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Let $p, q, r$ be primes and let $n$ be a positive integer such that $p^n + q^n = r^2$ . Prove that $n = 1$ .
Laurentiu Panaitopol | 1 | 6/8 |
Quadrilateral $ALEX,$ pictured below (but not necessarily to scale!)
can be inscribed in a circle; with $\angle LAX = 20^{\circ}$ and $\angle AXE = 100^{\circ}:$ | 80 | 2/8 |
Initially, a natural number \( N \) is written on the board. At any moment, Misha can choose a number \( a > 1 \) on the board, erase it, and write all natural divisors of \( a \) except \( a \) itself (the same number can appear multiple times on the board). After some time, it turns out that \( N^{2} \) numbers are written on the board. For which \( N \) could this happen? | 1 | 1/8 |
A fair six-sided die has faces numbered $1, 2, 3, 4, 5, 6$. The die is rolled four times, and the results are $a, b, c, d$. What is the probability that one of the numbers in the set $\{a, a+b, a+b+c, a+b+c+d\}$ is equal to 4? | $\frac{343}{1296}$ | 5/8 |
Consider a hyperbola with the equation $x^2 - y^2 = 9$. A line passing through the left focus $F_1$ of the hyperbola intersects the left branch of the hyperbola at points $P$ and $Q$. Let $F_2$ be the right focus of the hyperbola. If the length of segment $PQ$ is 7, then calculate the perimeter of $\triangle F_2PQ$. | 26 | 4/8 |
Find the sum of all prime numbers whose representation in base 14 has the form $101010...101$ (alternating ones and zeros). | 197 | 4/8 |
First, factorize 42 and 30 into prime factors, then answer the following questions:
(1) 42= , 30= .
(2) The common prime factors of 42 and 30 are .
(3) The unique prime factors of 42 and 30 are .
(4) The greatest common divisor (GCD) of 42 and 30 is .
(5) The least common multiple (LCM) of 42 and 30 is .
(6) From the answers above, you can conclude that . | 210 | 2/8 |
Given the sequence $\left\{a_{n}\right\}$ satisfying:
$$
a_{0}=\sqrt{6}, \quad a_{n+1}=\left[a_{n}\right]+\frac{1}{\left\{a_{n}\right\}} \text {, }
$$
where $[a]$ denotes the greatest integer less than or equal to the real number $a$, and $\{a\}=a-[a]$. Determine the value of $a_{2020}$. | 6060+\sqrt{6} | 2/8 |
Suppose $ABCD$ is a trapezoid with $AB\parallel CD$ and $AB\perp BC$ . Let $X$ be a point on segment $\overline{AD}$ such that $AD$ bisects $\angle BXC$ externally, and denote $Y$ as the intersection of $AC$ and $BD$ . If $AB=10$ and $CD=15$ , compute the maximum possible value of $XY$ . | 6 | 2/8 |
For how many integers \( n \) is \(\frac{2n^3 - 12n^2 - 2n + 12}{n^2 + 5n - 6}\) equal to an integer? | 32 | 5/8 |
Right triangle $ABC$ has leg lengths $AB=20$ and $BC=21$. Including $\overline{AB}$ and $\overline{BC}$, how many line segments with integer length can be drawn from vertex $B$ to a point on hypotenuse $\overline{AC}$? | 12 | 1/8 |
Point \( C \) moves along a line segment \( AB \) of length 4. There are two isosceles triangles \( ACD \) and \( BEC \) on the same side of the line passing through \( AB \) such that \( AD = DC \) and \( CE = EB \). Find the minimum length of segment \( DE \). | 2 | 5/8 |
ABCDE is a regular pentagon. The star ACEBD has an area of 1. AC and BE meet at P, BD and CE meet at Q. Find the area of APQD. | 1/2 | 1/8 |
Three young married couples were captured by cannibals. Before eating the tourists, the cannibals decided to weigh them. The total weight of all six people was not an integer, but the combined weight of all the wives was exactly 171 kg. Leon weighed the same as his wife, Victor weighed one and a half times more than his wife, and Maurice weighed twice as much as his wife. Georgette weighed 10 kg more than Simone, who weighed 5 kg less than Elizabeth. While the cannibals argued over who to eat first, five of the six young people managed to escape. The cannibals only ate Elizabeth's husband. How much did he weigh? | 85.5 | 5/8 |
