Found problems: 15925
2023 Tuymaada Olympiad, 3
Prove that for every positive integer $n \geq 2$, $$\frac{\sum_{1\leq i \leq n} \sqrt[3]{\frac{i}{n+1}}}{n} \leq \frac{\sum_{1\leq i \leq n-1} \sqrt[3]{\frac{i}{n}}}{n-1}.$$
2024 OMpD, 3
For each positive integer \( n \), let \( f(n) \) be the number of ordered triples \( (a, b, c) \) such that \( a, b, c \in \{1, 2, \ldots, n\} \) and that the two roots (possibly equal) of the quadratic equation \( ax^2 + bx + c = 0 \) are both integers.
(a) Prove that for every positive real number \( C \), there exists a positive integer \( n_C \) such that for all integers \( n \geq n_C \), we have \( f(n) > C \cdot n \).
(b) Prove that for every positive real number \( C \), there exists a positive integer \( n_C \) such that for all integers \( n \geq n_C \), we have \( f(n) < C \cdot n^{\frac{2025}{2024}} \).
2018 Middle European Mathematical Olympiad, 1
Let $Q^+$ denote the set of all positive rational number and let $\alpha\in Q^+.$ Determine all functions $f:Q^+ \to (\alpha,+\infty )$ satisfying $$f(\frac{ x+y}{\alpha}) =\frac{ f(x)+f(y)}{\alpha}$$
for all $x,y\in Q^+ .$
2011 Indonesia TST, 1
Find all $4$-tuple of real numbers $(x, y, z, w)$ that satisfy the following system of equations:
$$x^2 + y^2 + z^2 + w^2 = 4$$
$$\frac{1}{x^2} +\frac{1}{y^2} +\frac{1}{z^2 }+\frac{1}{w^2} = 5 -\frac{1}{(xyzw)^2}$$
2014 Federal Competition For Advanced Students, P2, 5
Show that the inequality $(x^2 + y^2z^2) (y^2 + x^2z^2) (z^2 + x^2y^2) \ge 8xy^2z^3$ is valid for all integers $x, y$ and $z$.When does equality apply?
2023 CMIMC Algebra/NT, 6
Compute the sum of all positive integers $N$ for which there exists a unique ordered triple of non-negative integers $(a,b,c)$ such that $2a+3b+5c=200$ and $a+b+c=N$.
[i]Proposed by Kyle Lee[/i]
1985 Vietnam Team Selection Test, 1
The sequence $ (x_n)$ of real numbers is defined by $ x_1\equal{}\frac{29}{10}$ and $ x_{n\plus{}1}\equal{}\frac{x_n}{\sqrt{x_n^2\minus{}1}}\plus{}\sqrt{3}$ for all $ n\ge 1$. Find a real number $ a$ (if exists) such that $ x_{2k\minus{}1}>a>x_{2k}$.
1967 IMO Shortlist, 6
Prove the following inequality:
\[\prod^k_{i=1} x_i \cdot \sum^k_{i=1} x^{n-1}_i \leq \sum^k_{i=1}
x^{n+k-1}_i,\] where $x_i > 0,$ $k \in \mathbb{N}, n \in
\mathbb{N}.$
2022 Grand Duchy of Lithuania, 1
Given a polynomial with integer coefficients $$P(x) = x^{20} + a_{19}x^{19} +... + a_1x + a_0,$$ having $20$ different real roots. Determine the maximum number of roots such a polynomial $P$ can have in the interval $(99, 100)$.
2012 239 Open Mathematical Olympiad, 4
For some positive numbers $a$, $b$, $c$ and $d$, we know that
$$ \frac{1}{a^3 + 1}+ \frac{1}{b^3 + 1}+ \frac{1}{c^3 + 1} + \frac{1}{d^3 + 1} = 2. $$
Prove that
$$ \frac{1 - a}{a^2 - a + 1} + \frac{1-b}{b^2 - b + 1} + \frac{1-c}{c^2 - c + 1} +\frac{1-d}{d^2 - d + 1} \geq 0. $$
2017 Balkan MO Shortlist, A2
Consider the sequence of rational numbers defined by $x_1=\frac{4}{3}$ and $x_{n+1}=\frac{x_n^2}{x_n^2-x_n+1}$ , $n\geq 1$.
