Found problems: 15925
Revenge EL(S)MO 2024, 4
Determine all triples of positive integers $(A,B,C)$ for which some function $f \colon \mathbb Z_{\geq 0} \to \mathbb Z_{\geq 0}$ satisfies
\[ f^{f(y)} (y + f(2x)) + f^{f(y)} (2y) = (Ax+By)^{C} \]
for all nonnegative integers $x$ and $y$, where $f^k$ as usual denotes $f$ composed $k$ times.
Proposed by [i]Benny Wang[/i]
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]
1977 Yugoslav Team Selection Test, Problem 1
Determine the set of all real numbers $\alpha$ with the following property: For each positive $c$ there exists a rational number $\frac mn~(m\in\mathbb Z,n\in\mathbb N)$ different than $\alpha$ such that
$$\left|\alpha-\frac mn\right|<\frac cn.$$
1957 Polish MO Finals, 4
Prove that if $ a \geq 0 $ and $ b \geq 0 $, then
$$ \sqrt{a^2 + b^2} \geq a + b - (2 - \sqrt{2}) \sqrt{ab}.$$
2001 IMO Shortlist, 1
Let $ T$ denote the set of all ordered triples $ (p,q,r)$ of nonnegative integers. Find all functions $ f: T \rightarrow \mathbb{R}$ satisfying
\[ f(p,q,r) = \begin{cases} 0 & \text{if} \; pqr = 0, \\
1 + \frac{1}{6}(f(p + 1,q - 1,r) + f(p - 1,q + 1,r) & \\
+ f(p - 1,q,r + 1) + f(p + 1,q,r - 1) & \\
+ f(p,q + 1,r - 1) + f(p,q - 1,r + 1)) & \text{otherwise} \end{cases}
\]
for all nonnegative integers $ p$, $ q$, $ r$.
1998 Junior Balkan MO, 3
Find all pairs of positive integers $ (x,y)$ such that
\[ x^y \equal{} y^{x \minus{} y}.
\]
[i]Albania[/i]
2018 Saudi Arabia BMO TST, 2
Find all functions $f : R \to R$ such that $f( 2x^3 + f (y)) = y + 2x^2 f (x)$ for all real numbers $x, y$.
1999 Portugal MO, 5
Each of the numbers $a_1,...,a_n$ is equal to $1$ or $-1$. If $a_1a_2 + a_2a_3 + ··· + a_{n-1}a_n + a_na_1 = 0$, proves that $n$ is divisible by $4$.
2022-23 IOQM India, 14
Let $x,y,z$ be complex numbers such that\\
$\hspace{ 2cm} \frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}=9$\\
$\hspace{ 2cm} \frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}=64$\\
$\hspace{ 2cm} \frac{x^3}{y+z}+\frac{y^3}{z+x}+\frac{z^3}{x+y}=488$\\
\\
If $\frac{x}{yz}+\frac{y}{zx}+\frac{z}{xy}=\frac{m}{n}$ where $m,n$ are positive integers with $GCD(m,n)=1$, find $m+n$.
2007 Cuba MO, 4
Find all functions $f : R_+ \to R_+$ such that $$x^2(f(x)+f(y)) = (x+y)f(f(x)y)$$ for all positive real $x, y$.
2016 Tournament Of Towns, 2
Do there exist integers $a$ and $b$ such that :
(a) the equation $x^2 + ax + b = 0$ has no real roots, and the equation $\lfloor x^2 \rfloor + ax + b = 0$ has at
least one real root?
[i](2 points)[/i]
(b) the equation $x^2 + 2ax + b$ = 0 has no real roots, and the equation $\lfloor x^2 \rfloor + 2ax + b = 0$ has at
least one real root?
[i]3 points[/i]
(By $\lfloor k \rfloor$ we denote the integer part of $k$, that is, the greatest integer not exceeding $k$.)
[i]Alexandr Khrabrov[/i]
2016 Poland - Second Round, 3
Determine, whether exists function $f$, which assigns each integer $k$, nonnegative integer $f(k)$ and meets the conditions:
$f(0) > 0$,
for each integer $k$ minimal number of the form $f(k - l) + f(l)$, where $l \in \mathbb{Z}$, equals $f(k)$.