Two identical rulers are placed together. Each ruler is exactly 10 cm long and is marked in centimeters from 0 to 10. The 3 cm mark on each ruler is aligned with the 4 cm mark on the other. The overall length is \( L \) cm. What is the value of \( L \)? | 13 | 1/8 |
In a right-angled triangle $LMN$, suppose $\sin N = \frac{5}{13}$ with $LM = 10$. Calculate the length of $LN$. | 26 | 4/8 |
The function \( f(n) \) is an integer-valued function defined on the integers which satisfies \( f(m + f(f(n))) = -f(f(m+1)) - n \) for all integers \( m \) and \( n \). The polynomial \( g(n) \) has integer coefficients and satisfies \( g(n) = g(f(n)) \) for all \( n \). Find \( f(1991) \) and determine the most general form for \( g \). | -1992 | 7/8 |
Given the sequence $\{a_{n}\}$ satisfying $a_{1}=1$, $a_{2}=4$, $a_{n}+a_{n+2}=2a_{n+1}+2$, find the sum of the first 2022 terms of the sequence $\{b_{n}\}$, where $\left[x\right)$ is the smallest integer greater than $x$ and $b_n = \left[\frac{n(n+1)}{a_n}\right)$. | 4045 | 1/8 |
A villager A and a motorcycle with a passenger, villager B, set off at the same time from the village to the station along a single road. Before reaching the station, the motorcyclist dropped off villager B and immediately went back towards the village, while villager B walked to the station. Upon meeting villager A, the motorcyclist picked him up and took him to the station. As a result, both villagers arrived at the station simultaneously. What fraction of the journey from the village to the station did villager A travel by motorcycle, considering that both villagers walked at the same speed, which was 9 times slower than the speed of the motorcycle? | \frac{5}{6} | 2/8 |
In the plane Cartesian coordinate system \(xOy\), point \(P\) is a moving point on the line \(y = -x - 2\). Two tangents to the parabola \(y = \frac{x^2}{2}\) are drawn through point \(P\), and the points of tangency are \(A\) and \(B\). Find the minimum area of the triangle \(PAB\). | 3\sqrt{3} | 6/8 |
Let $A B C D$ be a rectangle with $A B=8$ and $A D=20$. Two circles of radius 5 are drawn with centers in the interior of the rectangle - one tangent to $A B$ and $A D$, and the other passing through both $C$ and $D$. What is the area inside the rectangle and outside of both circles? | 112-25 \pi | 1/8 |
The Fibonacci numbers are defined by $F_{0}=0, F_{1}=1$, and $F_{n}=F_{n-1}+F_{n-2}$ for $n \geq 2$. There exist unique positive integers $n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}$ such that $\sum_{i_{1}=0}^{100} \sum_{i_{2}=0}^{100} \sum_{i_{3}=0}^{100} \sum_{i_{4}=0}^{100} \sum_{i_{5}=0}^{100} F_{i_{1}+i_{2}+i_{3}+i_{4}+i_{5}}=F_{n_{1}}-5 F_{n_{2}}+10 F_{n_{3}}-10 F_{n_{4}}+5 F_{n_{5}}-F_{n_{6}}$. Find $n_{1}+n_{2}+n_{3}+n_{4}+n_{5}+n_{6}$. | 1545 | 1/8 |
To celebrate the $20$ th LMT, the LHSMath Team bakes a cake. Each of the $n$ bakers places $20$ candles on the cake. When they count, they realize that there are $(n -1)!$ total candles on the cake. Find $n$ .
*Proposed by Christopher Cheng* | 6 | 5/8 |
Let \( S = \{1, 2, \cdots, n\} \), and let \( A \) be an arithmetic sequence with a positive common difference, containing at least two terms, all of which are elements of \( S \). Furthermore, adding any other elements from \( S \) to \( A \) must not result in an arithmetic sequence with the same common difference as \( A \). Find the number of such sequences \( A \). (An arithmetic sequence with only two terms is also considered an arithmetic sequence). | \lfloor\frac{n^2}{4}\rfloor | 2/8 |
Nine skiers started sequentially and covered the distance, each at their own constant speed. Could it be that each skier participated in exactly four overtakes? (In each overtake, exactly two skiers are involved - the one who overtakes and the one who is being overtaken.) | No | 2/8 |
Let $n$ be a nonnegative integer less than $2023$ such that $2n^2 + 3n$ is a perfect square. What is the sum of all possible $n$ ?
