Found problems: 592
2021 Brazil Team Selection Test, 4
[i]Version 1[/i]. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$,
\[
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x .
\]
[i]Version 2[/i]. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$,
\[
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x .
\]
2022 Thailand Mathematical Olympiad, 2
Define a function $f:\mathbb{N}\times \mathbb{N}\to\{-1,1\}$ such that
$$f(m,n)=\begin{cases} 1 &\text{if }m,n\text{ have the same parity, and} \\ -1 &\text{if }m,n\text{ have different parity}\end{cases}$$
for every positive integers $m,n$. Determine the minimum possible value of
$$\sum_{1\leq i<j\leq 2565} ijf(x_i,x_j)$$
across all permutations $x_1,x_2,x_3,\dots,x_{2565}$ of $1,2,\dots,2565$.
2018 Mexico National Olympiad, 4
Let $n\geq 2$ be an integer. For each $k$-tuple of positive integers $a_1, a_2, \ldots, a_k$ such that $a_1+a_2+\cdots +a_k=n$, consider the sums $S_i=1+2+\ldots +a_i$ for $1\leq i\leq k$. Determine, in terms of $n$, the maximum possible value of the product $S_1S_2\cdots S_k$.
[i]Proposed by Misael Pelayo[/i]
2017 Iran MO (2nd Round), 4
Let $x,y$ be two positive real numbers such that $x^4-y^4=x-y$. Prove that
$$\frac{x-y}{x^6-y^6}\leq \frac{4}{3}(x+y).$$
2000 Moldova National Olympiad, Problem 2
Show that if real numbers $x<1<y$ satisfy the inequality
$$2\log x+\log(1-x)\ge3\log y+\log(y-1),$$then $x^3+y^3<2$.
1989 Bundeswettbewerb Mathematik, 2
Find all pairs $(a,b)$ of real numbers such that
$$|\sqrt{1-x^2 }-ax-b| \leq \frac{\sqrt{2} -1}{2}$$
holds for all $x\in [0,1]$.
2011 Peru IMO TST, 2
Let $A_1A_2 \ldots A_n$ be a convex polygon. Point $P$ inside this polygon is chosen so that its projections $P_1, \ldots , P_n$ onto lines $A_1A_2, \ldots , A_nA_1$ respectively lie on the sides of the polygon. Prove that for arbitrary points $X_1, \ldots , X_n$ on sides $A_1A_2, \ldots , A_nA_1$ respectively,
\[\max \left\{ \frac{X_1X_2}{P_1P_2}, \ldots, \frac{X_nX_1}{P_nP_1} \right\} \geq 1.\]
[i]Proposed by Nairi Sedrakyan, Armenia[/i]
2010 Polish MO Finals, 3
Real number $C > 1$ is given. Sequence of positive real numbers $a_1, a_2, a_3, \ldots$, in which $a_1=1$ and $a_2=2$, satisfy the conditions
\[a_{mn}=a_ma_n, \] \[a_{m+n} \leq C(a_m + a_n),\]
for $m, n = 1, 2, 3, \ldots$. Prove that $a_n = n$ for $n=1, 2, 3, \ldots$.
2019 District Olympiad, 4
Find the smallest positive real number $\lambda$ such that for every numbers $a_1,a_2,a_3 \in \left[0, \frac{1}{2} \right]$ and $b_1,b_2,b_3 \in (0, \infty)$ with $\sum\limits_{i=1}^3a_i=\sum\limits_{i=1}^3b_i=1,$ we have $$b_1b_2b_3 \le \lambda (a_1b_1+a_2b_2+a_3b_3).$$
2002 Croatia National Olympiad, Problem 2
Prove that for any positive number $a,b,c$ and any nonnegative integer $p$
$$a^{p+2}+b^{p+2}+c^{p+2}\ge a^pbc+b^pca+c^pab.$$
2012 Kosovo National Mathematical Olympiad, 2
If $a>1,b>1$ are the legths of the catheti of an right triangle and $c$ the length of its hypotenuse, prove that
$a+b\leq c\sqrt 2$
2016 Belarus Team Selection Test, 1
Prove for positive $a,b,c$ that
$$ (a^2+\frac{b^2}{c^2})(b^2+\frac{c^2}{a^2})(c^2+\frac{a^2}{b^2}) \geq abc (a+\frac{1}{a})(b+\frac{1}{b})(c+\frac{1}{c})$$
Kvant 2021, M2663
For every positive integer $m$ prove the inquality
$|\{\sqrt{m}\} - \frac{1}{2}| \geq \frac{1}{8(\sqrt m+1)} $
(The integer part $[x]$ of the number $x$ is the largest integer not exceeding $x$. The fractional part of the number $x$ is a number $\{x\}$ such that $[x]+\{x\}=x$.)
