 4, 5, 6, 7, 13, 14, 15, 16, 17, 23, \ldots\}$. Find the number of positive integers less than or equal to $10,000$ which are simultaneously $7$-safe, $11$-safe, and $13$-safe. | 958 | 1/8 |
Let $n$ be a positive integer. Jadzia has to write all integers from $1$ to $2n-1$ on a board, and she writes each integer in blue or red color. We say that pair of numbers $i,j\in \{1,2,3,...,2n-1\}$ , where $i\leqslant j$ , is $\textit{good}$ if and only if number of blue numbers among $i,i+1,...,j$ is odd. Determine, in terms of $n$ , maximal number of good pairs. | n^2 | 4/8 |
Given a rectangle \(ABCD\). Point \(M\) is the midpoint of side \(AB\), and point \(K\) is the midpoint of side \(BC\). Segments \(AK\) and \(CM\) intersect at point \(E\). By what factor is the area of quadrilateral \(MBKE\) smaller than the area of quadrilateral \(AECD\)? | 4 | 2/8 |
There are \( R \) zeros at the end of \(\underbrace{99\ldots9}_{2009 \text{ of }} \times \underbrace{99\ldots9}_{2009 \text{ of } 9 \text{'s}} + 1 \underbrace{99\ldots9}_{2009 \text{ of } 9 \text{'s}}\). Find the value of \( R \). | 4018 | 7/8 |
Determine all real numbers $a$ such that \[4\lfloor an\rfloor =n+\lfloor a\lfloor an\rfloor \rfloor \; \text{for all}\; n \in \mathbb{N}.\] | 2+\sqrt{3} | 1/8 |
Through the use of theorems on logarithms
\[\log{\frac{a}{b}} + \log{\frac{b}{c}} + \log{\frac{c}{d}} - \log{\frac{ay}{dx}}\]
can be reduced to:
$\textbf{(A)}\ \log{\frac{y}{x}}\qquad \textbf{(B)}\ \log{\frac{x}{y}}\qquad \textbf{(C)}\ 1\qquad \\ \textbf{(D)}\ 140x-24x^2+x^3\qquad \textbf{(E)}\ \text{none of these}$ | \textbf{(B)}\\log{\frac{x}{y}} | 1/8 |
Let \( a_{1}, a_{2}, \cdots, a_{2014} \) be a permutation of the positive integers \( 1, 2, \cdots, 2014 \). Define
\[ S_{k} = a_{1} + a_{2} + \cdots + a_{k} \quad (k=1, 2, \cdots, 2014). \]
What is the maximum number of odd numbers among \( S_{1}, S_{2}, \cdots, S_{2014} \)? | 1511 | 1/8 |
Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a
running tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of 7. Play continues until either player wins, or else inde nitely. If Nathaniel goes fi rst, determine the probability that he ends up winning. | 5/11 | 2/8 |
In triangle $ABC$, $AB = AC$ and $\angle BAC = 36^\circ$. The angle bisector of $\angle BAC$ intersects the side $AC$ at point $D$. Let $O_1$ be the circumcenter of triangle $ABC$, $O_2$ be the circumcenter of triangle $BCD$, and $O_3$ be the circumcenter of triangle $ABD$.
a) Prove that triangle $O_1O_2O_3$ is similar to triangle $DBA$.
b) Calculate the exact ratio of $O_1O_2 : O_2O_3$. | \frac{\sqrt{5}-1}{2} | 1/8 |
Let $p$ be a prime with $p \equiv 1 \pmod{4}$ . Let $a$ be the unique integer such that \[p=a^{2}+b^{2}, \; a \equiv-1 \pmod{4}, \; b \equiv 0 \; \pmod{2}\] Prove that \[\sum^{p-1}_{i=0}\left( \frac{i^{3}+6i^{2}+i }{p}\right) = 2 \left( \frac{2}{p}\right),\] where $\left(\frac{k}{p}\right)$ denotes the Legendre Symbol. | 2(\frac{2}{p}) | 1/8 |
There is a five-digit positive odd number \( x \). In \( x \), all 2s are replaced with 5s, and all 5s are replaced with 2s, while other digits remain unchanged. This creates a new five-digit number \( y \). If \( x \) and \( y \) satisfy the equation \( y = 2(x + 1) \), what is the value of \( x \)?