*Proposed by Giacomo Rizzo* | 444 | 5/8 |
A cylinder of base radius 1 is cut into two equal parts along a plane passing through the center of the cylinder and tangent to the two base circles. Suppose that each piece's surface area is \( m \) times its volume. Find the greatest lower bound for all possible values of \( m \) as the height of the cylinder varies. | 3 | 2/8 |
A local community group sells 180 event tickets for a total of $2652. Some tickets are sold at full price, while others are sold at a discounted rate of half price. Determine the total revenue generated from the full-price tickets.
A) $960
B) $984
C) $1008
D) $1032 | 984 | 2/8 |
Four identical small rectangles are put together to form a large rectangle. The length of a shorter side of each small rectangle is 10 cm. What is the length of a longer side of the large rectangle?
A) 50 cm
B) 40 cm
C) 30 cm
D) 20 cm
E) 10 cm | 40\, | 1/8 |
Given the vectors $\overrightarrow{m}=(\cos x,\sin x)$ and $\overrightarrow{n}=(2 \sqrt {2}+\sin x,2 \sqrt {2}-\cos x)$, and the function $f(x)= \overrightarrow{m}\cdot \overrightarrow{n}$, where $x\in R$.
(I) Find the maximum value of the function $f(x)$;
(II) If $x\in(-\frac {3π}{2},-π)$ and $f(x)=1$, find the value of $\cos (x+\frac {5π}{12})$. | -\frac {3 \sqrt {5}+1}{8} | 1/8 |
There is a set of points \( M \) on a plane and seven different circles \( C_{1}, C_{2}, \cdots, C_{7} \). Circle \( C_{7} \) passes through exactly 7 points in \( M \), circle \( C_{6} \) passes through exactly 6 points in \( M \), and so on, with circle \( C_{1} \) passing through exactly 1 point in \( M \). Determine the minimum number of points in set \( M \). | 12 | 2/8 |
How many multiples of 7 between $10^{6}$ and $10^{9}$ are perfect squares? | 4375 | 7/8 |
Let $\mathcal{P}$ be a parabola with focus $F$ and directrix $\ell$. A line through $F$ intersects $\mathcal{P}$ at two points $A$ and $B$. Let $D$ and $C$ be the feet of the altitudes from $A$ and $B$ onto $\ell$, respectively. Given that $AB=20$ and $CD=14$, compute the area of $ABCD$. | 140 | 3/8 |
In a game, Jimmy and Jacob each randomly choose to either roll a fair six-sided die or to automatically roll a $1$ on their die. If the product of the two numbers face up on their dice is even, Jimmy wins the game. Otherwise, Jacob wins. The probability Jimmy wins $3$ games before Jacob wins $3$ games can be written as $\tfrac{p}{2^q}$ , where $p$ and $q$ are positive integers, and $p$ is odd. Find the remainder when $p+q$ is divided by $1000$ .
*Proposed by firebolt360* | 360 | 7/8 |
Given an equilateral triangle $\mathrm{ABC}$. Right isosceles triangles ABP and BCQ with right angles at $\angle \mathrm{ABP}$ and $\angle \mathrm{BCQ}$ are constructed externally on sides $\mathrm{AB}$ and $\mathrm{BC}$, respectively. Find the angle $\angle \mathrm{PAQ}$. | 90 | 5/8 |
Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures 3 by 4 by 5 units. Given that the volume of this set is $\displaystyle {{m+n\pi}\over
p}$, where $m$, $n$, and $p$ are positive integers, and $n$ and $p$ are relatively prime, find $m+n+p$. | 505 | 7/8 |
Use inversion to prove: In triangle ABC, the incenter and circumcenter are I and O, respectively, and the radii of the incircle and circumcircle are r and R, respectively. Then, \( IO^2 = R^2 - 2Rr \). | IO^2=R^2-2Rr | 1/8 |
Find the set of pairs of real numbers \((x, y)\) that satisfy the conditions:
$$
\left\{
\begin{array}{l}
3^{-x} y^{4}-2 y^{2}+3^{x} \leq 0 \\
27^{x}+y^{4}-3^{x}-1=0
\end{array}
\right.
