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|---|---|---|
Let $n$ be a natural number with the following property: If 50 different numbers are randomly chosen from the numbers $1, 2, \ldots, n$, there will necessarily be two numbers among them whose difference is 7. Find the maximum value of such $n$.
|
98
|
4/8
|
Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a function such that for all \( x, y \in \mathbb{R} \):
\[
f(f(x+y))=f(x+y)+f(x) f(y)-x y
\]
Let \(\alpha=f(0)\). Prove that:
(i) \(f(\alpha) f(-\alpha)=0\),
(ii) \(\alpha=0\),
(iii) \(f(x)=x\) for all \( x \in \mathbb{R} \).
|
f(x)=x
|
4/8
|
A certain lottery has tickets labeled with the numbers \(1, 2, 3, \ldots, 1000\). The lottery is run as follows: First, a ticket is drawn at random. If the number on the ticket is odd, the drawing ends; if it is even, another ticket is randomly drawn (without replacement). If this new ticket has an odd number, the drawing ends; if it is even, another ticket is randomly drawn (again without replacement), and so forth, until an odd number is drawn. Then, every person whose ticket number was drawn (at any point in the process) wins a prize.
You have ticket number 1000. What is the probability that you get a prize?
|
\frac{1}{501}
|
4/8
|
Six people stand in a row, with exactly two people between A and B. Calculate the number of different ways for them to stand.
|
144
|
7/8
|
A driver travels for $2$ hours at $60$ miles per hour, during which her car gets $30$ miles per gallon of gasoline. She is paid $$0.50$ per mile, and her only expense is gasoline at $$2.00$ per gallon. What is her net rate of pay, in dollars per hour, after this expense?
$\textbf{(A)}\ 20\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 25\qquad\textbf{(E)}\ 26$
|
\textbf{(E)}\26
|
1/8
|
When Cheenu was a young man, he could run 20 miles in 4 hours. In his middle age, he could jog 15 miles in 3 hours and 45 minutes. Now, as an older man, he walks 12 miles in 5 hours. What is the time difference, in minutes, between his current walking speed and his running speed as a young man?
|
13
|
7/8
|
Find all positive integers $n > 3$ such that there exist $n$ points $A_{1}, A_{2}, \cdots, A_{n}$ in the plane and real numbers $r_{1}, r_{2}, \cdots, r_{n}$ satisfying the following conditions:
(1) Any 3 points among $A_{1}, A_{2}, \cdots, A_{n}$ are not collinear;
(2) For each triplet of points $\left\{A_{i}, A_{j}, A_{k}\right\}$ (where $1 \leq i < j < k \leq n$), the area of triangle $\triangle A_{i} A_{j} A_{k}$, denoted by $S_{ijk}$, is equal to $r_{i} + r_{j} + r_{k}$.
(Note: This is a problem from the 36th IMO in 1995)
|
4
|
1/8
|
Find the number of permutations \((b_1, b_2, b_3, b_4, b_5, b_6)\) of \((1,2,3,4,5,6)\) such that
\[
\frac{b_1 + 6}{2} \cdot \frac{b_2 + 5}{2} \cdot \frac{b_3 + 4}{2} \cdot \frac{b_4 + 3}{2} \cdot \frac{b_5 + 2}{2} \cdot \frac{b_6 + 1}{2} > 6!.
\]
|
719
|
1/8
|
A person forgot the last digit of a phone number and dialed randomly. Calculate the probability of connecting to the call in no more than 3 attempts.
|
\dfrac{3}{10}
|
3/8
|
Describe how to place the vertices of a triangle in the faces of a cube in such a way that the shortest side of the triangle is the biggest possible.
|
\sqrt{2}
|
5/8
|
Let $ABCD$ be a parallelogram with $\angle ABC=135^\circ$, $AB=14$ and $BC=8$. Extend $\overline{CD}$ through $D$ to $E$ so that $DE=3$. If $\overline{BE}$ intersects $\overline{AD}$ at $F$, then find the length of segment $FD$.
