Found problems: 85335
Let $A_1,A_2,\cdots,A_n$ be $n$ distinct points on the plane ($n>1$). We consider all the segments $A_iA_j$ where $i<j\leq{n}$ and color the midpoints of them. What's the minimum number of colored points? (In fact, if $k$ colored points coincide, we count them $1$.)
You are standing at the edge of a river which is $1$ km wide. You have to go to your camp on the opposite bank . The distance to the camp from the point on the opposite bank directly across you is $1$ km . You can swim at $2$ km/hr and walk at $3$ km-hr . What is the shortest time you will take to reach your camp?(Ignore the speed of the river and assume that the river banks are straight and parallel).
Sergei skated around an ice rink, gliding along different paths. The gray lines in the figures below show four of the paths labeled $P$, $Q$, $R$, and $S$. What is the sorted order of the four paths from shortest to longest?
[center][img]https://wiki-images.artofproblemsolving.com/9/94/2024_AMC_8_Problem_6.png[/img][/center]
$\textbf{(A) }\text{P, Q, R, S}\qquad\textbf{(B) }\text{P, R, S, Q}\qquad\textbf{(C) }\text{Q, S, P, R}\qquad\textbf{(D) }\text{R, P, S, Q}\qquad\textbf{(E) }\text{R, S, P, Q}$
Let $P$ be a polygon that is convex and symmetric to some point $O$. Prove that for some parallelogram $R$ satisfying $P\subset R$ we have \[\frac{|R|}{|P|}\leq \sqrt 2\]
where $|R|$ and $|P|$ denote the area of the sets $R$ and $P$, respectively.
[i]Proposed by Witold Szczechla, Poland[/i]
We define a complex number $z=9+10i$
please find the maximum of a positive integer $n$ which satisfies $|z^n|\leq2023$
A $n+1$-tuple $\left(h_1,h_2, \cdots, h_{n+1}\right)$ where $h_i\left(x_1,x_2, \cdots , x_n\right)$ are $n$ variable polynomials with real coefficients is called [i]good[/i] if the following condition holds:
For any $n$ functions $f_1,f_2, \cdots ,f_n : \mathbb R \to \mathbb R$ if for all $1 \le i \le n+1$, $P_i(x)=h_i \left(f_1(x),f_2(x), \cdots, f_n(x) \right)$ is a polynomial with variable $x$, then $f_1(x),f_2(x), \cdots, f_n(x)$ are polynomials.
$a)$ Prove that for all positive integers $n$, there exists a [i]good[/i] $n+1$-tuple $\left(h_1,h_2, \cdots, h_{n+1}\right)$ such that the degree of all $h_i$ is more than $1$.
$b)$ Prove that there doesn't exist any integer $n>1$ that for which there is a [i]good[/i] $n+1$-tuple $\left(h_1,h_2, \cdots, h_{n+1}\right)$ such that all $h_i$ are symmetric polynomials.
[i]Proposed by Alireza Shavali[/i]
Let $k>2$ be a positive integer such that the $k$-digit number $n_k=133\dots 3$, consisting of a digit $1$ followed by $k-1$ digits $3$ is prime. Show that $24 \mid k(k+2)$.
There are $2014$ students from high schools nationwide communications sit around a round table in arbitrary manner. Then the organizers want to rearrange students from the same school sit next to each other by performing the following swapping: permutation view of two adjacent groups of students (see illustration). Find the smallest $k$ number so that a result can be obtained results as desired by the organizers with no more than $k$ swapping permits. Permission to change places like after
$...\underbrace{ABCD}_\text{1}\underbrace{EFG}_\text{2}... \to ...\underbrace{EFG}_\text{2}\underbrace{ABCD}_\text{1}...$
Vu The Khoi, Institute of Mathematics, Vietnam Academy of Science and Technology, Cau Giay, Hanoi.
Find all non-negative integers $n$ so that $\sqrt{n + 3}+ \sqrt{n +\sqrt{n + 3}} $ is an integer.
Show that, exist infinite triples $(a, b, c)$ where $a, b, c$ are natural numbers, such that:
$2a^2 + 3b^2 - 5c^2 = 1997$
Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38, 38,$ and $34.$ What is the distance between two adjacent parallel lines?
$\textbf{(A)}\ 5\frac{1}{2} \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 6\frac{1}{2} \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 7\frac{1}{2}$
For a point $ P $ situated in the plane determined by a triangle $ ABC, $ prove the following inequality:
$$ BC\cdot PB\cdot PC+AC\cdot PC\cdot PA +AB\cdot PA\cdot PB\ge AB\cdot BC\cdot CA $$
Given an acute-angled triangle $ABC$ and a point $P$ inside it such that $\angle PBA=\angle PCA$. The lines $PB$ and $PC$ intersect the circumcircles of triangles $PCA$ and $PAB$ secondly at points $M$ and $N$, respectively. Let the rays $MC$ and $NB$ intersect at a point $S$, $K$ is the center of the circumscribed circle of the triangle $SMN$. Prove that the lines $AK$ and $BC$ are perpendicular.