$$
Compute the values of the expression \(x_{k}^{3}+y_{k}^{3}\) for each solution \((x_{k}, y_{k})\) of the system and find the minimum among them. In the answer, specify the found minimum value, if necessary rounding it to two decimal places. If the original system has no solutions, write the digit 0 in the answer field. | -1 | 7/8 |
Given the sequence of positive integers \(\left\{a_{n}\right\}\) defined by \(a_{0}=m\) and \(a_{n+1}=a_{n}^{5}+487\) for \(n \geqslant 0\), find the value of \(m\) such that the number of perfect squares in the sequence \(\left\{a_{n}\right\}\) is maximized. | 9 | 5/8 |
If the radius of a circle is a rational number, its area is given by a number which is:
$\textbf{(A)\ } \text{rational} \qquad \textbf{(B)\ } \text{irrational} \qquad \textbf{(C)\ } \text{integral} \qquad \textbf{(D)\ } \text{a perfect square }\qquad \textbf{(E)\ } \text{none of these}$ | \textbf{(B)\} | 1/8 |
Compute the definite integral:
$$
\int_{\pi / 2}^{\pi} 2^{4} \cdot \sin ^{6} x \cos ^{2} x \, dx
$$ | \frac{5\pi}{16} | 5/8 |
A neighbor bought a certain quantity of beef at two shillings a pound, and the same quantity of sausages at eighteenpence a pound. If she had divided the same money equally between beef and sausages, she would have gained two pounds in the total weight. Determine the exact amount of money she spent. | 168 | 1/8 |
Determine all prime numbers \( p \) such that
$$
5^{p} + 4 \cdot p^{4}
$$
is a perfect square, i.e., the square of an integer. | 5 | 4/8 |
The Crackhams were supposed to make their first stop in Baglmintster and spend the night at a family friend's house. This family friend in turn was to leave home at the same time as them and stop in London at the Crackhams' house. Both the Crackhams and the friend were traveling on the same road, looking out for each other, and they met 40 km from Baglmintster. That same evening, George came up with the following small puzzle:
- I realized that if, upon arrival at our destinations, each of our cars immediately turned back and set off in the opposite direction, we would meet 48 km from London.
If George is correct, what is the distance from London to Baglmintster? | 72 | 3/8 |
Players A and B have a Go game match, agreeing that the first to win 3 games wins the match. After the match ends, assuming in a single game, the probability of A winning is 0.6, and the probability of B winning is 0.4, with the results of each game being independent. It is known that in the first 2 games, A and B each won 1 game.
(I) Calculate the probability of A winning the match;
(II) Let $\xi$ represent the number of games played from the 3rd game until the end of the match, calculate the distribution and the mathematical expectation of $\xi$. | 2.48 | 3/8 |
Rectangle \(ABCD\) has area 2016. Point \(Z\) is inside the rectangle and point \(H\) is on \(AB\) so that \(ZH\) is perpendicular to \(AB\). If \(ZH : CB = 4 : 7\), what is the area of pentagon \(ADCZB\)? | 1440 | 2/8 |
Let the number $x$ . Using multiply and division operations of any 2 given or already given numbers we can obtain powers with natural exponent of the number $x$ (for example, $x\cdot x=x^{2}$ , $x^{2}\cdot x^{2}=x^{4}$ , $x^{4}: x=x^{3}$ , etc). Determine the minimal number of operations needed for calculating $x^{2006}$ . | 17 | 1/8 |
A certain high school has three mathematics teachers. For the convenience of the students, they arrange for a math teacher to be on duty every day from Monday to Friday, and two teachers are scheduled to be on duty on Monday. If each teacher is on duty for two days per week, there are ________ possible duty arrangements for the week. | 36 | 3/8 |
It is known that the number 400000001 is the product of two prime numbers \( p \) and \( q \). Find the sum of the natural divisors of the number \( p+q-1 \). | 45864 | 4/8 |
For points \( A_{1}, \ldots, A_{5} \) on the sphere of radius 1, what is the maximum value that \( \min _{1 \leq i, j \leq 5} A_{i} A_{j} \) can take? Determine all configurations for which this maximum is attained. | \sqrt{2} | 4/8 |
Find all positive integer solutions \((x, y, z, t)\) to the equation \(2^{y} + 2^{z} \times 5^{t} - 5^{x} = 1\). | (2,4,1,1) | 4/8 |
The sum of the squares of the first ten binomial coefficients ${C}_{2}^{2}+{C}_{3}^{2}+{C}_{4}^{2}+\cdots +{C}_{10}^{2}$ can be found. | 165 | 4/8 |
There are 7 parking spaces in a row in a parking lot, and now 4 cars need to be parked. If 3 empty spaces need to be together, calculate the number of different parking methods. | 120 | 7/8 |
The adjoining figure shows two intersecting chords in a circle, with $B$ on minor arc $AD$. Suppose that the radius of the circle is $5$, that $BC=6$, and that $AD$ is bisected by $BC$. Suppose further that $AD$ is the only chord starting at $A$ which is bisected by $BC$. It follows that the sine of the central angle of minor arc $AB$ is a rational number. If this number is expressed as a fraction $\frac{m}{n}$ in lowest terms, what is the product $mn$?