A. Golovanov
2017 Azerbaijan BMO TST, 1
Let $a, b,c$ be positive real numbers.
Prove that $ \sqrt{a^3b+a^3c}+\sqrt{b^3c+b^3a}+\sqrt{c^3a+c^3b}\ge \frac43 (ab+bc+ca)$
1985 IMO Shortlist, 7
The positive integers $x_1, \cdots , x_n$, $n \geq 3$, satisfy $x_1 < x_2 <\cdots< x_n < 2x_1$. Set $P = x_1x_2 \cdots x_n.$ Prove that if $p$ is a prime number, $k$ a positive integer, and $P$ is divisible by $pk$, then $\frac{P}{p^k} \geq n!.$
2021 China Team Selection Test, 4
Suppose $x_1,x_2,...,x_{60}\in [-1,1]$ , find the maximum of
$$ \sum_{i=1}^{60}x_i^2(x_{i+1}-x_{i-1}),$$
where $x_{i+60}=x_i$.
1985 IMO Longlists, 63
Let $x_n = \sqrt[2]{2+\sqrt[3]{3+\cdots+\sqrt[n]{n}}}.$ Prove that
\[x_{n+1}-x_n <\frac{1}{n!} \quad n=2,3,\cdots\]
2017 Vietnamese Southern Summer School contest, Problem 1
Let $x,y,z$ be the non-negative real numbers satisfying $xy+yz+zx\leq 1$. Prove that:
$$1-xy-yz-zx\leq (6-2\sqrt{6})(1-\min\{x,y,z\}).$$
India EGMO 2021 TST, 2
Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of
$$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$
[i]Israel[/i]
1985 IMO Longlists, 22
The positive integers $x_1, \cdots , x_n$, $n \geq 3$, satisfy $x_1 < x_2 <\cdots< x_n < 2x_1$. Set $P = x_1x_2 \cdots x_n.$ Prove that if $p$ is a prime number, $k$ a positive integer, and $P$ is divisible by $pk$, then $\frac{P}{p^k} \geq n!.$
1971 IMO, 1
Let \[ E_n=(a_1-a_2)(a_1-a_3)\ldots(a_1-a_n)+(a_2-a_1)(a_2-a_3)\ldots(a_2-a_n)+\ldots+(a_n-a_1)(a_n-a_2)\ldots(a_n-a_{n-1}). \] Let $S_n$ be the proposition that $E_n\ge0$ for all real $a_i$. Prove that $S_n$ is true for $n=3$ and $5$, but for no other $n>2$.
2011 Germany Team Selection Test, 1
A sequence $x_1, x_2, \ldots$ is defined by $x_1 = 1$ and $x_{2k}=-x_k, x_{2k-1} = (-1)^{k+1}x_k$ for all $k \geq 1.$ Prove that $\forall n \geq 1$ $x_1 + x_2 + \ldots + x_n \geq 0.$
[i]Proposed by Gerhard Wöginger, Austria[/i]
1977 Germany Team Selection Test, 1
We consider two sequences of real numbers $x_{1} \geq x_{2} \geq \ldots \geq x_{n}$ and $\ y_{1} \geq y_{2} \geq \ldots \geq y_{n}.$ Let $z_{1}, z_{2}, .\ldots, z_{n}$ be a permutation of the numbers $y_{1}, y_{2}, \ldots, y_{n}.$ Prove that $\sum \limits_{i=1}^{n} ( x_{i} -\ y_{i} )^{2} \leq \sum \limits_{i=1}^{n}$ $( x_{i} - z_{i})^{2}.$
2022 JBMO Shortlist, N2
Let $a < b < c < d < e$ be positive integers. Prove that
$$\frac{1}{[a, b]} + \frac{1}{[b, c]} + \frac{1}{[c, d]} + \frac{2}{[d, e]} \le 1$$
where $[x, y]$ is the least common multiple of $x$ and $y$ (e.g., $[6, 10] = 30$). When does equality hold?
2024 Junior Balkan Team Selection Tests - Moldova, 10
Let $a \geq b \geq 1 \geq c \geq 0$ be real numbers such that $a+b+c=3$. Show that
$$3 \left( \frac{a}{b}+\frac{b}{a} \right ) \geq 4c^2+\frac{a^2}{b}+\frac{b^2}{a}$$