(Chinese Junior High School Mathematics Competition, 1987) | 29995 | 1/8 |
Regular octagon $ABCDEFGH$ has its center at $J$. Each of the vertices and the center are to be associated with one of the digits $1$ through $9$, with each digit used once, in such a way that the sums of the numbers on the lines $AJE$, $BJF$, $CJG$, and $DJH$ are all equal. In how many ways can this be done?
[asy]
pair A,B,C,D,E,F,G,H,J;
A=(20,20(2+sqrt(2)));
B=(20(1+sqrt(2)),20(2+sqrt(2)));
C=(20(2+sqrt(2)),20(1+sqrt(2)));
D=(20(2+sqrt(2)),20);
E=(20(1+sqrt(2)),0);
F=(20,0);
G=(0,20);
H=(0,20(1+sqrt(2)));
J=(10(2+sqrt(2)),10(2+sqrt(2)));
draw(A--B);
draw(B--C);
draw(C--D);
draw(D--E);
draw(E--F);
draw(F--G);
draw(G--H);
draw(H--A);
dot(A);
dot(B);
dot(C);
dot(D);
dot(E);
dot(F);
dot(G);
dot(H);
dot(J);
label("$A$",A,NNW);
label("$B$",B,NNE);
label("$C$",C,ENE);
label("$D$",D,ESE);
label("$E$",E,SSE);
label("$F$",F,SSW);
label("$G$",G,WSW);
label("$H$",H,WNW);
label("$J$",J,SE);
size(4cm);
[/asy] | 1152 | 7/8 |
Let \( A = \left(a_1, a_2, \ldots, a_{2001}\right) \) be a sequence of positive integers. Let \( m \) be the number of 3-element subsequences \( (a_i, a_j, a_k) \) with \( 1 \leq i < j < k \leq 2001 \) such that \( a_j = a_i + 1 \) and \( a_k = a_j + 1 \). Considering all such sequences \( A \), find the greatest value of \( m \). | 667^3 | 1/8 |
A geometric sequence $\left\{a_{n}\right\}$ has the first term $a_{1} = 1536$ and the common ratio $q = -\frac{1}{2}$. Let $\Pi_{n}$ represent the product of its first $n$ terms. For what value of $n$ is $\Pi_{n}$ maximized? | 11 | 1/8 |
In rectangle \(ABCD\), side \(AB\) is 6 and side \(BC\) is 11. From vertices \(B\) and \(C\), angle bisectors are drawn intersecting side \(AD\) at points \(X\) and \(Y\) respectively. Find the length of segment \(XY\). | 1 | 6/8 |
The distance between location A and location B originally required a utility pole to be installed every 45m, including the two poles at both ends, making a total of 53 poles. Now, the plan has been changed to install a pole every 60m. Excluding the two poles at both ends, how many poles in between do not need to be moved? | 12 | 6/8 |
Find the maximum value of the function
$$
f(x)=\sin (x+\sin x)+\sin (x-\sin x)+\left(\frac{\pi}{2}-2\right) \sin (\sin x)
$$ | \frac{\pi - 2}{\sqrt{2}} | 1/8 |
Sasa wants to make a pair of butterfly wings for her Barbie doll. As shown in the picture, she first drew a trapezoid and then drew two diagonals, which divided the trapezoid into four triangles. She cut off the top and bottom triangles, and the remaining two triangles are exactly a pair of beautiful wings. If the areas of the two triangles that she cut off are 4 square centimeters and 9 square centimeters respectively, then the area of the wings that Sasa made is $\qquad$ square centimeters. | 12 | 6/8 |
Suppose that $n$ is a positive integer number. Consider a regular polygon with $2n$ sides such that one of its largest diagonals is parallel to the $x$ -axis. Find the smallest integer $d$ such that there is a polynomial $P$ of degree $d$ whose graph intersects all sides of the polygon on points other than vertices.
Proposed by Mohammad Ahmadi | n | 1/8 |
There are two straight lines, each of which passes through four points of the form $(1,0,a), (b,1,0), (0,c,1),$ and $(6d,6d,-d),$ where $a,b,c,$ and $d$ are real numbers, not necessarily in that order. Enter all possible values of $d,$ separated by commas. | \frac{1}{3}, \frac{1}{8} | 1/8 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.