problem
stringlengths 10
7.44k
| answer
stringlengths 1
270
| difficulty
stringclasses 8
values |
|---|---|---|
Three intersecting lines form 12 angles, with $n$ of them being equal. What can be the maximum value of $n$? | 6 | 3/8 |
Given that $y$ is a multiple of $45678$, what is the greatest common divisor of $g(y)=(3y+4)(8y+3)(14y+9)(y+14)$ and $y$? | 1512 | 1/8 |
Given a triangular pyramid \(ABCD\) with right angles at vertex \(D\), where \(CD = AD + DB\), prove that the sum of the plane angles at vertex \(C\) is \(90^\circ\). | 90 | 2/8 |
In a country there are infinitely many towns and for every pair of towns there is one road connecting them. Initially there are $n$ coin in each city. Every day traveller Hong starts from one town and moves on to another, but if Hong goes from town $A$ to $B$ on the $k$ -th day, he has to send $k$ coins from $B$ to $A$ , and he can no longer use the road connecting $A$ and $B$ . Prove that Hong can't travel for more than $n+2n^\frac{2}{3}$ days. | n+2n^{\frac{2}{3}} | 1/8 |
In square \( A B C D \), \( P \) and \( Q \) are points on sides \( C D \) and \( B C \), respectively, such that \( \angle A P Q = 90^\circ \). If \( A P = 4 \) and \( P Q = 3 \), find the area of \( A B C D \). | \frac{256}{17} | 4/8 |
Values \( a_{1}, \ldots, a_{2013} \) are chosen independently and at random from the set \( \{1, \ldots, 2013\} \). What is the expected number of distinct values in the set \( \{a_{1}, \ldots, a_{2013}\} \)? | 2013(1-(\frac{2012}{2013})^{2013}) | 6/8 |
Let \( P \) be a point inside the acute angle \( \angle BAC \). Suppose that \( \angle ABP = \angle ACP = 90^\circ \). The points \( D \) and \( E \) are on the segments \( BA \) and \( CA \), respectively, such that \( BD = BP \) and \( CP = CE \). The points \( F \) and \( G \) are on the segments \( AC \) and \( AB \), respectively, such that \( DF \) is perpendicular to \( AB \) and \( EG \) is perpendicular to \( AC \). Show that \( PF = PG \). | PF=PG | 1/8 |
Let $\triangle ABC$ be an isosceles. $(AB=AC)$ .Let $D$ and $E$ be two points on the side $BC$ such that $D\in BE$ , $E\in DC$ and $2\angle DAE = \angle BAC$ .Prove that we can construct a triangle $XYZ$ such that $XY=BD$ , $YZ=DE$ and $ZX=EC$ .Find $\angle BAC + \angle YXZ$ . | 180 | 1/8 |
How many of the numbers from the set $\{1,\ 2,\ 3,\ldots,\ 100\}$ have a perfect square factor other than one? | 39 | 3/8 |
Let $ABCDEF$ be a regular octahedron with unit side length, such that $ABCD$ is a square. Points $G, H$ are on segments $BE, DF$ respectively. The planes $AGD$ and $BCH$ divide the octahedron into three pieces, each with equal volume. Compute $BG$ . | \frac{9-\sqrt{57}}{6} | 1/8 |
In a plane, there are $n$ arbitrary points that do not lie on a single straight line. A line is drawn through each pair of these points. Prove that among the lines drawn, there exists at least one that passes through exactly two points. | 3 | 1/8 |
Teacher Li plans to buy 25 souvenirs for students from a store that has four types of souvenirs: bookmarks, postcards, notebooks, and pens, with 10 pieces available for each type (souvenirs of the same type are identical). Teacher Li intends to buy at least one piece of each type. How many different purchasing plans are possible? (Answer in numeric form.). | 592 | 7/8 |
For a given positive integer $k$ find, in terms of $k$, the minimum value of $N$ for which there is a set of $2k + 1$ distinct positive integers that has sum greater than $N$ but every subset of size $k$ has sum at most $\tfrac{N}{2}.$ | 2k^3 + 3k^2 + 3k | 1/8 |
Mateo's 300 km trip from Edmonton to Calgary included a 40 minute break in Red Deer. He started in Edmonton at 7 a.m. and arrived in Calgary at 11 a.m. Not including the break, what was his average speed for the trip?
