Found problems: 191
Let $x,y$ be positive numbers, $s$ -- the least of $$\{ x, (y+ 1/x), 1/y\}$$ What is the greatest possible value of $s$? To what $x$ and $y$ does it correspond?
The positive numbers $a, b,c, d, p, q$ are such that $(x+a)(x+b)(x+c)(x+d) = x^4+4px^3+6x^2+4qx+1$ holds for all real numbers $x$. Find the smallest value of $p$ or the largest value of $q$.
Given a triangle $ABC$ with the are $1$. Let $A',B'$ and $C' $ are the midpoints of the sides $[BC], [CA]$ and $[AB]$ respectively. What is the minimal possible area of the common part of two triangles $A'B'C'$ and $KLM$, if the points $K,L$ and $M$ are lying on the segments $[AB'], [CA']$ and $[BC']$ respectively?
Let $A_1$ and $B_1$ be internal points lying on the sides $BC$ and $AC$ of the triangle $ABC$ respectively and segments $AA_1$ and $BB_1$ meet at $O$. The areas of the triangles $AOB_1,AOB$ and $BOA_1$ are distinct prime numbers and the area of the quadrilateral $A_1OB_1C$ is an integer. Find the least possible value of the area of the triangle $ABC$, and argue the existence of such a triangle.
In a mathematical olympiad students received marks for any of the four areas: algebra, geometry, number theory and combinatorics. Any two of the students have distinct marks for all four areas. A group of students is called [i]nice [/i] if all students in the group can be ordered in increasing order simultaneously of at least two of the four areas. Find the least positive integer N, such that among any N students there exist a [i]nice [/i] group of ten students.
On the legs of $\angle AOB$, the segments $OA$ and $OB$ lie, $OA > OB$. Points $M$ and $N$ on lines $OA$ and $OB$, respectively, are such that $AM = BN = x$. Find $x$ for which the length of $MN$ is minimal.
Let $a$ and $b$ be positive whole numbers such that $\frac{4.5}{11}<\frac{a}{b}<\frac{5}{11}$.
Find the fraction $\frac{a}{b}$ for which the sum $a+b$ is as small as possible.
Justify your answer
Let $x, y$ be positive reals such that $x \ne y$. Find the minimum possible value of $(x + y)^2 + \frac{54}{xy(x-y)^2}$ .
Find the minimum of the expression $\frac{a + b + c}{2} -\frac{[a, b] + [b, c] + [c, a]}{a + b + c}$ where the variables $a, b, c$ are any integers greater than $1$ and $[x, y]$ denotes the least common multiple of numbers $x, y$.
(a) On each square of a squared sheet of paper of size $20 \times 20$ there is a soldier. Vanya chooses a number $d$ and Petya moves the soldiers to new squares in such a way that each soldier is moved through a distance of at least $d$ (the distance being measured between the centres of the initial and the new squares) and each square is occupied by exactly one soldier. For which $d$ is this possible?
(Give the maximum possible $d$, prove that it is possible to move the soldiers through distances not less than $d$ and prove that there is no greater $d$ for which this procedure may be carried out.)
(b) Answer the same question as (a), but with a sheet of size $21 \times 21$.
(SS Krotov, Moscow)
The $A$ set consists of integers only. Its minimal element is $1$ and its maximal element is $100$. Every element of $A$ except $1$ equals to the sum of two (may be equal) numbers being contained in $A$. What is the least possible number of elements in $A$?
Write $2013$ as a sum of $m$ prime numbers. The smallest value of $m$ is:
(A): $2$, (B): $3$, (C): $4$, (D): $1$, (E): None of the above.
Let $f$ be a one-to-one function from the set of natural numbers to itself such that $f(mn) = f(m)f(n)$ for all natural numbers $m$ and $n$. What is the least possible value of $f (999)$ ?
Find the minimum value of $S = |x + 1| + |x + 5|+ |x + 14| + |x + 97| + |x + 1920|$.
Let $x_1, x_2, ..., x_n$ be positive reals with sum $1$. Show that $$\frac{x_1^2}{x_1 + x_2}+ \frac{x_2^2}{x_2 + x_3} +... + \frac{x_{n-1}^2}{x_{n-1} + x_n} + \frac{x_n^2}{x_n + x_1} \ge \frac12$$
Let $a, b, c$ be real numbers satisfying the condition $18ab + 9ca + 29bc = 1$.
Find the minimum value of the expression $T = 42a^2 + 34b^2 + 43c^2$.
What is the smallest possible natural number $n$ for which the equation $x^2 -nx + 2014 = 0$ has integer roots?
What is the least possible number of the checkers being required
a) for the $8\times 8$ chess-board,
b) for the $n\times n$ chess-board,
to provide the property:
[i]Every line (of the chess-board fields) parallel to the side or diagonal is occupied by at least one checker[/i] ?
Find minimal value of $a \in \mathbb{R}$ such that system $$\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}=a-1$$ $$\sqrt{x+1}+\sqrt{y+1}+\sqrt{z+1}=a+1$$ has solution in set of real numbers
The point $M$ belongs to the side $[AC]$ of the acute-angle triangle $ABC$. Two circles are circumscribed around triangles $ABM$ and $BCM$ . What $M$ position corresponds to the minimal area of those circles intersection?
Let $ABC$ a right isosceles triangle with hypotenuse $AB=\sqrt2$ . Determine the positions of the points $X,Y,Z$ on the sides $BC,CA,AB$ respectively so that the triangle $XYZ$ is isosceles, right, and with minimum area.
In the triangle $ABC$, $\angle C$ is obtuse and $D$ is a fixed point on the side $BC$, different from $B$ and $C$. For any point $M$ on the side $BC$, different from $D$, the ray $AM$ intersects the circumcircle $S$ of $ABC$ at $N$. The circle through $M, D$ and $N$ meets $S$ again at $P$, different from $N$. Find the location of the point $M$ which minimises $MP$.
What is the smallest perimeter of the convex $32$-gon, having all the vertices in the nodes of cross-lined paper with the sides of its squares equal to $1$?
What minimal number of numbers from the set $\{1,2,...,1982\}$ should be deleted to provide the property:
[i]none of the remained numbers equals to the product of two other remained numbers[/i]?
Given points $A_1 (x_1, y_1, z_1), A_2 (x_2, y_2, z_2), .., A_n (x_n, y_n, z_n)$ let $P (x, y, z)$ be the point which minimizes $\Sigma ( |x - x_i| + |y -y_i| + |z -z_i| )$. Give an example (for each $n > 4$) of points $A_i $ for which the point $P$ lies outside the convex hull of the points $A_i$.