[asy]size(100); defaultpen(linewidth(.8pt)+fontsize(11pt)); dotfactor=1; pair O1=(0,0); pair A=(-0.91,-0.41); pair B=(-0.99,0.13); pair C=(0.688,0.728); pair D=(-0.25,0.97); path C1=Circle(O1,1); draw(C1); label("$A$",A,W); label("$B$",B,W); label("$C$",C,NE); label("$D$",D,N); draw(A--D); draw(B--C); pair F=intersectionpoint(A--D,B--C); add(pathticks(A--F,1,0.5,0,3.5)); add(pathticks(F--D,1,0.5,0,3.5)); [/asy] | 175 | 2/8 |
A beautiful maiden with radiant eyes told me a number. If this number is multiplied by 3, then $\frac{3}{4}$ of the product is added, the result is divided by 7, reduced by $\frac{1}{3}$ of the quotient, multiplied by itself, reduced by 52, the square root is extracted, 8 is added, and then divided by 10, the result will be 2. | 28 | 7/8 |
Given the function $f(x)=4\cos(3x+\phi)(|\phi|<\frac{\pi}{2})$, its graph is symmetrical about the line $x=\frac{11\pi}{12}$. When $x_1,x_2\in(-\frac{7\pi}{12},-\frac{\pi}{12})$, $x_1\neq x_2$, and $f(x_1)=f(x_2)$, find $f(x_1+x_2)$. | 2\sqrt{2} | 6/8 |
In the diagram below, $ABCD$ is a rectangle with side lengths $AB=3$ and $BC=11$, and $AECF$ is a rectangle with side lengths $AF=7$ and $FC=9,$ as shown. The area of the shaded region common to the interiors of both rectangles is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [asy] pair A, B, C, D, E, F; A=(0,3); B=(0,0); C=(11,0); D=(11,3); E=foot(C, A, (9/4,0)); F=foot(A, C, (35/4,3)); draw(A--B--C--D--cycle); draw(A--E--C--F--cycle); filldraw(A--(9/4,0)--C--(35/4,3)--cycle,gray*0.5+0.5*lightgray); dot(A^^B^^C^^D^^E^^F); label("$A$", A, W); label("$B$", B, W); label("$C$", C, (1,0)); label("$D$", D, (1,0)); label("$F$", F, N); label("$E$", E, S); [/asy] | 109 | 1/8 |
Tetrahedron $A B C D$ has side lengths $A B=6, B D=6 \sqrt{2}, B C=10, A C=8, C D=10$, and $A D=6$. The distance from vertex $A$ to face $B C D$ can be written as $\frac{a \sqrt{b}}{c}$, where $a, b, c$ are positive integers, $b$ is square-free, and $\operatorname{gcd}(a, c)=1$. Find $100 a+10 b+c$. | 2851 | 6/8 |
Given two lines $l_{1}$: $x+my+6=0$, and $l_{2}$: $(m-2)x+3y+2m=0$, if the lines $l_{1}\parallel l_{2}$, then $m=$_______. | -1 | 5/8 |
The total number of toothpicks used to build a rectangular grid 15 toothpicks high and 12 toothpicks wide, with internal diagonal toothpicks, is calculated by finding the sum of the toothpicks. | 567 | 1/8 |
Suppose $a$, $b,$ and $c$ are positive numbers satisfying: \begin{align*}
a^2/b &= 1, \\
b^2/c &= 2, \text{ and}\\
c^2/a &= 3.