(A) 83 km/h
(B) 94 km/h
(C) 90 km/h
(D) 95 km/h
(E) 64 km/h | 90 | 1/8 |
Circle $T$ has a circumference of $12\pi$ inches, and segment $XY$ is a diameter. If the measure of angle $TXZ$ is $60^{\circ}$, what is the length, in inches, of segment $XZ$?
[asy]
size(150);
draw(Circle((0,0),13),linewidth(1));
draw((-12,-5)--(-5,-12)--(12,5)--cycle,linewidth(1));
dot((0,0));
label("T",(0,0),N);
label("X",(-12,-5),W);
label("Z",(-5,-12),S);
label("Y",(12,5),E);
[/asy] | 6 | 6/8 |
A sequence $(c_n)$ is defined as follows: $c_1 = 1$, $c_2 = \frac{1}{3}$, and
\[c_n = \frac{2 - c_{n-1}}{3c_{n-2}}\] for all $n \ge 3$. Find $c_{100}$. | \frac{1}{3} | 1/8 |
Four cars $A$, $B$, $C$, and $D$ start simultaneously from the same point on a circular track. Cars $A$ and $B$ travel clockwise, while cars $C$ and $D$ travel counterclockwise. All cars move at constant but distinct speeds. Exactly 7 minutes after the race starts, $A$ meets $C$ for the first time, and at the same moment, $B$ meets $D$ for the first time. After another 46 minutes, $A$ and $B$ meet for the first time. How long after the race starts will $C$ and $D$ meet for the first time? | 53 | 1/8 |
At the moment when Pierrot left the "Commercial" bar, heading to the "Theatrical" bar, Jeannot was leaving the "Theatrical" bar on his way to the "Commercial" bar. They were walking at constant (but different) speeds. When the vagabonds met, Pierrot proudly noted that he had walked 200 meters more than Jeannot. After their fight ended, they hugged and continued on their paths but at half their previous speeds due to their injuries. Pierrot then took 8 minutes to reach the "Theatrical" bar, and Jeannot took 18 minutes to reach the "Commercial" bar. What is the distance between the bars? | 1000 | 7/8 |
A clueless ant makes the following route: starting at point $ A $ goes $ 1$ cm north, then $ 2$ cm east, then $ 3$ cm south, then $ 4$ cm west, immediately $ 5$ cm north, continues $ 6$ cm east, and so on, finally $ 41$ cm north and ends in point $ B $ . Calculate the distance between $ A $ and $ B $ (in a straight line).
| 29 | 7/8 |
Let $x$ be a real number. Find the maximum value of $2^{x(1-x)}$. | \sqrt[4]{2} | 6/8 |
Consider a large square divided into a grid of \(5 \times 5\) smaller squares, each with side length \(1\) unit. A shaded region within the large square is formed by connecting the centers of four smaller squares, creating a smaller square inside. Calculate the ratio of the area of the shaded smaller square to the area of the large square. | \frac{2}{25} | 6/8 |
Every 1 kilogram of soybeans can produce 0.8 kilograms of soybean oil. With 20 kilograms of soybeans, you can produce \_\_\_\_\_\_ kilograms of soybean oil. To obtain 20 kilograms of soybean oil, you need \_\_\_\_\_\_ kilograms of soybeans. | 25 | 7/8 |
On each side of an equilateral triangle with side length $n$ units, where $n$ is an integer, $1 \leq n \leq 100$ , consider $n-1$ points that divide the side into $n$ equal segments. Through these points, draw lines parallel to the sides of the triangle, obtaining a net of equilateral triangles of side length one unit. On each of the vertices of these small triangles, place a coin head up. Two coins are said to be adjacent if the distance between them is 1 unit. A move consists of flipping over any three mutually adjacent coins. Find the number of values of $n$ for which it is possible to turn all coins tail up after a finite number of moves. | 67 | 2/8 |
Let \( a, b, c, d \) be odd integers such that \( 0 < a < b < c < d \) and \( ad = bc \). Prove that if \( a + d = 2k \) and \( b + c = 2m \) for some integers \( k \) and \( m \), then \( a = 1 \). | 1 | 2/8 |
Given numbers $5, 6, 7, 8, 9, 10, 11, 12, 13$ are written in a $3\times3$ array, with the condition that two consecutive numbers must share an edge. If the sum of the numbers in the four corners is $32$, calculate the number in the center of the array. | 13 | 1/8 |