A) $\frac{6}{17}$
B) $\frac{18}{17}$
C) $\frac{24}{17}$
D) $\frac{30}{17}$
|
\frac{24}{17}
|
1/8
|
We place flower pots with masses $1 \mathrm{~kg}$, $2 \mathrm{~kg}$, and $3 \mathrm{~kg}$ at the corners of an equilateral weightless triangular board. Where should we suspend the board so that the flower pots do not fall off?
|
(\frac{7}{12},\frac{\sqrt{3}}{4})
|
4/8
|
On every kilometer of the highway between the villages Yolkino and Palkino, there is a post with a sign. On one side of the sign, the distance to Yolkino is written, and on the other side, the distance to Palkino is written. Borya noticed that on each post, the sum of all the digits is equal to 13. What is the distance from Yolkino to Palkino?
|
49
|
4/8
|
Evaluate \(\left(a^a - a(a-2)^a\right)^a\) when \( a = 4 \).
|
1358954496
|
7/8
|
It is known that there are a total of $n$ students in the first grade of Shuren High School, with $550$ male students. They are divided into layers based on gender, and $\frac{n}{10}$ students are selected to participate in a wetland conservation knowledge competition. It is given that there are $10$ more male students than female students among the participants. Find the value of $n$.
|
1000
|
4/8
|
The function $f: N \to N_0$ is such that $f (2) = 0, f (3)> 0, f (6042) = 2014$ and $f (m + n)- f (m) - f (n) \in\{0,1\}$ for all $m,n \in N$ . Determine $f (2014)$ . $N_0=\{0,1,2,...\}$
|
671
|
7/8
|
Let \( M, K \), and \( L \) be points on \( AB \), \( BC \), and \( CA \), respectively. Prove that the area of at least one of the three triangles \( \triangle MAL \), \( \triangle KBM \), and \( \triangle LCK \) is less than or equal to one-fourth the area of \( \triangle ABC \).
|
1
|
1/8
|
Compute
\[\sum_{n = 2}^\infty \frac{4n^3 - n^2 - n + 1}{n^6 - n^5 + n^4 - n^3 + n^2 - n}.\]
|
1
|
2/8
|
We rotate the square \(ABCD\) with a side length of 1 around its vertex \(C\) by \(90^\circ\). What area does side \(AB\) sweep out?
|
\frac{\pi}{4}
|
3/8
|
In $\triangle BAC$, $\angle BAC=40^\circ$, $AB=10$, and $AC=6$. Points $D$ and $E$ lie on $\overline{AB}$ and $\overline{AC}$ respectively. What is the minimum possible value of $BE+DE+CD$?
|
14
|
6/8
|
A triangular array of squares has one square in the first row, two in the second, and in general, $k$ squares in the $k$th row for $1 \leq k \leq 11.$ With the exception of the bottom row, each square rests on two squares in the row immediately below (illustrated in the given diagram). In each square of the eleventh row, a $0$ or a $1$ is placed. Numbers are then placed into the other squares, with the entry for each square being the sum of the entries in the two squares below it. For how many initial distributions of $0$'s and $1$'s in the bottom row is the number in the top square a multiple of $3$?
[asy] for (int i=0; i<12; ++i){ for (int j=0; j<i; ++j){ //dot((-j+i/2,-i)); draw((-j+i/2,-i)--(-j+i/2+1,-i)--(-j+i/2+1,-i+1)--(-j+i/2,-i+1)--cycle); } } [/asy]
|
640
|
6/8
|
Given a triangle ABC, let the lengths of the sides opposite to angles A, B, C be a, b, c, respectively. If a, b, c satisfy $a^2 + c^2 - b^2 = \sqrt{3}ac$,
(1) find angle B;
(2) if b = 2, c = $2\sqrt{3}$, find the area of triangle ABC.
|
2\sqrt{3}
|
3/8
|
Let $\overline{MN}$ be a diameter of a circle with diameter 1. Let $A$ and $B$ be points on one of the semicircular arcs determined by $\overline{MN}$ such that $A$ is the midpoint of the semicircle and $MB=\frac{3}5$. Point $C$ lies on the other semicircular arc. Let $d$ be the length of the line segment whose endpoints are the intersections of diameter $\overline{MN}$ with chords $\overline{AC}$ and $\overline{BC}$. The largest possible value of $d$ can be written in the form $r-s\sqrt{t}$, where $r, s$ and $t$ are positive integers and $t$ is not divisible by the square of any prime. Find $r+s+t$.
|
14
|
1/8
|
Seven natural numbers are written in a circle. Prove that there are two adjacent numbers whose sum is even.
|
2
|
6/8
|
The function \( g(x) \) satisfies
\[ g(x) - 2 g \left( \frac{1}{x} \right) = 3^x + x \]
for all \( x \neq 0 \). Find \( g(2) \).
