Found problems: 6530
2014 Putnam, 5
Let $P_n(x)=1+2x+3x^2+\cdots+nx^{n-1}.$ Prove that the polynomials $P_j(x)$ and $P_k(x)$ are relatively prime for all positive integers $j$ and $k$ with $j\ne k.$
2016 Azerbaijan JBMO TST, 1
Let $a,b,c \ge 0$ and $a+b+c=3$. Prove that:
$2(ab+bc+ca)-3abc\ge \sum_{cyc}^{}a\sqrt{\frac{b^2+c^2}{2}}$
1967 IMO Shortlist, 3
Prove that for arbitrary positive numbers the following inequality holds
\[\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \leq \frac{a^8 + b^8 + c^8}{a^3b^3c^3}.\]
2012 Germany Team Selection Test, 3
Let $a,b,c$ be positive real numbers with $a^2+b^2+c^2 \geq 3$. Prove that:
$$\frac{(a+1)(b+2)}{(b+1)(b+5)}+\frac{(b+1)(c+2)}{(c+1)(c+5)}+\frac{(c+1)(a+2)}{(a+1)(a+5)} \geq \frac{3}{2}.$$
1995 IMC, 6
Let $p>1$. Show that there exists a constant $K_{p} >0$ such that for every $x,y\in \mathbb{R}$
with $|x|^{p}+|y|^{p}=2$, we have
$$(x-y)^{2} \leq K_{p}(4-(x+y)^{2}).$$
1974 Bulgaria National Olympiad, Problem 3
(a) Find all real numbers $p$ for which the inequality
$$x_1^2+x_2^2+x_3^2\ge p(x_1x_2+x_2x_3)$$
is true for all real numbers $x_1,x_2,x_3$.
(b) Find all real numbers $q$ for which the inequality
$$x_1^2+x_2^2+x_3^2+x_4^2\ge q(x_1x_2+x_2x_3+x_3x_4)$$
is true for all real numbers $x_1,x_2,x_3,x_4$.
[i]I. Tonov[/i]
2006 Regional Competition For Advanced Students, 1
Let $ 0 < x <y$ be real numbers. Let
$ H\equal{}\frac{2xy}{x\plus{}y}$ , $ G\equal{}\sqrt{xy}$ , $ A\equal{}\frac{x\plus{}y}{2}$ , $ Q\equal{}\sqrt{\frac{x^2\plus{}y^2}{2}}$
be the harmonic, geometric, arithmetic and root mean square (quadratic mean) of $ x$ and $ y$. As generally known $ H<G<A<Q$. Arrange the intervals $ [H,G]$ , $ [G,A]$ and $ [A,Q]$ in ascending order by their length.
2011 China Western Mathematical Olympiad, 1
Given that $0 < x,y < 1$, find the maximum value of $\frac{xy(1-x-y)}{(x+y)(1-x)(1-y)}$
1970 Bulgaria National Olympiad, Problem 6
In space, we are given the points $A,B,C$ and a sphere with center $O$ and radius $1$. Find the point $X$ from the sphere for which the sum $f(X)=|XA|^2+|XB|^2+|XC|^2$ attains its maximal and minimal value. Prove that if the segments $OA,OB,OC$ are pairwise perpendicular and $d$ is the distance from the center $O$ to the centroid of the triangle $ABC$ then:
(a) the maximum of $f(X)$ is equal to $9d^2+3+6d$;
(b) the minimum of $f(X)$ is equal to $9d^2+3-6d$.
[i]K. Dochev and I. Dimovski[/i]
2013 Moldova Team Selection Test, 4
Prove that for any positive real numbers $a_i,b_i,c_i$ with $i=1,2,3$,
$(a_1^3+b_1^3+c_1^3+1)(a_2^3+b_2^3+c_2^3+1)(a_3^3+b_3^3+c_3^3+1)\geq \frac{3}{4} (a_1+b_1+c_1)(a_2+b_2+c_2)(a_3+b_3+c_3)$
2022-2023 OMMC, 21
Define the minimum real $C$ where for any reals $0 = a_0 < a_{1} < \dots < a_{1000}$ then $$\min_{0 \le k \le 1000} (a_{k}^2 + (1000-k)^2) \le C(a_1+ \dots + a_{1000})$$ holds. Find $\lfloor 100C \rfloor.$
2014 Iran MO (2nd Round), 3
Let $ x,y,z $ be three non-negative real numbers such that \[x^2+y^2+z^2=2(xy+yz+zx). \] Prove that \[\dfrac{x+y+z}{3} \ge \sqrt[3]{2xyz}.\]
1992 Polish MO Finals, 1
The functions $f_0, f_1, f_2, ...$ are defined on the reals by $f_0(x) = 8$ for all $x$, $f_{n+1}(x) = \sqrt{x^2 + 6f_n(x)}$. For all $n$ solve the equation $f_n(x) = 2x$.