Find all functions $f$ defined on the set of positive reals which take positive real values and satisfy: $f(xf(y))=yf(x)$ for all $x,y$; and $f(x)\to0$ as $x\to\infty$. | f(x)=\frac1x | 1/8 |
Given the polynomial
$$
\begin{aligned}
f(x)= & a_{2020} x^{2020}+a_{2019} x^{2019}+\cdots+ \\
& a_{2018} x^{2018}+a_{1} x+a_{0},
\end{aligned}
$$
where $a_{i} \in \mathbf{Z}$ for $i=0,1, \cdots, 2020$, determine the number of ordered pairs $(n, p)$ (where $p$ is a prime) such that $p^{2}<n<2020$ and for any $i=0, 1, \cdots, n$, it holds that $\mathrm{C}_{n}^{i}+f(i) \equiv 0 \pmod{p}$. | 8 | 1/8 |
Of the integers from 1 to \(8 \cdot 10^{20}\) (inclusive), which are more numerous, and by how much: those containing only even digits in their representation or those containing only odd digits? | \frac{5^{21}-5}{4} | 1/8 |
What is the length of side $y$ in the following diagram?
[asy]
import olympiad;
draw((0,0)--(2,0)--(0,2*sqrt(3))--cycle); // modified triangle lengths
draw((0,0)--(-2,0)--(0,2*sqrt(3))--cycle);
label("10",(-1,2*sqrt(3)/2),NW); // changed label
label("$y$",(2/2,2*sqrt(3)/2),NE);
draw("$30^{\circ}$",(2.5,0),NW); // modified angle
draw("$45^{\circ}$",(-1.9,0),NE);
draw(rightanglemark((0,2*sqrt(3)),(0,0),(2,0),4));
[/asy] | 10\sqrt{3} | 1/8 |
A sequence $y_1,y_2,\dots,y_k$ of real numbers is called \emph{zigzag} if $k=1$, or if $y_2-y_1, y_3-y_2, \dots, y_k-y_{k-1}$ are nonzero and alternate in sign. Let $X_1,X_2,\dots,X_n$ be chosen independently from the uniform distribution on $[0,1]$. Let $a(X_1,X_2,\dots,X_n)$ be the largest value of $k$ for which there exists an increasing sequence of integers $i_1,i_2,\dots,i_k$ such that $X_{i_1},X_{i_2},\dots,X_{i_k}$ is zigzag. Find the expected value of $a(X_1,X_2,\dots,X_n)$ for $n \geq 2$. | \frac{2n+2}{3} | 1/8 |
Abbot writes the letter $A$ on the board. Every minute, he replaces every occurrence of $A$ with $A B$ and every occurrence of $B$ with $B A$, hence creating a string that is twice as long. After 10 minutes, there are $2^{10}=1024$ letters on the board. How many adjacent pairs are the same letter? | 341 | 1/8 |
There are three bags. One bag contains three green candies and one red candy. One bag contains two green candies and two red candies. One bag contains one green candy and three red candies. A child randomly selects one of the bags, randomly chooses a first candy from that bag, and eats the candy. If the first candy had been green, the child randomly chooses one of the other two bags and randomly selects a second candy from that bag. If the first candy had been red, the child randomly selects a second candy from the same bag as the first candy. If the probability that the second candy is green is given by the fraction $m/n$ in lowest terms, find $m + n$ . | 217 | 5/8 |
In the coordinate plane, a rectangle has vertices with coordinates $(34,0), (41,0), (34,9), (41,9)$. Find the smallest value of the parameter $a$ such that the line $y = ax$ divides this rectangle into two parts where the area of one part is twice the area of the other. If the answer is not an integer, write it as a decimal. | 0.08 | 2/8 |
Lesha's summer cottage has the shape of a nonagon with three pairs of equal and parallel sides. Lesha knows that the area of the triangle with vertices at the midpoints of the remaining sides of the nonagon is 12 sotkas. Help him find the area of the entire summer cottage. | 48 | 2/8 |
The sum of the first $15$ positive even integers is also the sum of six consecutive even integers. What is the smallest of these six integers? | 35 | 1/8 |
Using the 3 vertices of a triangle and 7 points inside it (a total of 10 points), how many smaller triangles can the original triangle be divided into?
