Found problems: 15925
2001 USA Team Selection Test, 2
Express \[ \sum_{k=0}^n (-1)^k (n-k)!(n+k)! \] in closed form.
2021 South East Mathematical Olympiad, 1
A sequence $\{a_n\}$ is defined recursively by $a_1=\frac{1}{2}, $ and for $n\ge 2,$ $0<a_n\leq a_{n-1}$ and
\[a_n^2(a_{n-1}+1)+a_{n-1}^2(a_n+1)-2a_na_{n-1}(a_na_{n-1}+a_n+1)=0.\]
$(1)$ Determine the general formula of the sequence $\{a_n\};$
$(2)$ Let $S_n=a_1+\cdots+a_n.$ Prove that for $n\ge 1,$ $\ln\left(\frac{n}{2}+1\right)<S_n<\ln(n+1).$
2023 Thailand TST, 1
Let $(a_n)_{n\geq 1}$ be a sequence of positive real numbers with the property that
$$(a_{n+1})^2 + a_na_{n+2} \leq a_n + a_{n+2}$$
for all positive integers $n$. Show that $a_{2022}\leq 1$.
2021 Mexico National Olympiad, 1
The real positive numbers $a_1, a_2,a_3$ are three consecutive terms of an arithmetic progression, and similarly, $b_1, b_2, b_3$ are distinct real positive numbers and consecutive terms of an arithmetic progression. Is it possible to use three segments of lengths $a_1, a_2, a_3$ as bases, and other three segments of lengths $b_1, b_2, b_3$ as altitudes, to construct three rectangles of equal area ?
2002 Switzerland Team Selection Test, 9
For each real number $a$ and integer $n \ge 1$ prove the inequality $a^n +\frac{1}{a^n} -2 \ge n^2 \left(a +\frac{1}{a} -2\right)$ and find the cases of equality.
2018 Nepal National Olympiad, 1c
[b]Problem Section #1
c) Find all pairs $(m, n)$ of non-negative integers for which $m^2+2.3^n=m(2^{n+1}-1).$
2016 CCA Math Bonanza, T6
Consider the polynomials $P\left(x\right)=16x^4+40x^3+41x^2+20x+16$ and $Q\left(x\right)=4x^2+5x+2$. If $a$ is a real number, what is the smallest possible value of $\frac{P\left(a\right)}{Q\left(a\right)}$?
[i]2016 CCA Math Bonanza Team #6[/i]
2007 IMC, 6
Let $ f \ne 0$ be a polynomial with real coefficients. Define the sequence $ f_{0}, f_{1}, f_{2}, \ldots$ of polynomials by $ f_{0}= f$ and $ f_{n+1}= f_{n}+f_{n}'$ for every $ n \ge 0$. Prove that there exists a number $ N$ such that for every $ n \ge N$, all roots of $ f_{n}$ are real.
2023 Romanian Master of Mathematics, 3
Let $n\geq 2$ be an integer and let $f$ be a $4n$-variable polynomial with real coefficients. Assume that, for any $2n$ points $(x_1,y_1),\dots,(x_{2n},y_{2n})$ in the Cartesian plane, $f(x_1,y_1,\dots,x_{2n},y_{2n})=0$ if and only if the points form the vertices of a regular $2n$-gon in some order, or are all equal.
Determine the smallest possible degree of $f$.
(Note, for example, that the degree of the polynomial $$g(x,y)=4x^3y^4+yx+x-2$$ is $7$ because $7=3+4$.)
[i]Ankan Bhattacharya[/i]
2010 Postal Coaching, 2
Find all non-negative integers $m,n,p,q$ such that \[ p^mq^n = (p+q)^2 +1 . \]
2014 Hanoi Open Mathematics Competitions, 13
Let $a,b, c > 0$ and $abc = 1$. Prove that $\frac{a - 1}{c}+\frac{c - 1}{b}+\frac{b - 1}{a} \ge 0$
2018 Hong Kong TST, 2
Find all polynomials $f$ such that $f$ has non-negative integer coefficients, $f(1)=7$ and $f(2)=2017$.
2021 New Zealand MO, 2
Prove that $$x^2 +\frac{8}{xy}+ y^2 \ge 8$$ for all positive real numbers $x$ and $y$.