Denote by $ M$ midpoint of side $ BC$ in an isosceles triangle $ \triangle ABC$ with $ AC = AB$. Take a point $ X$ on a smaller arc $ \overarc{MA}$ of circumcircle of triangle $ \triangle ABM$. Denote by $ T$ point inside of angle $ BMA$ such that $ \angle TMX = 90$ and $ TX = BX$.
Prove that $ \angle MTB - \angle CTM$ does not depend on choice of $ X$.
[i]Author: Farzan Barekat, Canada[/i]
Straight lines $OA$ and $OB$ are perpendicular. Find the locus of endpoints $M$ of all broken lines $OM$ of length $\ell$ which intersect each line parallel to $OA$ or $OB$ at not more than one point.
Let $ \mathcal{F}$ be a family of hexagons $ H$ satisfying the following properties:
i) $ H$ has parallel opposite sides.
ii) Any 3 vertices of $ H$ can be covered with a strip of width 1.
Determine the least $ \ell\in\mathbb{R}$ such that every hexagon belonging to $ \mathcal{F}$ can be covered with a strip of width $ \ell$.
Note: A strip is the area bounded by two parallel lines separated by a distance $ \ell$. The lines belong to the strip, too.
Let $a$, $b$ and $c$ be any positive real numbers. Prove that $$\dfrac{a^2+b^2}{2c}+\dfrac{a^2+c^2}{2b}+\dfrac{b^2+c^2}{2a} \geqslant a+b+c.$$
Show that the triangle whose angles satisfy the equality
\[\frac{\sin^2A+\sin^2B+\sin^2C}{\cos^2A+\cos^2B+\cos^2C} = 2\]
is right angled.
The ratio of $w$ to $x$ is $4 : 3$, the ratio of $y$ to $z$ is $3 : 2$, and the ratio of $z$ to $x$ is $1 : 6$. What is the ratio of $w$ to $y$?
$\textbf{(A) }4:3 \qquad \textbf{(B) }3:2 \qquad \textbf{(C) } 8:3 \qquad \textbf{(D) } 4:1 \qquad \textbf{(E) } 16:3 $
Let $\vartriangle ABC$ be a triangle with orthocenter $H$ and altitudes $AD$, $BE$ and $CF$. Let $D'$, $E' $ and $F'$ be the intersections of the heights $AD$, $BE$ and $CF$, respectively, with the circumcircle of $\vartriangle ABC $, so that they are different points from the vertices of triangle $\vartriangle ABC$. Let $L$, $M$ and $N$ be the midpoints of $BC$, $AC$ and $AB$, respectively. Let $ P$, $Q$ and $R$ be the intersections of the circumcircle with $LH$, $MH$ and $NH$, respectively, such that $ P$ and $ A$ are on opposite sides of $BC$, $Q$ and $A$ are on opposite sides of $AC$ and $R$ and $C$ are on opposite sides of $AB$. Show that there exists a triangle whose sides have the lengths of the segments $D' P$, $E'Q$, and $F'R$.
Determine all positive integers $n$ such that $5^n - 1$ can be written as a product of an even number of consecutive integers.
Let $a, b$, and $c$ be positive real numbers. Find the largest integer $n$ such that $$\frac{1}{ax + b + c}
+\frac{1}{a + bx + c}+\frac{1}{a + b + cx} \ge \frac{n}{a + b + c},$$
for all $ x \in [0, 1]$ .
There are $14$ students with different heights. At each step, two adjacent students will be swapped. Whatever the first arrangement is, in at least how many steps the students can be lined up?
$\textbf{(A)}\ 42 \qquad\textbf{(B)}\ 43 \qquad\textbf{(C)}\ 45 \qquad\textbf{(D)}\ 52 \qquad\textbf{(E)}\ \text{None}$
Let $a, b, c$ be integers such that \[\frac ab+\frac bc+\frac ca= 3\] Prove that $abc$ is a cube of an integer.
Let $n$ be a positive integer. Initially, a $2n \times 2n$ grid has $k$ black cells and the rest white cells. The following two operations are allowed :
(1) If a $2\times 2$ square has exactly three black cells, the fourth is changed to a black cell;
(2) If there are exactly two black cells in a $2 \times 2$ square, the black cells are changed to white and white to black.
Find the smallest positive integer $k$ such that for any configuration of the $2n \times 2n$ grid with $k$ black cells, all cells can be black after a finite number of operations.