(1985 Shanghai Junior High School Math Competition, China;
1988 Jiangsu Province Junior High School Math Competition, China) | 15 | 7/8 |
Calculate the angle \(\theta\) for the expression \[ e^{11\pi i/60} + e^{23\pi i/60} + e^{35\pi i/60} + e^{47\pi i/60} + e^{59\pi i/60} \] in the form \( r e^{i \theta} \), where \( 0 \leq \theta < 2\pi \). | \frac{7\pi}{12} | 4/8 |
A standard $n$-sided die has $n$ sides labeled 1 to $n$. Luis, Luke, and Sean play a game in which they roll a fair standard 4-sided die, a fair standard 6-sided die, and a fair standard 8-sided die, respectively. They lose the game if Luis's roll is less than Luke's roll, and Luke's roll is less than Sean's roll. Compute the probability that they lose the game. | \frac{1}{4} | 6/8 |
If the domain of the function $f(x)=x^{2}$ is $D$, and its range is ${0,1,2,3,4,5}$, then there are \_\_\_\_\_\_ such functions $f(x)$ (answer with a number). | 243 | 7/8 |
A palindrome is a positive integer which is unchanged if you reverse the order of its digits. For example, 23432. If all palindromes are written in increasing order, what possible prime values can the difference between successive palindromes take? | 11 | 5/8 |
The circle touches the sides $AB$ and $BC$ of triangle $ABC$ at points $D$ and $E$ respectively. Find the height of triangle $ABC$ dropped from point $A$, given that $AB = 5$, $AC = 2$, and points $A$, $D$, $E$, and $C$ lie on the same circle. | \frac{4\sqrt{6}}{5} | 2/8 |
A creative contest at the institute consisted of four tasks. There were 70 applicants in total. 35 applicants successfully completed the first task, 48 completed the second task, 64 completed the third task, and 63 completed the fourth task. No one completed all four tasks. Applicants who passed both the third and fourth tasks were admitted to the institute. How many applicants were admitted? | 57 | 7/8 |
What three-digit integer is equal to the sum of the factorials of its digits, where one of the digits is `3`, contributing `3! = 6` to the sum? | 145 | 1/8 |
It is necessary to erect some public welfare billboards on one side of a road. The first billboard is erected at the beginning of the road, and then one billboard is erected every 5 meters, so that exactly one billboard can be erected at the end of the road. In this case, there are 21 billboards missing. If one billboard is erected every 5.5 meters, also exactly one billboard can be erected at the end of the road, in this case, there is only 1 billboard missing. Then, there are billboards in total, and the length of the road is meters. | 1100 | 5/8 |
Suppose $f:\mathbb{R} \to \mathbb{R}$ is a function given by $$ f(x) =\begin{cases} 1 & \mbox{if} \ x=1 e^{(x^{10}-1)}+(x-1)^2\sin\frac1{x-1} & \mbox{if} \ x\neq 1\end{cases} $$ (a) Find $f'(1)$ (b) Evaluate $\displaystyle \lim_{u\to\infty} \left[100u-u\sum_{k=1}^{100} f\left(1+\frac{k}{u}\right)\right]$ . | -50500 | 7/8 |
Let $n$ be a given positive integer. How many solutions are there in ordered positive integer pairs $(x,y)$ to the equation $$ \frac{xy}{x+y}=n? $$ | (n^2) | 7/8 |
Determine all pairs of positive integers \((n, p)\) that satisfy:
- \(p\) is a prime,
- \(n \leq 2p\), and
- \((p-1)^n + 1\) is divisible by \(n^{p-1}\). | (3,3) | 2/8 |
Let $a_{1}=1$, and let $a_{n}=\left\lfloor n^{3} / a_{n-1}\right\rfloor$ for $n>1$. Determine the value of $a_{999}$. | 999 | 5/8 |
During an underwater archaeological activity, a diver needs to dive $50$ meters to the bottom of the water for archaeological work. The oxygen consumption consists of the following three aspects: $(1)$ The average diving speed is $x$ meters/minute, and the oxygen consumption per minute is $\frac{1}{100}x^{2}$ liters; $(2)$ The working time at the bottom of the water is between $10$ and $20$ minutes, and the oxygen consumption per minute is $0.3$ liters; $(3)$ When returning to the water surface, the average speed is $\frac{1}{2}x$ meters/minute, and the oxygen consumption per minute is $0.32$ liters. The total oxygen consumption of the diver in this archaeological activity is $y$ liters.