|
-4 - \frac{2\sqrt{3}}{3}
|
7/8
|
After watching a movie, viewers rated the movie sequentially with an integer score from 0 to 10. At any given moment, the movie rating was calculated as the sum of all given scores divided by their number. At a certain moment $T$, the rating turned out to be an integer, and then with each new voting viewer, it decreased by one unit. What is the maximum number of viewers who could have voted after moment $T ?$
|
5
|
3/8
|
In triangle \(ABC\), \(AB = 15\), \(BC = 12\), and \(AC = 18\). In what ratio does the center \(O\) of the incircle of the triangle divide the angle bisector \(CM\)?
|
2:1
|
7/8
|
Let $n$ be the product of the first $10$ primes, and let $$ S=\sum_{xy\mid n} \varphi(x) \cdot y, $$ where $\varphi(x)$ denotes the number of positive integers less than or equal to $x$ that are relatively prime to $x$ , and the sum is taken over ordered pairs $(x, y)$ of positive integers for which $xy$ divides $n$ . Compute $\tfrac{S}{n}.$
|
1024
|
7/8
|
If $r$ is the remainder when each of the numbers $1059,~1417$, and $2312$ is divided by $d$, where $d$ is an integer greater than $1$, then $d-r$ equals
$\textbf{(A) }1\qquad \textbf{(B) }15\qquad \textbf{(C) }179\qquad \textbf{(D) }d-15\qquad \textbf{(E) }d-1$
|
\textbf{(B)}15
|
1/8
|
Given positive real numbers \( x \) and \( y \) satisfy:
\[
\left(2 x+\sqrt{4 x^{2}+1}\right)\left(\sqrt{y^{2}+4}-2\right) \geqslant y
\]
then the minimum value of \( x + y \) is ______.
|
2
|
6/8
|
How many ways can one color the squares of a \(6 \times 6\) grid red and blue such that the number of red squares in each row and column is exactly 2?
|
67950
|
1/8
|
A jar contains $5$ different colors of gumdrops. $30\%$ are blue, $20\%$ are brown, $15\%$ are red, $10\%$ are yellow, and other $30$ gumdrops are green. If half of the blue gumdrops are replaced with brown gumdrops, how many gumdrops will be brown?
$\textbf{(A)}\ 35\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 64$
|
\textbf{(C)}\42
|
1/8
|
From a point \( M \) on the ellipse \(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1\), two tangent lines are drawn to the circle with the minor axis of the ellipse as its diameter. The points of tangency are \( A \) and \( B \). The line \( AB \) intersects the \(x\)-axis and \(y\)-axis at points \( P \) and \( Q \) respectively. Find the minimum value of \(|PQ|\).
|
10/3
|
7/8
|
Find the largest positive integer $n$ such that the number $(2n)!$ ends with $10$ more zeroes than the number $n!$ .
*Proposed by Andy Xu*
|
42
|
1/8
|
Among the numbers $85_{(9)}$, $210_{(6)}$, $1000_{(4)}$, and $111111_{(2)}$, the smallest number is __________.
|
111111_{(2)}
|
6/8
|
Let \( A \) and \( B \) be two points in a plane, and let \( (d) \) be a line that does not intersect the segment \([A B]\). Determine (geometrically) the point \( M \) on \( (d) \) for which the angle \(\widehat{A M B}\) is maximal.
|
M
|
4/8
|
Add together all natural numbers less than 1980 for which the sum of their digits is even!
|
979605
|
1/8
|
Say a real number $r$ is \emph{repetitive} if there exist two distinct complex numbers $z_1,z_2$ with $|z_1|=|z_2|=1$ and $\{z_1,z_2\}\neq\{-i,i\}$ such that
\[
z_1(z_1^3+z_1^2+rz_1+1)=z_2(z_2^3+z_2^2+rz_2+1).
\]
There exist real numbers $a,b$ such that a real number $r$ is \emph{repetitive} if and only if $a < r\le b$ . If the value of $|a|+|b|$ can be expressed in the form $\frac{p}{q}$ for relatively prime positive integers $p$ and $q$ , find $100p+q$ .
*Proposed by James Lin*
|
2504
|
1/8
|
Niko counted a total of 60 birds perching in three trees. Five minutes later, 6 birds had flown away from the first tree, 8 birds had flown away from the second tree, and 4 birds had flown away from the third tree. He noticed that there was now the same number of birds in each tree. How many birds were originally perched in the second tree?
