Found problems: 191
Let $a, b, c$ be the side-lengths of triangle $ABC$ with $a+b+c = 12$.
Determine the smallest value of $M =\frac{a}{b + c - a}+\frac{4b}{c + a - b}+\frac{9c}{a + b - c}$.
A triangle $ABC$ is inscribed in a circle $G$. For any point $M$ inside the triangle, $A_1$ denotes the intersection of the ray $AM$ with $G$. Find the locus of point $M$ for which $\frac{BM\cdot CM}{MA_1}$ is minimal, and find this minimum value.
Let $a$ and $b$ be two positive reals such that the following inequality
\[ ax^3 + by^2 \geq xy - 1 \] is satisfied for any positive reals $x, y \geq 1$. Determine the smallest possible value of $a^2 + b$.
[i]Proposed by Fajar Yuliawan[/i]
Let $a, b,c$ and $d$ be real numbers such that $a + b + c + d = 2$ and $ab + bc + cd + da + ac + bd = 0$.
Find the minimum value and the maximum value of the product $abcd$.
A convex polygon $P$ lies on a flat wooden table. You are allowed to drive some nails into the table. The nails must not go through $P$, but they may touch its boundary. We say that a set of nails blocks $P$ if the nails make it impossible to move $P$ without lifting it off the table. What is the minimum number of nails that suffices to block any convex polygon $P$?
(N. Beluhov, S. Gerdgikov)
Of all parallelograms of a given area find the one with the shortest possible longer diagonal.
Let $a$ and $b$ be natural numbers such that $2a-b$, $a-2b$ and $a+b$ are all distinct squares. What is the smallest possible value of $b$ ?
Let $n$ a positive integer and let $x_1, \ldots, x_n, y_1, \ldots, y_n$ real positive numbers such that $x_1+\ldots+x_n=y_1+\ldots+y_n=1$. Prove that:
$$|x_1-y_1|+\ldots+|x_n-y_n|\leq 2-\underset{1\leq i\leq n}{min} \;\dfrac{x_i}{y_i}-\underset{1\leq i\leq n}{min} \;\dfrac{y_i}{x_i}$$
Let $a, b$ be two real numbers that satisfy $a^3 + b^3 = 8-6ab$.
Find the maximum value and the minimum value that $a + b$ can take.
Given an infinite sheet of square ruled paper. Some of the squares contain a piece. A move consists of a piece jumping over a piece on a neighbouring square (which shares a side) onto an empty square and removing the piece jumped over. Initially, there are no pieces except in an $m x n$ rectangle ($m, n > 1$) which has a piece on each square. What is the smallest number of pieces that can be left after a series of moves?
$4$ men stand at the entrance of a dark tunnel. Man $A$ needs $10$ minutes to pass through the tunnel, man $B$ needs $5$ minutes, man $C$ needs $2$ minutes and man $D$ needs $1$ minute. There is only one torch, that may be used from anyone that passes through the tunnel. Additionaly, at most $2$ men can pass through at the same time using the existing torch.
Determine the smallest possible time the four men need to reach the exit of the tunnel.
Show that $1 \le n^{1/n} \le 2$ for all positive integers $n$.
Find the smallest $k$ such that $1 \le n ^{1/n} \le k$ for all positive integers $n$.
Alex is going to make a set of cubical blocks of the same size and to write a digit on each of their faces so that it would be possible to form every $30$-digit integer with these blocks. What is the minimal number of blocks in a set with this property? (The digits $6$ and $9$ do not turn one into another.)
Points $A_1,A_2,...,A_n$ are selected from the equilateral triangle with a side that is equal to $1$. Denote by $d_k$ the least distance from $A_k$ to all other selected points. Prove that $d_1^2+...+d_n^2 \le 3,5$.
There are $21$ towns. Each airline runs direct flights between every pair of towns in a group of five. What is the minimum number of airlines needed to ensure that at least one airline runs direct flights between every pair of towns?
Two grasshoppers sit at opposite ends of the interval $[0, 1]$. A finite number of points (greater than zero) in the interval are marked. A move is for a grasshopper to select a marked point and jump over it to the equidistant point the other side. This point must lie in the interval for the move to be allowed, but it does not have to be marked. What is the smallest $n$ such that if each grasshopper makes $n$ moves or less, then they end up with no marked points between them?
Consider a rectangular array of single digits $d_{i,j}$ with 10 rows and 7 columns, such that $d_{i+1,j}-d_{i,j}$ is always 1 or -9 for all $1 \leq i \leq 9$ and all $1 \leq j \leq 7$, as in the example below. For $1 \leq i \leq 10$, let $m_i$ be the median of $d_{i,1}$, ..., $d_{i,7}$. Determine the least and greatest possible values of the mean of $m_1$, $m_2$, ..., $m_{10}$.
Example:
[img]https://cdn.artofproblemsolving.com/attachments/8/a/b77c0c3aeef14f0f48d02dde830f979eca1afb.png[/img]
Let $n$ be a fixed positive integer and let $x, y$ be positive integers such that $xy = nx+ny$.
Determine the minimum and the maximum of $x$ in terms of $n$.
A triangle $ABC$ and two lines $\ell_1, \ell_2$ are given. Through an arbitrary point $D$ on the side $AB$, a line parallel to $\ell_1$ intersects the $AC$ at point $E$ and a line parallel to $\ell_2$ intersects the $BC$ at point $F$. Construct a point $D$ for which the segment $EF$ has the smallest length.
NASA has proposed populating Mars with $2,004$ settlements. The only way to get from one settlement to another will be by a connecting tunnel. A bored bureaucrat draws on a map of Mars, randomly placing $N$ tunnels connecting the settlements in such a way that no two settlements have more than one tunnel connecting them. What is the smallest value of $N$ that guarantees that, no matter how the tunnels are drawn, it will be possible to travel between any two settlements?
Find minimum of $x+y+z$ where $x$, $y$ and $z$ are real numbers such that $x \geq 4$, $y \geq 5$, $z \geq 6$ and $x^2+y^2+z^2 \geq 90$
A village is constructed in the form of a square, consisting of $9$ blocks , each of side length $\ell$, in a $3 \times 3$ formation . Each block is bounded by a bitumen road . If we commence at a corner of the village, what is the smallest distance we must travel along bitumen roads , if we are to pass along each section of bitumen road at least once and finish at the same corner?
(Muscovite folklore)
The real numbers $x, y$ and $z$ are such that $x^2 + y^2 + z^2 = 1$.
a) Determine the smallest and the largest possible values of $xy + yz - xz$.
b) Prove that there does not exist a triple $(x, y, z)$ of rational numbers, which attains any of the two values in a).
Convex quadrilateral $ABCD$ is given. Lines $BC$ and $AD$ intersect at point $O$, with $B$ lying on the segment $OC$, and $A$ on the segment $OD$. $I$ is the center of the circle inscribed in the $OAB$ triangle, $J$ is the center of the circle exscribed in the triangle $OCD$ touching the side of $CD$ and the extensions of the other two sides. The perpendicular from the midpoint of the segment $IJ$ on the lines $BC$ and $AD$ intersect the corresponding sides of the quadrilateral (not the extension) at points $X$ and $Y$. Prove that the segment $XY$ divides the perimeter of the quadrilateral$ABCD$ in half, and from all segments with this property and ends on $BC$ and $AD$, segment $XY$ has the smallest length.
Find the least possible value of $(x + y)(y + z)$ for positive reals satisfying $(x + y + z) xyz = 1$.