$(1)$ If the working time at the bottom of the water is $10$ minutes, express $y$ as a function of $x$;
$(2)$ If $x \in [6, 10]$, the working time at the bottom of the water is $20$ minutes, find the range of total oxygen consumption $y$;
$(3)$ If the diver carries $13.5$ liters of oxygen, how many minutes can the diver stay underwater at most (round to the nearest whole number)? | 18 | 1/8 |
Read the following material: Expressing a fraction as the sum of two fractions is called expressing the fraction as "partial fractions."<br/>Example: Express the fraction $\frac{{1-3x}}{{{x^2}-1}}$ as partial fractions. Solution: Let $\frac{{1-3x}}{{{x^2}-1}}=\frac{M}{{x+1}}+\frac{N}{{x-1}}$, cross multiply on the right side of the equation, we get $\frac{{M(x-1)+N(x+1)}}{{(x+1)(x-1)}}=\frac{{(M+N)x+(N-M)}}{{{x^2}-1}}$. According to the question, we have $\left\{\begin{array}{l}M+N=3\\ N-M=1\end{array}\right.$, solving this system gives $\left\{\begin{array}{l}M=-2\\ N=-1\end{array}\right.$, so $\frac{{1-3x}}{{{x^2}-1}}=\frac{{-2}}{{x+1}}+\frac{{-1}}{{x-1}}$. Please use the method learned above to solve the following problems:<br/>$(1)$ Express the fraction $\frac{{2n+1}}{{{n^2}+n}}$ as partial fractions;<br/>$(2)$ Following the pattern in (1), find the value of $\frac{3}{{1×2}}-\frac{5}{{2×3}}+\frac{7}{{3×4}}-\frac{9}{{4×5}}+⋯+\frac{{39}}{{19×20}}-\frac{{41}}{{20×21}}$. | \frac{20}{21} | 7/8 |
Compute \[ \left\lfloor \dfrac {2005^3}{2003 \cdot 2004} - \dfrac {2003^3}{2004 \cdot 2005} \right\rfloor,\]where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x.$ | 8 | 7/8 |
[b]Problem Section #1
a) A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are $189, 320, 287, 264, x$ , and y. Find the greatest possible value of: $x + y$ .
<span style="color:red">NOTE: There is a high chance that this problems was copied.</span> | 761 | 3/8 |
In the Cartesian coordinate system xOy, the equation of line l is given as x+1=0, and curve C is a parabola with the coordinate origin O as the vertex and line l as the axis. Establish a polar coordinate system with the coordinate origin O as the pole and the non-negative semi-axis of the x-axis as the polar axis.