A) 14
B) 18
C) 20
D) 21
E) 22
|
22
|
6/8
|
For positive constant $a$ , let $C: y=\frac{a}{2}(e^{\frac{x}{a}}+e^{-\frac{x}{a}})$ . Denote by $l(t)$ the length of the part $a\leq y\leq t$ for $C$ and denote by $S(t)$ the area of the part bounded by the line $y=t\ (a<t)$ and $C$ . Find $\lim_{t\to\infty} \frac{S(t)}{l(t)\ln t}.$
|
a
|
6/8
|
Let \( f_{1} = 9 \) and
\[ f_{n} = \begin{cases}
f_{n-1} + 3 & \text{if } n \text{ is a multiple of } 3 \\
f_{n-1} - 1 & \text{if } n \text{ is not a multiple of } 3
\end{cases} \]
If \( B \) is the number of possible values of \( k \) such that \( f_{k} < 11 \), determine the value of \( B \).
|
5
|
4/8
|
The numbers $a, b, c, d$ belong to the interval $[-6.5, 6.5]$. Find the maximum value of the expression $a + 2b + c + 2d - ab - bc - cd - da$.
|
182
|
4/8
|
A traffic light cycles as follows: green for 45 seconds, yellow for 5 seconds, then red for 50 seconds. Felix chooses a random five-second interval to observe the light. What is the probability that the color changes while he is observing?
|
\frac{3}{20}
|
7/8
|
From a sheet of squared paper measuring $29 \times 29$ cells, 99 squares have been cut out, each consisting of four cells. Prove that it is possible to cut out one more square.
|
1
|
7/8
|
Let $n$ points be given inside a rectangle $R$ such that no two of them lie on a line parallel to one of the sides of $R$ . The rectangle $R$ is to be dissected into smaller rectangles with sides parallel to the sides of $R$ in such a way that none of these rectangles contains any of the given points in its interior. Prove that we have to dissect $R$ into at least $n + 1$ smaller rectangles.
*Proposed by Serbia*
|
n+1
|
4/8
|
Let $ f(x)\equal{}\sin 3x\plus{}\cos x,\ g(x)\equal{}\cos 3x\plus{}\sin x.$
(1) Evaluate $ \int_0^{2\pi} \{f(x)^2\plus{}g(x)^2\}\ dx$ .
(2) Find the area of the region bounded by two curves $ y\equal{}f(x)$ and $ y\equal{}g(x)\ (0\leq x\leq \pi).$
|
\frac{8\sqrt{2+\sqrt{2}}-4}{3}
|
1/8
|
Given the equations \( x^{2} + px + q = 0 \) and \( x^{2} + rx + s = 0 \) where one root of the first equation is \(\alpha\) and one root of the second equation is \(\beta\), and \(\alpha / \beta = k\). Prove the following equality:
\[ \left(q - k^{2} s\right)^{2} + k(p - k r)(k p s - q r) = 0 \]
|
(k^{2})^{2}+k(p-kr)(kp-)=0
|
1/8
|
Place four small spheres, each with a radius of 1, into a larger sphere. What is the minimum possible radius of the larger sphere?
|
1+\frac{\sqrt{6}}{2}
|
7/8
|
**
How many non-similar regular 500-pointed stars are there?
**
|
99
|
7/8
|
What fraction of the volume of a parallelepiped is the volume of a tetrahedron whose vertices are the centroids of the tetrahedra cut off by the planes of a tetrahedron inscribed in the parallelepiped?
|
1/24
|
1/8
|
How many of the smallest 216 positive integers written in base 6 use the digit 3 or 4 (or both) as a digit?
|
168
|
1/8
|
Find any solution to the rebus
$$
\overline{A B C A}=182 \cdot \overline{C D}
$$
where \( A, B, C, D \) are four distinct non-zero digits (the notation \(\overline{X Y \ldots Z}\) denotes the decimal representation of a number).
As an answer, write the four-digit number \(\overline{A B C D}\).
|
2916
|
3/8
|
In a triangular pyramid \( SABC \), the lateral edge \( SC \) is equal to the edge \( AB \) and is inclined to the plane of the base \( ABC \) at an angle of \( 60^\circ \). It is known that the vertices \( A, B, C \) and the midpoints of the lateral edges of the pyramid are located on a sphere with a radius of 1. Prove that the center of this sphere lies on the edge \( AB \) and find the height of the pyramid.
|
\sqrt{3}
|
4/8
|
All integers from 1 to \(2n\) are written in a row. Then, the position number is added to each of these integers.