\end{align*} Find $a$. | 12^{1/7} | 1/8 |
Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively in $\triangle ABC$, and vectors $\overrightarrow{m}=(1-\cos (A+B),\cos \frac {A-B}{2})$ and $\overrightarrow{n}=( \frac {5}{8},\cos \frac {A-B}{2})$ with $\overrightarrow{m}\cdot \overrightarrow{n}= \frac {9}{8}$,
(1) Find the value of $\tan A\cdot\tan B$;
(2) Find the maximum value of $\frac {ab\sin C}{a^{2}+b^{2}-c^{2}}$. | -\frac {3}{8} | 4/8 |
The first and twentieth terms of an arithmetic sequence are 3 and 63, respectively. What is the fortieth term? | 126 | 1/8 |
Let $ABCDE$ be a convex pentagon, and let $G_A, G_B, G_C, G_D, G_E$ denote the centroids of triangles $BCDE, ACDE, ABDE, ABCE, ABCD$, respectively. Find the ratio $\frac{[G_A G_B G_C G_D G_E]}{[ABCDE]}$. | \frac{1}{16} | 3/8 |
In the plane quadrilateral \(ABCD\), points \(E\) and \(F\) are the midpoints of sides \(AD\) and \(BC\) respectively. Given that \(AB = 1\), \(EF = \sqrt{2}\), and \(CD = 3\), and that \(\overrightarrow{AD} \cdot \overrightarrow{BC} = 15\), find \(\overrightarrow{AC} \cdot \overrightarrow{BD}\). | 16 | 6/8 |
The numerator of a fraction is $6x + 1$, then denominator is $7 - 4x$, and $x$ can have any value between $-2$ and $2$, both included. The values of $x$ for which the numerator is greater than the denominator are: | \frac{3}{5} < x \le 2 | 1/8 |
Find the volume of a rectangular parallelepiped, the areas of whose diagonal sections are equal to $\sqrt{13}$, $2 \sqrt{10}$, and $3 \sqrt{5}$. | 6 | 5/8 |
1. Given that the terminal side of angle $\alpha$ passes through point $P(4, -3)$, find the value of $2\sin\alpha + \cos\alpha$.
2. Given that the terminal side of angle $\alpha$ passes through point $P(4a, -3a)$ ($a \neq 0$), find the value of $2\sin\alpha + \cos\alpha$.
3. Given that the ratio of the distance from a point $P$ on the terminal side of angle $\alpha$ to the x-axis and to the y-axis is 3:4, find the value of $2\sin\alpha + \cos\alpha$. | -\frac{2}{5} | 3/8 |
Define the sequence $f_{1}, f_{2}, \ldots:[0,1) \rightarrow \mathbb{R}$ of continuously differentiable functions by the following recurrence: $$ f_{1}=1 ; \quad f_{n+1}^{\prime}=f_{n} f_{n+1} \quad \text { on }(0,1), \quad \text { and } \quad f_{n+1}(0)=1 $$ Show that \(\lim _{n \rightarrow \infty} f_{n}(x)\) exists for every $x \in[0,1)$ and determine the limit function. | \frac{1}{1-x} | 7/8 |
Given the ellipse \( C: \frac{x^{2}}{3} + y^{2} = 1 \) with the upper vertex as \( A \), a line \( l \) that does not pass through \( A \) intersects the ellipse \( C \) at points \( P \) and \( Q \). Additionally, \( A P \perp A Q \). Find the maximum area of triangle \( \triangle A P Q \). | \frac{9}{4} | 2/8 |
Let \( S = \left\{ A = \left(a_{1}, a_{2}, \cdots, a_{8}\right) \mid a_{i} = 0 \text{ or } 1, i = 1,2, \cdots, 8 \right\} \). For any two elements \( A = \left(a_{1}, a_{2}, \cdots, a_{8}\right) \) and \( B = \left(b_{1}, b_{2}, \cdots, b_{8}\right) \) in \( S \), define \( d(A, B) = \sum_{i=1}^{8} \left| a_{i} - b_{i} \right| \), which is called the distance between \( A \) and \( B \). What is the maximum number of elements that can be chosen from \( S \) such that the distance between any two of them is at least \( 5 \)? | 4 | 1/8 |
Suppose $\triangle ABC$ is such that $AB=13$ , $AC=15$ , and $BC=14$ . It is given that there exists a unique point $D$ on side $\overline{BC}$ such that the Euler lines of $\triangle ABD$ and $\triangle ACD$ are parallel. Determine the value of $\tfrac{BD}{CD}$ . (The $\textit{Euler}$ line of a triangle $ABC$ is the line connecting the centroid, circumcenter, and orthocenter of $ABC$ .) | \frac{93+56\sqrt{3}}{33} | 2/8 |
The Vasilyev family's budget consists of the following income items:
- Parents' salary after income tax deduction: 71,000 rubles;
- Income from renting out real estate: 11,000 rubles;
- Daughter's scholarship: 2,600 rubles.
The average monthly expenses of the family include:
- Utility payments: 8,400 rubles;
- Food expenses: 18,000 rubles;
- Transportation expenses: 3,200 rubles;
- Tutor fees: 2,200 rubles;
- Miscellaneous expenses: 18,000 rubles.
10 percent of the remaining amount is allocated to a deposit for an emergency fund. Determine the maximum amount the Vasilyev family can pay monthly for a car loan.