1. Find the polar coordinate equations for line l and curve C respectively.
2. Point A is a moving point on curve C in the first quadrant, and point B is a moving point on line l in the second quadrant. If ∠AOB=$$\frac {π}{4}$$, find the maximum value of $$\frac {|OA|}{|OB|}$$. | \frac { \sqrt {2}}{2} | 1/8 |
Alice thinks of four positive integers $a \leq b \leq c \leq d$ satisfying $\{a b+c d, a c+b d, a d+b c\}=\{40,70,100\}$. What are all the possible tuples $(a, b, c, d)$ that Alice could be thinking of? | (1,4,6,16) | 1/8 |
Anna and Berta are playing a game in which they alternately take marbles from a table. Anna makes the first move. If at the beginning of a turn there are $n \geq 1$ marbles on the table, the player whose turn it is takes $k$ marbles, where $k \geq 1$ is either an even number with $k \leq \frac{n}{2}$ or an odd number with $\frac{n}{2} \leq k \leq n$. A player wins the game if they take the last marble from the table.
Determine the smallest number $N \geq 100000$ such that Berta can force a win if there are initially exactly $N$ marbles on the table.
(Gerhard Woeginger) | 131070 | 3/8 |
The sum of the largest number and the smallest number of a triple of positive integers $(x,y,z)$ is the power of the triple. Compute the sum of powers of all triples $(x,y,z)$ where $x,y,z \leq 9$. | 7290 | 7/8 |
Find the least integer value of $x$ for which $3|x| - 2 > 13$. | -6 | 3/8 |
Inside a circle with a radius of 15 cm, a point \( M \) is taken at a distance of 13 cm from the center. A chord is drawn through the point \( M \) with a length of 18 cm. Find the lengths of the segments into which the point \( M \) divides the chord. | 14 | 1/8 |
Find the minimum positive integer $n\ge 3$, such that there exist $n$ points $A_1,A_2,\cdots, A_n$ satisfying no three points are collinear and for any $1\le i\le n$, there exist $1\le j \le n (j\neq i)$, segment $A_jA_{j+1}$ pass through the midpoint of segment $A_iA_{i+1}$, where $A_{n+1}=A_1$ | 6 | 1/8 |
There are 40 identical gas cylinders with unknown and potentially different gas pressures. It is permitted to connect any number of cylinders together, not exceeding a given natural number $k$, and then disconnect them; during the connection, the gas pressure in the connected cylinders becomes equal to the arithmetic mean of their pressures before the connection. What is the smallest $k$ for which there exists a way to equalize the pressures in all 40 cylinders, regardless of the initial distribution of pressures in the cylinders? | 5 | 1/8 |
The product of two positive integers $m$ and $n$ is divisible by their sum. Prove that $m + n \le n^2$ .
(Boris Frenkin) | +n\len^2 | 7/8 |
A sphere is inscribed in a right cone with base radius \(15\) cm and height \(30\) cm. Determine the radius of the sphere if it can be expressed as \(b\sqrt{d} - b\) cm. What is the value of \(b + d\)? | 12.5 | 1/8 |
Solve the equations:
(1) 2x^2 - 5x + 1 = 0;
(2) 3x(x - 2) = 2(2 - x). | -\frac{2}{3} | 2/8 |
Given the complex number \( z \) satisfies
\[
|z-1| = |z-\mathrm{i}|.