Prove that among the obtained sums, there are at least two that give the same remainder when divided by \(2n\).
|
2
|
6/8
|
Given that the sine and cosine values of angle $α$ are both negative, and $\cos(75^{\circ}+α)=\frac{1}{3}$, find the value of $\cos(105^{\circ}-α)+\sin(α-105^{\circ})$ = \_\_\_\_\_\_.
|
\frac{2\sqrt{2}-1}{3}
|
7/8
|
Given a sequence $\{x_n\}$ that satisfies $x_{n+2}=|x_{n+2}-x_n|$ (where $n \in \mathbb{N}^*$), if $x_1=1$, $x_2=a$ (where $a \leqslant 1$ and $a \neq 0$), and $x_{n+3}=x_n$ for any positive integer $n$, then the sum of the first 2017 terms of the sequence $\{x_n\}$ is ______.
|
1345
|
6/8
|
Given a four-digit number $\overline{A B C D}$ that satisfies the following properties: $\overline{A B}$, $\overline{B C}$, and $\overline{C D}$ are all perfect squares (a perfect square is a number that can be expressed as the square of an integer, such as $4 = 2^2$ and $81 = 9^2$). What is the sum of all four-digit numbers that satisfy this property?
|
13462
|
7/8
|
There are 4 boys and 3 girls standing in a row. (You must write down the formula before calculating the result to score points)
(Ⅰ) If the 3 girls must stand together, how many different arrangements are there?
(Ⅱ) If no two girls are next to each other, how many different arrangements are there?
(Ⅲ) If there are exactly three people between person A and person B, how many different arrangements are there?
|
720
|
7/8
|
A snail is crawling forward at a non-uniform speed (without moving backward). Several people observe its crawling in a 6-minute span, each starting their observation before the previous person finishes. Each observer watches the snail for exactly 1 minute. It is known that every observer finds that the snail crawled 1 meter in the minute they observed. Prove that the distance the snail crawled in these 6 minutes does not exceed 10 meters.
|
10
|
1/8
|
The sequence ${a_{n}}$ $n\in \mathbb{N}$ is given in a recursive way with $a_{1}=1$ , $a_{n}=\prod_{i=1}^{n-1} a_{i}+1$ , for all $n\geq 2$ .
Determine the least number $M$ , such that $\sum_{n=1}^{m} \frac{1}{a_{n}} <M$ for all $m\in \mathbb{N}$
|
2
|
7/8
|
A solid right prism $PQRSTU$ has a height of 20, as shown. Its bases are equilateral triangles with side length 10. Points $V$, $W$, and $X$ are the midpoints of edges $PR$, $RQ$, and $QT$, respectively. Determine the perimeter of triangle $VWX$.
|
5 + 10\sqrt{5}
|
1/8
|
Let $A$ be a set with $n$ elements. The $m$ subsets $A_{1}, A_{2}, \cdots, A_{m}$ of $A$ are pairwise disjoint. Prove that:
(1) $\sum_{i=1}^{m} \frac{1}{C_{n}^{|A_{i}|}} \leq 1$;
(2) $\sum_{i=1}^{m} C_{n}^{|A_{i}|} \geq m^{2}$.
|
1
|
1/8
|
The ternary sequence $\left(x_{n}, y_{n}, z_{n}\right), n \in \mathbb{N}$ is determined by the following relations: $x_{1}=2, y_{1}=4, z_{1}=\frac{6}{7}$, $x_{n-1}=\frac{2 x_{n}}{x_{n}^{2}-1}, y_{n+1}=\frac{2 y_{n}}{y_{n}^{2}-1}, z_{n+1}=\frac{2 z_{n}}{z_{n}^{2}-1}$.
1. Prove that the process of forming the ternary sequence can continue indefinitely.
2. Is it possible that at some step, the obtained ternary sequence $\left(x_{n}, y_{n}, z_{n}\right)$ satisfies the equation $x_{n}+y_{n}+z_{n}=0$?
|
No
|
1/8
|
Assume a deck of 27 cards where each card features one of three symbols (star, circle, or square), each symbol painted in one of three colors (red, yellow, or blue), and each color applied in one of three intensities (light, medium, or dark). Each symbol-color-intensity combination is unique across the cards. A set of three cards is defined as complementary if:
i. Each card has a different symbol or all have the same symbol.
ii. Each card has a different color or all have the same color.
iii. Each card has a different intensity or all have the same intensity.