2016 Tournament Of Towns, 5
On a blackboard, several polynomials of degree $37$ are written, each of them has the leading coefficient equal to $1$. Initially all coefficients of each polynomial are non-negative. By one move it is allowed to erase any pair of polynomials $f, g$ and replace it by another pair of polynomials $f_1, g_1$ of degree $37$ with the leading coefficients equal to $1$ such that either $f_1+g_1 = f+g$ or $f_1g_1 = fg$. Prove that it is impossible that after some move each polynomial
on the blackboard has $37$ distinct positive roots. [i](8 points)[/i]
[i]Alexandr Kuznetsov[/i]
2000 Tournament Of Towns, 3
Prove the inequality $$ 1^k+2^k+...+n^k \le \frac{n^{2k}-(n-1)^k}{n^k-(n-1)^k}$$
(L Emelianov)
2003 Singapore Senior Math Olympiad, 3
(i) Find a formula for $S_n = -1^2 \times 2 + 2^2 \times 3 - 3^2 \times 4 + 4^2 \times 5 -... + (-l)^n n^2 \times (n + 1)$ in terms of the positive integer $n$. Justify your answer.
(As an example, one has $1 + 2 + 3 +...+n = \frac{n(n+1)}{2}$)
(ii) Using your formula in (i), find the value of
$ -1^2 \times 2 + 2^2 \times 3 - 3^2 \times 4 + 4^2 \times 5 -... + (-l)^{100} 100^2 \times (100 + 1)$
2008 Bulgaria Team Selection Test, 3
Let $\mathbb{R}^{+}$ be the set of positive real numbers. Find all real numbers $a$ for which there exists a function $f :\mathbb{R}^{+} \to \mathbb{R}^{+}$ such that $3(f(x))^{2}=2f(f(x))+ax^{4}$, for all $x \in \mathbb{R}^{+}$.
2019 Pan-African Shortlist, A1
Let $(a_n)_{n=0}^{\infty}$ be a sequence of real numbers defined as follows:
[list]
[*] $a_0 = 3$, $a_1 = 2$, and $a_2 = 12$; and
[*] $2a_{n + 3} - a_{n + 2} - 8a_{n + 1} + 4a_n = 0$ for $n \geq 0$.
[/list]
Show that $a_n$ is always a strictly positive integer.
2018 Pan African, 3
For any positive integer $x$, we set
$$
g(x) = \text{ largest odd divisor of } x,
$$
$$
f(x) = \begin{cases}
\frac{x}{2} + \frac{x}{g(x)} & \text{ if } x \text{ is even;} \\
2^{\frac{x+1}{2}} & \text{ if } x \text{ is odd.}
\end{cases}
$$
Consider the sequence $(x_n)_{n \in \mathbb{N}}$ defined by $x_1 = 1$, $x_{n + 1} = f(x_n)$. Show that the integer $2018$ appears in this sequence, determine the least integer $n$ such that $x_n = 2018$, and determine whether $n$ is unique or not.
1996 Tuymaada Olympiad, 7
In the set of all positive real numbers define the operation $a * b = a^b$ .
Find all positive rational numbers for which $a * b = b * a$.
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 \[\dfrac{x+y+z}{3} \ge \sqrt[3]{2xyz}.\]
2016 Greece Junior Math Olympiad, 2
Given is that $x, y, z$ are real numbers, different from 0, $x$ and $z$ are different, such that
$(x+y) ^2+(2-xy)=9$ and $(y+z) ^2-(3+yz)=4$
Find the value of $A=(x/y+y^2/x^2+z^3/x^2y)(y/z+z^2/y^2+x^3/y^2z)(z/x+x^2/z^2+y^3/z^2x)=?$
2012 Iran MO (3rd Round), 4
Suppose $f(z)=z^n+a_1z^{n-1}+...+a_n$ for which $a_1,a_2,...,a_n\in \mathbb C$. Prove that the following polynomial has only one positive real root like $\alpha$
\[x^n+\Re(a_1)x^{n-1}-|a_2|x^{n-2}-...-|a_n|\]
and the following polynomial has only one positive real root like $\beta$
\[x^n-\Re(a_1)x^{n-1}-|a_2|x^{n-2}-...-|a_n|.\]
And roots of the polynomial $f(z)$ satisfy $-\beta \le \Re(z) \le \alpha$.
1989 Greece National Olympiad, 1
Find all real solutions of $$ \begin{matrix}
\sqrt{9+x_1}+ \sqrt{9+x_2}+...+ \sqrt{9+x_{100}}=100\sqrt{10}\\
\sqrt{16-x_1}+ \sqrt{16-x_2}+...+ \sqrt{16-x_{100}}=100\sqrt{15}
\end{matrix}$$
2022-IMOC, A3
Find all functions $f:\mathbb R\to \mathbb R$ such that $$xy(f(x+y)-f(x)-f(y))=2f(xy)$$ for all $x,y\in \mathbb R.$
[i]Proposed by USJL[/i]
2022 JHMT HS, 1
If three of the roots of the quartic polynomial $f(x) = x^4 + ax^3 + bx^2 + cx + d$ are $0$, $2$, and $4$, and the sum of $a$, $b$, and $c$ is at most $12$, then find the largest possible value of $f(1)$.