In the answer, specify only the number without units of measurement. | 31320 | 2/8 |
Kolya and his sister Masha went to visit someone. After walking a quarter of the way, Kolya remembered that they had forgotten the gift at home and turned back, while Masha continued walking. Masha arrived at the visit 20 minutes after leaving home. How many minutes later did Kolya arrive, given that they walked at the same speeds all the time? | 10 | 7/8 |
A solid wooden rectangular prism measures $3 \times 5 \times 12$. The prism is cut in half by a vertical cut through four vertices, creating two congruent triangular-based prisms. What is the surface area of one of these triangular-based prisms? | 150 | 1/8 |
Given the sequence defined by \( a_{0}=3, a_{1}=9, a_{n}=4a_{n-1}-3a_{n-2}-4n+2 \) for \( n \geq 2 \), find the general term formula for the sequence \(\{a_n\}_{n \geqslant 0}\). | a_n=3^n+n^2+3n+2 | 3/8 |
Given that a parabola \( P \) has the center of the ellipse \( E \) as its focus, \( P \) passes through the two foci of \( E \), and \( P \) intersects \( E \) at exactly three points, determine the eccentricity of the ellipse. | \frac{2\sqrt{5}}{5} | 5/8 |
Let \( p_1, p_2, \ldots, p_{100} \) be one hundred distinct prime numbers. The natural numbers \( a_1, \ldots, a_k \), each greater than 1, are such that each of the numbers \( p_{1} p_{2}^{3}, p_{2} p_{3}^{3}, \ldots, p_{99} p_{100}^{3}, p_{100} p_{1}^{3} \) is the product of some two of the numbers \( a_1, \ldots, a_k \). Prove that \( k \geq 150 \). | 150 | 1/8 |
Using the trapezoidal rule with an accuracy of 0.01, calculate $\int_{2}^{3} \frac{d x}{x-1}$. | 0.6956 | 3/8 |
In triangle $ABC$ we have $|AB| \ne |AC|$ . The bisectors of $\angle ABC$ and $\angle ACB$ meet $AC$ and $AB$ at $E$ and $F$ , respectively, and intersect at I. If $|EI| = |FI|$ find the measure of $\angle BAC$ . | 60 | 2/8 |
Vehicle A and Vehicle B start from points A and B, respectively, at the same time and travel towards each other. They meet after 3 hours, at which point Vehicle A turns back towards point A, and Vehicle B continues forward. After Vehicle A reaches point A, it turns around and heads towards point B. Half an hour later, it meets Vehicle B again. How many hours does it take for Vehicle B to travel from A to B? | 7.2 | 1/8 |
Arrange the positive integers in the following number matrix:
\begin{tabular}{lllll}
1 & 2 & 5 & 10 & $\ldots$ \\
4 & 3 & 6 & 11 & $\ldots$ \\
9 & 8 & 7 & 12 & $\ldots$ \\
16 & 15 & 14 & 13 & $\ldots$ \\
$\ldots$ & $\ldots$ & $\ldots$ & $\ldots$ & $\ldots$
\end{tabular}
What is the number in the 21st row and 21st column? | 421 | 5/8 |
Find the minimum of the expression $\frac{x}{\sqrt{1-x}}+\frac{y}{\sqrt{1-y}}$ where $x$ and $y$ are strictly positive real numbers such that $x + y = 1$. | \sqrt{2} | 7/8 |
Given the parabola $C: x^{2}=8y$ and its focus $F$, the line $PQ$ and $MN$ intersect the parabola $C$ at points $P$, $Q$, and $M$, $N$, respectively. If the slopes of the lines $PQ$ and $MN$ are $k_{1}$ and $k_{2}$, and satisfy $\frac{1}{{k_1^2}}+\frac{4}{{k_2^2}}=1$, then the minimum value of $|PQ|+|MN|$ is ____. | 88 | 1/8 |
Find the number of triangles whose sides are formed by the sides and the diagonals of a regular heptagon (7-sided polygon). (Note: The vertices of triangles need not be the vertices of the heptagon.) | 287 | 1/8 |
A school has $100$ students and $5$ teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are $50, 20, 20, 5,$ and $5$. Let $t$ be the average value obtained if a teacher is picked at random and the number of students in their class is noted. Let $s$ be the average value obtained if a student was picked at random and the number of students in their class, including the student, is noted. What is $t-s$? | -13.5 | 6/8 |