Determine the number of different complementary three-card sets available.
|
117
|
6/8
|
Point \(P\) is inside an equilateral \(\triangle ABC\) such that the measures of \(\angle APB, \angle BPC, \angle CPA\) are in the ratio 5:6:7. Determine the ratio of the measures of the angles of the triangle formed by \(PA, PB, PC\) (in increasing order).
|
2: 3: 4
|
1/8
|
In a right triangle \(ABC\) (with \(\angle C = 90^\circ\)), height \(CD\) is drawn. The radii of the circles inscribed in triangles \(ACD\) and \(BCD\) are 0.6 cm and 0.8 cm, respectively. Find the radius of the circle inscribed in triangle \(ABA\).
|
1
|
6/8
|
Let \( \triangle ABC \) be an acute triangle, with \( M \) being the midpoint of \( \overline{BC} \), such that \( AM = BC \). Let \( D \) and \( E \) be the intersection of the internal angle bisectors of \( \angle AMB \) and \( \angle AMC \) with \( AB \) and \( AC \), respectively. Find the ratio of the area of \( \triangle DME \) to the area of \( \triangle ABC \).
|
\frac{2}{9}
|
5/8
|
Prove that if a pair of values for the variables \( x \) and \( y \) satisfies the equations
$$
x^{2}-3 x y+2 y^{2}+x-y=0
$$
and
$$
x^{2}-2 x y+y^{2}-5 x+7 y=0,
$$
then this pair also satisfies the equation
$$
x y-12 x+15 y=0.
$$
|
xy-12x+15y=0
|
1/8
|
Let $\triangle ABC$ be a triangle with circumcenter $O$ satisfying $AB=13$ , $BC = 15$ , and $AC = 14$ . Suppose there is a point $P$ such that $PB \perp BC$ and $PA \perp AB$ . Let $X$ be a point on $AC$ such that $BX \perp OP$ . What is the ratio $AX/XC$ ?
*Proposed by Thomas Lam*
|
\frac{169}{225}
|
3/8
|
John and Mary select a natural number each and tell that to Bill. Bill wrote their sum and product in two papers hid one paper and showed the other to John and Mary.
John looked at the number (which was $2002$ ) and declared he couldn't determine Mary's number. Knowing this Mary also said she couldn't determine John's number as well.
What was Mary's Number?
|
1001
|
1/8
|
Let \( n \) be an odd number greater than 1. Given:
\[
\begin{aligned}
x_{0} & = \left(x_{1}^{(0)}, x_{2}^{(0)}, \ldots, x_{n}^{(0)}\right) = (1, 0, \ldots, 0, 1) \\
\text{Let} \quad x_{i}^{k} & = \begin{cases}
0, & \text{if } x_{i}^{(k-1)} = x_{i+1}^{(k-1)} \\
1, & \text{if } x_{i}^{(k-1)} \neq x_{i+1}^{(k-1)}
\end{cases}, \quad i = 1, 2, \ldots, n
\end{aligned}
\]
where \( x_{n+1}^{k-1} = x_{1}^{k-1} \). Define:
\[
x_{k} = \left(x_{1}^{(k)}, x_{2}^{(k)}, \ldots, x_{n}^{(k)}\right), \quad k = 1, 2, \ldots
\]
If a positive integer \( m \) satisfies \( x_{m} = x_{0} \), prove that \( m \) is a multiple of \( n \).
|
ismultipleofn
|
1/8
|
There is a rectangle $ABCD$ such that $AB=12$ and $BC=7$ . $E$ and $F$ lie on sides $AB$ and $CD$ respectively such that $\frac{AE}{EB} = 1$ and $\frac{CF}{FD} = \frac{1}{2}$ . Call $X$ the intersection of $AF$ and $DE$ . What is the area of pentagon $BCFXE$ ?
Proposed by Minseok Eli Park (wolfpack)
|
47
|
4/8
|
On the sides \(BC\) and \(AC\) of an isosceles triangle \(ABC\) (\(AB = AC\)), points \(D\) and \(E\) are found such that \(AE = AD\) and \(\angle EDC = 18^\circ\). Find the measure of the angle \(\angle BAD\).
|
36
|
2/8
|
It is known that the graph of the function \(y = x^2 + px + q\) passes through points \(A(1, 1)\) and \(B(3, 1)\). Does this graph pass through point \(C(4, 5)\)?
|
0
|
1/8
|
Given that \( G \) is a simple graph with 20 vertices and 100 edges, we can find 4050 ways to identify a pair of non-intersecting edges. Prove that \( G \) is regular.