1995 South africa National Olympiad, 3
Suppose that $a_1,a_2,\dots,a_n$ are the numbers $1,2,3,\dots,n$ but written in any order. Prove that
\[(a_1-1)^2+(a_2-2)^2+\cdots+(a_n-n)^2\]
is always even.
2007 AIME Problems, 8
The polynomial $P(x)$ is cubic. What is the largest value of $k$ for which the polynomials $Q_{1}(x) = x^{2}+(k-29)x-k$ and $Q_{2}(x) = 2x^{2}+(2k-43)x+k$ are both factors of $P(x)$?
2004 Gheorghe Vranceanu, 4
Prove that $ \left\{ (x,y)\in\mathbb{C}^2 |x^2+y^2=1 \right\} =\{ (1,0)\}\cup \left\{ \left( \frac{z^2-1}{z^2+1} ,\frac{2z}{z^2+1} \right) | z\in\mathbb{C}\setminus \{\pm \sqrt{-1}\} \right\} . $
2013 IMO, 5
Let $\mathbb Q_{>0}$ be the set of all positive rational numbers. Let $f:\mathbb Q_{>0}\to\mathbb R$ be a function satisfying the following three conditions:
(i) for all $x,y\in\mathbb Q_{>0}$, we have $f(x)f(y)\geq f(xy)$;
(ii) for all $x,y\in\mathbb Q_{>0}$, we have $f(x+y)\geq f(x)+f(y)$;
(iii) there exists a rational number $a>1$ such that $f(a)=a$.
Prove that $f(x)=x$ for all $x\in\mathbb Q_{>0}$.
[i]Proposed by Bulgaria[/i]
2003 Purple Comet Problems, 16
Find the largest real number $x$ such that \[\left(\dfrac{x}{x-1}\right)^2+\left(\dfrac{x}{x+1}\right)^2=\dfrac{325}{144}.\]
2011 Morocco National Olympiad, 2
Solve in $(\mathbb{R}_{+}^{*})^{4}$ the following system :
$\left\{\begin{matrix}
x+y+z+t=4\\
\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{t}=5-\frac{1}{xyzt}
\end{matrix}\right.$
2004 Balkan MO, 1
The sequence $\{a_n\}_{n\geq 0}$ of real numbers satisfies the relation:
\[ a_{m+n} + a_{m-n} - m + n -1 = \frac12 (a_{2m} + a_{2n}) \]
for all non-negative integers $m$ and $n$, $m \ge n$. If $a_1 = 3$ find $a_{2004}$.
2024 Bulgaria MO Regional Round, 12.2
Let $N$ be a positive integer. The sequence $x_1, x_2, \ldots$ of non-negative reals is defined by $$x_n^2=\sum_{i=1}^{n-1} \sqrt{x_ix_{n-i}}$$ for all positive integers $n>N$. Show that there exists a constant $c>0$, such that $x_n \leq \frac{n} {2}+c$ for all positive integers $n$.
2014 Belarus Team Selection Test, 2
Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Prove that $$\frac{a^2}{(b+c)^3}+\frac{b^2}{(c+a)^3}+\frac{c^2}{(a+b)^3}\geq \frac98$$
2017 Caucasus Mathematical Olympiad, 1
Two points $A$ and $B$ lie on two branches of hyperbola given by equation $y=\frac1x$. Let $A_x$ and $A_y$ be projections of $A$ onto coordinate axis, similarly, let $B_x$ and $B_y$ be projections of $B$ onto coordinate axis. Prove that triangles $AB_xB_y$ and $BA_xA_y$ have equal areas.
2012 Balkan MO Shortlist, A6
Let $k$ be a positive integer. Find the maximum value of \[a^{3k-1}b+b^{3k-1}c+c^{3k-1}a+k^2a^kb^kc^k,\] where $a$, $b$, $c$ are non-negative reals such that $a+b+c=3k$.
2021 Lusophon Mathematical Olympiad, 4
Let $x_1, x_2, x_3, x_4, x_5\in\mathbb{R}^+$ such that
$$x_1^2-x_1x_2+x_2^2=x_2^2-x_2x_3+x_3^2=x_3^2-x_3x_4+x_4^2=x_4^2-x_4x_5+x_5^2=x_5^2-x_5x_1+x_1^2$$
Prove that $x_1=x_2=x_3=x_4=x_5$.