1967 Miklós Schweitzer, 4
Let $ a_1,a_2,...,a_N$ be positive real numbers whose sum equals $ 1$. For a natural number $ i$, let $ n_i$ denote the number
of $ a_k$ for which $ 2^{1-i} \geq a_k \geq 2^{-i}$ holds. Prove that
\[ \sum_{i=1}^{\infty} \sqrt{n_i2^{-i}} \leq 4+\sqrt{\log_2 N}.\]
[i]L. Leinder[/i]
2008 JBMO Shortlist, 7
Let $a, b$ and $c$ be positive real numbers such that $abc = 1$. Prove the inequality
$\Big(ab + bc +\frac{1}{ca}\Big)\Big(bc + ca +\frac{1}{ab}\Big)\Big(ca + ab +\frac{1}{bc}\Big)\ge (1 + 2a)(1 + 2b)(1 + 2c)$.
2018 China Northern MO, 7
If $a$,$b$,$c$ are positive reals, prove that
$$\frac{a+bc}{a+a^2}+\frac{b+ca}{b+b^2}+\frac{c+ab}{c+c^2} \geq 3$$
1972 Bulgaria National Olympiad, Problem 6
It is given a tetrahedron $ABCD$ for which two points of opposite edges are mutually perpendicular. Prove that:
(a) the four altitudes of $ABCD$ intersects at a common point $H$;
(b) $AH+BH+CH+DH<p+2R$, where $p$ is the sum of the lengths of all edges of $ABCD$ and $R$ is the radii of the sphere circumscribed around $ABCD$.
[i]H. Lesov[/i]
1962 Poland - Second Round, 4
Prove that if the sides $ a $, $ b $, $ c $ of a triangle satisfy the inequality
$$a < b < c$$then the angle bisectors $ d_a $, $ d_b $, $ d_c $ of opposite angles satisfy the inequality
$$ d_a > d_b > d_c.$$
2019 Belarus Team Selection Test, 1.4
Let the sequence $(a_n)$ be constructed in the following way:
$$
a_1=1,\mbox{ }a_2=1,\mbox{ }a_{n+2}=a_{n+1}+\frac{1}{a_n},\mbox{ }n=1,2,\ldots.
$$
Prove that $a_{180}>19$.
[i](Folklore)[/i]
2007 Stars of Mathematics, 3
Let $ n\ge 3 $ be a natural number and $ A_0A_1...A_{n-1} $ a regular polygon. Consider $ B_0 $ on the segment $ A_0A_1 $ such that $ A_0B_0<\frac{1}{2}A_0A_1; B_1 $ on $ A_1A_2 $ so that $ A_1B_1<\frac{1}{2} A_1A_2; $ etc.; $ B_{n-2} $ on $ A_{n-2}A_{n-1} $ so that $ A_{n-2}B_{n-2} <\frac{1}{2} A_{n-2}A_{n-1} , $ and $ B_{n-1} $ on $ A_{n-1}A_0 $ with $ A_{n-1}B_{n-1} <\frac{1}{2} A_{n-1}A_{0} . $
Show that the perimeter of any ploygon that has its vertices on the segments $ A_1B_1,A_2B_2,...,A_{n-1}B_{n-1}, $ is equal or greater than the perimeter of $ B_0B_1...B_{n-1} . $
1992 Vietnam Team Selection Test, 3
Let $ABC$ a triangle be given with $BC = a$, $CA = b$, $AB = c$ ($a \neq b \neq c \neq a$). In plane ($ABC$) take the points $A'$, $B'$, $C'$ such that:
[b]I.[/b] The pairs of points $A$ and $A'$, $B$ and $B'$, $C$ and $C'$ either all lie in one side either all lie in different sides under the lines $BC$, $CA$, $AB$ respectively;
[b]II.[/b] Triangles $A'BC$, $B'CA$, $C'AB$ are similar isosceles triangles.
Find the value of angle $A'BC$ as function of $a, b, c$ such that lengths $AA', BB', CC'$ are not sides of an triangle. (The word "triangle" must be understood in its ordinary meaning: its vertices are not collinear.)
1992 Turkey Team Selection Test, 2
There are $n$ boxes which is numbere from $1$ to $n$. The box with number $1$ is open, and the others are closed. There are $m$ identical balls ($m\geq n$). One of the balls is put into the open box, then we open the box with number $2$. Now, we put another ball to one of two open boxes, then we open the box with number $3$. Go on until the last box will be open. After that the remaining balls will be randomly put into the boxes. In how many ways this arrangement can be done?
1979 IMO Longlists, 3
Is it possible to partition $3$-dimensional Euclidean space into $1979$ mutually isometric subsets?
2012-2013 SDML (Middle School), 5
If $a$ and $b$ are positive integers such that $\frac{1}{a}+\frac{1}{b}=\frac{1}{9}$, what is the greatest possible value of $a+b$?
2024 Bundeswettbewerb Mathematik, 2
Determine the set of all real numbers $r$ for which there exists an infinite sequence $a_1,a_2,\dots$ of positive integers satisfying the following three properties:
(1) No number occurs more than once in the sequence.
(2) The sum of two different elements of the sequence is never a power of two.
(3) For all positive integers $n$, we have $a_n<r \cdot n$.