(2004 Iran Mathematics Olympiad)
|
G
|
6/8
|
Non-negative real numbers \( x, y, z \) satisfy \( x^{2} + y^{2} + z^{2} = 10 \). Find the maximum and minimum values of \( \sqrt{6 - x^{2}} + \sqrt{6 - y^{2}} + \sqrt{6 - z^{2}} \).
|
\sqrt{6}+\sqrt{2}
|
2/8
|
Let \( f(n) \) be the number of real solutions of \( x^3 = \lfloor x^3 \rfloor + (x - \lfloor x \rfloor)^3 \) in the range \( 1 \le x < n \). Find \( f(n) \).
|
n^3-n
|
1/8
|
On the board, the natural number \( N \) was written nine times (one below the other). Petya added a non-zero digit to the left or right of each of the 9 numbers; all added digits are distinct. What is the largest possible number of prime numbers that could result from these 9 new numbers?
(I. Efremov)
|
6
|
1/8
|
Let \( M = \{1, 2, 3, \cdots, 1995\} \) and \( A \subseteq M \), with the constraint that if \( x \in A \), then \( 19x \notin A \). Find the maximum value of \( |A| \).
|
1890
|
1/8
|
Given \(\alpha\) and \(\beta\) satisfy the equations
\[
\begin{array}{c}
\alpha^{3}-3 \alpha^{2}+5 \alpha-4=0, \\
\beta^{3}-3 \beta^{2}+5 \beta-2=0 .
\end{array}
\]
find \(\alpha + \beta\).
|
2
|
7/8
|
Find all the pairs of positive integers $(m,n)$ such that the numbers $A=n^2+2mn+3m^2+3n$ , $B=2n^2+3mn+m^2$ , $C=3n^2+mn+2m^2$ are consecutive in some order.
|
(,n)=(k,k+1)
|
1/8
|
Given \(\triangle ABC\) is an acute triangle, \(b = 2c\), and \(\sin B - \sin(A+B) = 2 \sin C \cos A\). Find the range of values for \((\cos B + \sin B)^{2} + \sin 2C\).
|
1+\frac{\sqrt{3}}{2}
|
1/8
|
A rectangular pasture is to be fenced off on three sides using part of a 100 meter rock wall as the fourth side. Fence posts are to be placed every 15 meters along the fence including at the points where the fence meets the rock wall. Given the dimensions of the pasture are 36 m by 75 m, find the minimum number of posts required.
|
14
|
1/8
|
Ten distinct positive real numbers are given and the sum of each pair is written (So 45 sums). Between these sums there are 5 equal numbers. If we calculate product of each pair, find the biggest number $k$ such that there may be $k$ equal numbers between them.
|
4
|
1/8
|
Let \(\triangle ABC\) be a scalene triangle. Let \(h_{a}\) be the locus of points \(P\) such that \(|PB - PC| = |AB - AC|\). Let \(h_{b}\) be the locus of points \(P\) such that \(|PC - PA| = |BC - BA|\). Let \(h_{c}\) be the locus of points \(P\) such that \(|PA - PB| = |CA - CB|\). In how many points do all of \(h_{a}, h_{b}\), and \(h_{c}\) concur?
|
2
|
1/8
|
Let $H$ be a convex, equilateral heptagon whose angles measure (in degrees) $168^\circ$ , $108^\circ$ , $108^\circ$ , $168^\circ$ , $x^\circ$ , $y^\circ$ , and $z^\circ$ in clockwise order. Compute the number $y$ .
|
132
|
1/8
|
Integers less than $4010$ but greater than $3000$ have the property that their units digit is the sum of the other digits and also the full number is divisible by 3. How many such integers exist?
|
12
|
3/8
|
How many four-digit numbers starting with the digit $2$ and having exactly three identical digits are there?
|
27
|
1/8
|
Let $S$ be a finite set of points in the plane. A linear partition of $S$ is an unordered pair $\{A,B\}$ of subsets of $S$ such that $A\cup B=S,\ A\cap B=\emptyset,$ and $A$ and $B$ lie on opposite sides of some straight line disjoint from $S$ ( $A$ or $B$ may be empty). Let $L_{S}$ be the number of linear partitions of $S.$ For each positive integer $n,$ find the maximum of $L_{S}$ over all sets $S$ of $n$ points.
|
\frac{n(n-1)}{2}+1
|
1/8
|
Someone observed that $6! = 8 \cdot 9 \cdot 10$. Find the largest positive integer $n$ for which $n!$ can be expressed as the product of $n - 3$ consecutive positive integers.
|
23
|
1/8
|
Consider a square ABCD with side length 4 units. Points P and R are the midpoints of sides AB and CD, respectively. Points Q is located at the midpoint of side BC, and point S is located at the midpoint of side AD. Calculate the fraction of the square's total area that is shaded when triangles APQ and CSR are shaded.