\]
If \( z - \frac{z-6}{z-1} \) is a positive real number, then \( z = \) ______. | 2+2i | 7/8 |
Consider an isosceles triangle $T$ with base 10 and height 12. Define a sequence $\omega_{1}, \omega_{2}, \ldots$ of circles such that $\omega_{1}$ is the incircle of $T$ and $\omega_{i+1}$ is tangent to $\omega_{i}$ and both legs of the isosceles triangle for $i>1$. Find the ratio of the radius of $\omega_{i+1}$ to the radius of $\omega_{i}$. | \frac{4}{9} | 7/8 |
Compute
\[\frac{\lfloor \sqrt{1} \rfloor \cdot \lfloor \sqrt{2} \rfloor \cdot \lfloor \sqrt{3} \rfloor \cdot \lfloor \sqrt{5} \rfloor \dotsm \lfloor \sqrt{15} \rfloor}{\lfloor \sqrt{2} \rfloor \cdot \lfloor \sqrt{4} \rfloor \cdot \lfloor \sqrt{6} \rfloor \dotsm \lfloor \sqrt{16} \rfloor}.\] | \frac{3}{8} | 5/8 |
Prove that the graphs of the quadratic polynomials \( y = ax^2 - bx + c \), \( y = bx^2 - cx + a \), and \( y = cx^2 - ax + b \) have a common point. | (-1,) | 5/8 |
Altitudes $\overline{AX}$ and $\overline{BY}$ of acute triangle $ABC$ intersect at $H$. If $\angle BAC = 61^\circ$ and $\angle ABC = 73^\circ$, then what is $\angle CHX$? | 73^\circ | 6/8 |
From a single point, two tangents are drawn to a circle. The length of each tangent is 12, and the distance between the points of tangency is 14.4. Find the radius of the circle. | 9 | 6/8 |
The given quadratic polynomial \( f(x) \) has two distinct roots. Can it happen that the equation \( f(f(x)) = 0 \) has three distinct roots, while the equation \( f(f(f(x))) = 0 \) has seven distinct roots? | No | 1/8 |
For how many integer values of $a$ does the equation $$x^2 + ax + 12a = 0$$ have integer solutions for $x$? | 16 | 5/8 |
Given that \(MN\) is a moving chord of the circumscribed circle of an equilateral triangle \( \triangle ABC \) with side length \( 2\sqrt{6} \), and \(MN = 4\), \(P\) is a moving point on the sides of \( \triangle ABC \). Find the maximum value of \( \overrightarrow{MP} \cdot \overrightarrow{PN} \). | 4 | 1/8 |
What is the maximum number of 1x1 squares that can be placed next to a given unit square \( K \) such that no two of these squares overlap? | 8 | 3/8 |
Points $A, B, C$ lie on a circle \omega such that $B C$ is a diameter. $A B$ is extended past $B$ to point $B^{\prime}$ and $A C$ is extended past $C$ to point $C^{\prime}$ such that line $B^{\prime} C^{\prime}$ is parallel to $B C$ and tangent to \omega at point $D$. If $B^{\prime} D=4$ and $C^{\prime} D=6$, compute $B C$. | \frac{24}{5} | 5/8 |
If \(\frac{5+6+7+8}{4} = \frac{2014+2015+2016+2017}{N}\), calculate the value of \(N\). | 1240 | 1/8 |
Given that $\sin 2α - 2 = 2 \cos 2α$, find the value of ${\sin}^{2}α + \sin 2α$. | \frac{8}{5} | 6/8 |
Given that $\dfrac{\pi}{2} < \alpha < \beta < \dfrac{3\pi}{4}, \cos(\alpha - \beta) = \dfrac{12}{13}, \sin(\alpha + \beta) = -\dfrac{3}{5}$, find the value of $\sin 2\alpha$. | -\dfrac{56}{65} | 1/8 |
Let \(\{a_{n}\}\) be a geometric sequence with each term greater than 1, then the value of \(\lg a_{1} \lg a_{2012} \sum_{i=1}^{20111} \frac{1}{\lg a_{i} \lg a_{i+1}}\) is ________ . | 2011 | 7/8 |
Find all positive integers \( n \) for which we can find one or more integers \( m_1, m_2, ..., m_k \), each at least 4, such that:
1. \( n = m_1 m_2 ... m_k \)
2. \( n = 2^M - 1 \), where \( M = \frac{(m_1 - 1)(m_2 - 1) ... (m_k - 1)}{2^k} \) | 7 | 2/8 |
For any set $A = \{x_1, x_2, x_3, x_4, x_5\}$ of five distinct positive integers denote by $S_A$ the sum of its elements, and denote by $T_A$ the number of triples $(i, j, k)$ with $1 \le i < j < k \le 5$ for which $x_i + x_j + x_k$ divides $S_A$ .