A literary and art team went to a nursing home for a performance. Originally, there were 6 programs planned, but at the request of the elderly, they decided to add 3 more programs. However, the order of the original six programs remained unchanged, and the added 3 programs were neither at the beginning nor at the end. Thus, there are a total of different orders for this performance. | 210 | 1/8 |
Points \( A, B, C, \) and \( D \) lie on a straight line in that order. For a point \( E \) outside the line,
\[ \angle AEB = \angle BEC = \angle CED = 45^\circ. \]
Let \( F \) be the midpoint of segment \( AC \), and \( G \) be the midpoint of segment \( BD \). What is the measure of angle \( FEG \)? | 90 | 1/8 |
How many rows of Pascal's Triangle contain the number $43$? | 1 | 7/8 |
Vasya wrote twenty threes in a row on a board. By placing "+" signs between some of them, Vasya found that the sum equals 600. How many plus signs did Vasya place? | 9 | 7/8 |
Find the least positive integer \( N \) with the following property: If all lattice points in \([1,3] \times [1,7] \times [1,N]\) are colored either black or white, then there exists a rectangular prism, whose faces are parallel to the \( xy \), \( xz \), and \( yz \) planes, and whose eight vertices are all colored in the same color. | 127 | 1/8 |
Given any \( n > 3 \), show that for any positive reals \( x_1, x_2, \ldots, x_n \), we have \( \frac{x_1}{x_n + x_2} + \frac{x_2}{x_1 + x_3} + \ldots + \frac{x_n}{x_{n-1} + x_1} \geq 2 \). Show that the number 2 cannot be replaced by a smaller value. | 2 | 1/8 |
Given a triangle \( ABC \) with the longest side \( BC \). The bisector of angle \( C \) intersects the altitudes \( AA_1 \) and \( BB_1 \) at points \( P \) and \( Q \) respectively, and the circumcircle of \( ABC \) at point \( L \). Find \(\angle ACB\), given that \( AP = LQ \). | 60 | 1/8 |
In the diagram, points $A$, $B$, $C$, $D$, $E$, and $F$ lie on a straight line with $AB=BC=CD=DE=EF=3$. Semicircles with diameters $AF$, $AB$, $BC$, $CD$, $DE$, and $EF$ create a shape as depicted. What is the area of the shaded region underneath the largest semicircle that exceeds the areas of the other semicircles combined, given that $AF$ is after the diameters were tripled compared to the original configuration?
[asy]
size(5cm); defaultpen(fontsize(9));
pair one = (0.6, 0);
pair a = (0, 0); pair b = a + one; pair c = b + one; pair d = c + one; pair e = d + one; pair f = e + one;
path region = a{up}..{down}f..{up}e..{down}d..{up}c..{down}b..{up}a--cycle;
filldraw(region, gray(0.75), linewidth(0.75));
draw(a--f, dashed + linewidth(0.75));
// labels
label("$A$", a, W); label("$F$", f, E);
label("$B$", b, 0.8 * SE); label("$D$", d, 0.8 * SE);
label("$C$", c, 0.8 * SW); label("$E$", e, 0.8 * SW);
[/asy] | \frac{45}{2}\pi | 7/8 |
Find the smallest real number \( A \) such that for every quadratic polynomial \( f(x) \) satisfying the condition
\[
|f(x)| \leq 1 \quad (0 \leq x \leq 1),
\]
the inequality \( f^{\prime}(0) \leq A \) holds. | 8 | 2/8 |
We can find sets of 13 distinct positive integers that add up to 2142. Find the largest possible greatest common divisor of these 13 distinct positive integers. | 21 | 7/8 |
How many non-similar regular 1200-pointed stars are there, considering the definition of a regular $n$-pointed star provided in the original problem? | 160 | 1/8 |
Find the number of real solutions to the equation
\[\frac{1}{x - 1} + \frac{2}{x - 2} + \frac{3}{x - 3} + \dots + \frac{100}{x - 100} = x.\] | 101 | 7/8 |
Given that the polynomial $P(x) = x^5 - x^2 + 1$ has $5$ roots $r_1, r_2, r_3, r_4, r_5$ . Find the value of the product $Q(r_1)Q(r_2)Q(r_3)Q(r_4)Q(r_5)$ , where $Q(x) = x^2 + 1$ . | 5 | 7/8 |
Let \( A \cong \{0, 1, 2, \cdots, 29\} \) satisfy: for any integer \( k \) and any numbers \( a, b \) in \( A \) ( \( a \) and \( b \) can be the same), \( a + b + 30k \) is not the product of two consecutive integers. Determine all sets \( A \) with the maximum number of elements. | {2,5,8,11,14,17,20,23,26,29} | 1/8 |
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