[asy]
filldraw((0,0)--(4,0)--(4,4)--(0,4)--(0,0)--cycle,gray,linewidth(1));
filldraw((0,2)--(2,4)--(0,4)--(0,2)--cycle,white,linewidth(1));
filldraw((4,2)--(2,0)--(4,0)--(4,2)--cycle,white,linewidth(1));
label("P",(0,2),W);
label("Q",(2,4),N);
label("R",(4,2),E);
label("S",(2,0),S);
[/asy]
|
\frac{1}{4}
|
6/8
|
In a new diagram, $A$ is the center of a circle with radii $AB=AC=8$. The sector $BOC$ is shaded except for a triangle $ABC$ within it, where $B$ and $C$ lie on the circle. If the central angle of $BOC$ is $240^\circ$, what is the perimeter of the shaded region?
|
16 + \frac{32}{3}\pi
|
7/8
|
Given an arithmetic sequence $\\{a_{n}\\}$, let $S_{n}$ denote the sum of its first $n$ terms. If $a_{4}=-12$ and $a_{8}=-4$:
$(1)$ Find the general term formula for the sequence;
$(2)$ Find the minimum value of $S_{n}$ and the corresponding value of $n$.
|
-90
|
7/8
|
Three chiefs of Indian tribes are sitting by a fire with three identical pipes. They are holding a war council and smoking. The first chief can finish a whole pipe in ten minutes, the second in thirty minutes, and the third in an hour. How should the chiefs exchange the pipes among themselves in order to prolong their discussion for as long as possible?
|
20
|
1/8
|
Given that \(a_{1}=1\), \(a_{2}=3\), \(a_{n}=4a_{n-1}-a_{n-2}\) for \(n \geq 3\); \(b_{1}=1\), \(b_{2}=3\), \(b_{n}=\frac{b_{n-1}^{2}+2}{b_{n-2}}\) for \(n \geq 3\); and \(c_{1}=1\), \(c_{n+1}=2c_{n}+\sqrt{3c_{n}^{2}-2}\) for \(n \geq 3\). Prove that for all positive integers \(n\), \(a_{n}=b_{n}=c_{n}\).
|
a_n=b_n=c_n
|
4/8
|
On a table, there are 10 number cards labeled $0$ to $9$. Three people, A, B, and C, each take three of these cards. They then compute the sum of all different three-digit numbers that can be formed with the three cards they took. The results for A, B, and C are $1554$, $1688$, and $4662$ respectively. What is the remaining card? (Note: $6$ and $9$ cannot be considered as each other.)
|
9
|
5/8
|
A certain district's education department wants to send 5 staff members to 3 schools for earthquake safety education. Each school must receive at least 1 person and no more than 2 people. How many different arrangements are possible? (Answer with a number)
|
90
|
6/8
|
Four princesses thought of two-digit numbers, and Ivan thought of a four-digit number. After they wrote their numbers in a row in some order, the result was 132040530321. Find Ivan's number.
|
5303
|
1/8
|
On the island of Truth and Lies, there are knights who always tell the truth, and liars who always lie. One day, 20 inhabitants of the island lined up in order of their height (from tallest to shortest, the tallest being the first) for a game. Each of them had to say one of the following phrases: "There is a liar below me" or "There is a knight above me". As a result, those standing in positions from the third to the seventh said the first phrase, while the others said the second phrase. How many knights were there among these 20 people, if all the inhabitants have different heights?
|
17
|
1/8
|
The points $A=\left(4, \frac{1}{4}\right)$ and $B=\left(-5,-\frac{1}{5}\right)$ lie on the hyperbola $x y=1$. The circle with diameter $A B$ intersects this hyperbola again at points $X$ and $Y$. Compute $X Y$.
|
\sqrt{\frac{401}{5}}
|
1/8
|
Subsets and Splits
Filtered Answers A-D
Retrieves 100 rows where the answer is a single letter from A to D, providing basic filtering of the dataset.