Find the largest possible value of $T_A$ . | 4 | 1/8 |
What is the smallest positive integer $n$ for which $11n - 3$ and $8n + 2$ share a common factor greater than $1$? | 19 | 1/8 |
What is the maximum number of lattice points (i.e. points with integer coordinates) in the plane that can be contained strictly inside a circle of radius 1? | 4 | 2/8 |
We wrote an even number in binary. By removing the trailing $0$ from this binary representation, we obtain the ternary representation of the same number. Determine the number! | 10 | 1/8 |
Let the sum $\sum_{n=1}^{9} \frac{1}{n(n+1)(n+2)}$ written in its lowest terms be $\frac{p}{q}$ . Find the value of $q - p$ . | 83 | 4/8 |
What is the maximum number of distinct numbers from 1 to 1000 that can be selected so that the difference between any two selected numbers is not equal to 4, 5, or 6? | 400 | 1/8 |
January 3, 1911 was an odd date as its abbreviated representation, \(1/3/1911\), can be written using only odd digits (note all four digits are written for the year). To the nearest month, how many months will have elapsed between the most recent odd date and the next odd date (today is \(3/3/2001\), an even date)? | 13333 | 1/8 |
What percent of the palindromes between 1000 and 2000 contain at least one 3 or 5, except in the first digit? | 36\% | 2/8 |
For some integer $m$, the polynomial $x^3 - 2011x + m$ has the three integer roots $a$, $b$, and $c$. Find $|a| + |b| + |c|$. | 98 | 6/8 |
Let $n$ be a positive integer and let $d_{1},d_{2},,\ldots ,d_{k}$ be its divisors, such that $1=d_{1}<d_{2}<\ldots <d_{k}=n$ . Find all values of $n$ for which $k\geq 4$ and $n=d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}$ . | 130 | 1/8 |
Carolina has a box of 30 matches. She uses matchsticks to form the number 2022. How many matchsticks will be left in the box when she has finished?
A) 20
B) 19
C) 10
D) 9
E) 5 | 9 | 7/8 |
The wristwatch is 5 minutes slow per hour; 5.5 hours ago, it was set to the correct time. It is currently 1 PM on a clock that shows the correct time. How many minutes will it take for the wristwatch to show 1 PM? | 30 | 6/8 |
Let
$$
\begin{array}{c}
A=\left(\binom{2010}{0}-\binom{2010}{-1}\right)^{2}+\left(\binom{2010}{1}-\binom{2010}{0}\right)^{2}+\left(\binom{2010}{2}-\binom{2010}{1}\right)^{2} \\
+\cdots+\left(\binom{2010}{1005}-\binom{2010}{1004}\right)^{2}
\end{array}
$$
Determine the minimum integer \( s \) such that
$$
s A \geq \binom{4020}{2010}
$$ | 2011 | 1/8 |
There are six clearly distinguishable frogs sitting in a row. Two are green, three are red, and one is blue. Green frogs refuse to sit next to the red frogs, for they are highly poisonous. In how many ways can the frogs be arranged? | 24 | 1/8 |
If 1983 were expressed as a sum of distinct powers of 2, with the requirement that at least five distinct powers of 2 are used, what would be the least possible sum of the exponents of these powers? | 55 | 1/8 |
The hare and the tortoise had a race over 100 meters, in which both maintained constant speeds. When the hare reached the finish line, it was 75 meters in front of the tortoise. The hare immediately turned around and ran back towards the start line. How far from the finish line did the hare and the tortoise meet? | 60 | 7/8 |
Given that a person can click four times in sequence and receive one of three types of red packets each time, with the order of appearance corresponding to different prize rankings, calculate the number of different prize rankings that can be obtained if all three types of red packets are collected in any order before the fourth click. | 18 | 5/8 |
In the arithmetic sequence $\{a_n\}$, $a_1 > 0$, and $S_n$ is the sum of the first $n$ terms, and $S_9=S_{18}$, find the value of $n$ at which $S_n$ is maximized. | 13 | 2/8 |
Vasya has three cans of paint of different colors. In how many different ways can he paint a fence of 10 boards such that any two adjacent boards are of different colors, and all three colors are used? | 1530 | 5/8 |
John scores 93 on this year's AHSME. Had the old scoring system still been in effect, he would score only 84 for the same answers.
How many questions does he leave unanswered? | 9 | 1/8 |
Forty-two cards are labeled with the natural numbers 1 through 42 and randomly shuffled into a stack. One by one, cards are taken off the top of the stack until a card labeled with a prime number is removed. How many cards are removed on average? | \frac{43}{14} | 7/8 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.