Show that the numerator of the lowest term expression of each sum $\sum_{k=1}^{n}x_k$ is a perfect square.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
2004 Regional Olympiad - Republic of Srpska, 1
Find all real solutions of the equation \[\sqrt[3]{x-1}+\sqrt[3]{3x-1}=\sqrt[3]{x+1}.\]
2002 China Team Selection Test, 1
Given that $ a_1\equal{}1$, $ a_2\equal{}5$, $ \displaystyle a_{n\plus{}1} \equal{} \frac{a_n \cdot a_{n\minus{}1}}{\sqrt{a_n^2 \plus{} a_{n\minus{}1}^2 \plus{} 1}}$. Find a expression of the general term of $ \{ a_n \}$.
2021 All-Russian Olympiad, 2
Let $P(x)$ be a nonzero polynomial of degree $n>1$ with nonnegative coefficients such that function $y=P(x)$ is odd. Is that possible thet for some pairwise distinct points $A_{1}, A_{2}, \dots A_{n}$ on the graph $G: y = P(x)$ the following conditions hold: tangent to $G$ at $A_{1}$ passes through $A_{2}$, tangent to $G$ at $A_{2}$ passes through $A_{3}$, $\dots$, tangent to $G$ at $A_{n}$ passes through $A_{1}$?
1998 Estonia National Olympiad, 3
A function $f$ satisfies the conditions $f (x) \ne 0$ and $f (x+2) = f (x-1) f (x+5)$ for all real x. Show that $f (x+18) = f (x)$ for any real $x$.
1993 China Team Selection Test, 2
Let $S = \{(x,y) | x = 1, 2, \ldots, 1993, y = 1, 2, 3, 4\}$. If $T \subset S$ and there aren't any squares in $T.$ Find the maximum possible value of $|T|.$ The squares in T use points in S as vertices.
1997 IMC, 5
Let $X$ be an arbitrary set and $f$ a bijection from $X$ to $X$. Show that there exist bijections $g,\ g':X\to X$ s.t. $f=g\circ g',\ g\circ g=g'\circ g'=1_X$.
1988 IMO Longlists, 65
The Fibonacci sequence is defined by \[ a_{n+1} = a_n + a_{n-1}, n \geq 1, a_0 = 0, a_1 = a_2 = 1. \] Find the greatest common divisor of the 1960-th and 1988-th terms of the Fibonacci sequence.
2005 AMC 10, 16
The quadratic equation $x^2+mx+n=0$ has roots that are twice those of $x^2+px+m=0$, and none of $m$, $n$, and $p$ is zero. What is the value of $\frac{n}{p}$?
$\text{(A)} \ 1 \qquad \text{(B)} \ 2 \qquad \text{(C)} \ 4 \qquad \text{(D)} \ 8\qquad \text{(E)} \ 16$
2000 Saint Petersburg Mathematical Olympiad, 9.1
On the two sides of the road two lines of trees are planted. On every tree, the number of oaks among itself and its neighbors is written. (For the first and last trees, this is the number of oaks among itself and its only neighbor). Prove that if the two sequences of numbers on the trees are equal, then sequnces of trees on the two sides of the road are equal
[I]Proposed by A. Khrabrov, D. Rostovski[/i]
2006 India Regional Mathematical Olympiad, 3
If $ a,b,c$ are three positive real numbers, prove that $ \frac {a^{2}\plus{}1}{b\plus{}c}\plus{}\frac {b^{2}\plus{}1}{c\plus{}a}\plus{}\frac {c^{2}\plus{}1}{a\plus{}b}\ge 3$
2013 Kazakhstan National Olympiad, 1
Find maximum value of
$|a^2-bc+1|+|b^2-ac+1|+|c^2-ba+1|$ when $a,b,c$ are reals in $[-2;2]$.
2021 Switzerland - Final Round, 4
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]
2022 IFYM, Sozopol, 2
We say that a rectangle and a triangle are [i]similar[/i], if they have the same area and the same perimeter. Let $P$ be a rectangle for which the ratio of the longer to the shorter side is at least $\lambda -1+\sqrt{\lambda (\lambda -2)}$ where $\lambda =\frac{3\sqrt{3}}{2}$. Prove that there exists a tringle that is [i]similar[/i] to $P$.
2013 IFYM, Sozopol, 5
Find all polynomilals $P$ with real coefficients, such that
$(x+1)P(x-1)+(x-1)P(x+1)=2xP(x)$