Found problems: 4776
2019 Jozsef Wildt International Math Competition, W. 30
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[*] Prove that $$\lim \limits_{n \to \infty} \left(n+\frac{1}{4}-\zeta(3)-\zeta(5)-\cdots -\zeta(2n+1)\right)=0$$
[*] Calculate $$\sum \limits_{n=1}^{\infty} \left(n+\frac{1}{4}-\zeta(3)-\zeta(5)-\cdots -\zeta(2n+1)\right)$$
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2000 AMC 12/AHSME, 15
Let $ f$ be a function for which $ f(x/3) \equal{} x^2 \plus{} x \plus{} 1$. Find the sum of all values of $ z$ for which $ f(3z) \equal{} 7$.
$ \textbf{(A)}\ \minus{} 1/3 \qquad \textbf{(B)}\ \minus{} 1/9 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ 5/9 \qquad \textbf{(E)}\ 5/3$
2018 Israel Olympic Revenge, 4
Let $F:\mathbb R^{\mathbb R}\to\mathbb R^{\mathbb R}$ be a function (from the set of real-valued functions to itself) such that
$$F(F(f)\circ g+g)=f\circ F(g)+F(F(F(g)))$$
for all $f,g:\mathbb R\to\mathbb R$.
Prove that there exists a function $\sigma:\mathbb R\to\mathbb R$ such that
$$F(f)=\sigma\circ f\circ\sigma$$
for all $f:\mathbb R\to\mathbb R$.
2016 Balkan MO Shortlist, A3
Find all injective functions $f: \mathbb R \rightarrow \mathbb R$ such that for every real number $x$ and every positive integer $n$,$$ \left|\sum_{i=1}^n i\left(f(x+i+1)-f(f(x+i))\right)\right|<2016$$
[i](Macedonia)[/i]
1983 AIME Problems, 6
Let $a_n = 6^n + 8^n$. Determine the remainder on dividing $a_{83}$ by 49.
2000 Poland - Second Round, 5
Decide whether exists function $f: \mathbb{N} \rightarrow \mathbb{N}$, such that for each $n \in \mathbb{N}$ is $f(f(n) )= 2n$.
2015 Costa Rica - Final Round, F1
A function $f$ defined on integers such that
$f (n) =n + 3$ if $n$ is odd
$f (n) = \frac{n}{2}$ if $n$ is even
If $k$ is an odd integer, determine the values for which $f (f (f (k))) = k$.
2016 District Olympiad, 4
Let $ I $ be an open real interval, and let be two functions $ f,g:I\longrightarrow\mathbb{R} $ satisfying the identity:
$$ x,y\in I\wedge x\neq y\implies\frac{f(x)-g(y)}{x-y} +|x-y|\ge 0. $$
[b]a)[/b] Prove that $ f,g $ are nondecreasing.
[b]b)[/b] Give a concrete example for $ f\neq g. $
1990 Bulgaria National Olympiad, Problem 3
Let $n=p_1p_2\cdots p_s$, where $p_1,\ldots,p_s$ are distinct odd prime numbers.
(a) Prove that the expression
$$F_n(x)=\prod\left(x^{\frac n{p_{i_1}\cdots p_{i_k}}}-1\right)^{(-1)^k},$$where the product goes over all subsets $\{p_{i_1},\ldots,p_{i_k}\}$ or $\{p_1,\ldots,p_s\}$ (including itself and the empty set), can be written as a polynomial in $x$ with integer coefficients.
(b) Prove that if $p$ is a prime divisor of $F_n(2)$, then either $p\mid n$ or $n\mid p-1$.
2004 Harvard-MIT Mathematics Tournament, 7
If $x$, $y$, $k$ are positive reals such that \[3=k^2\left(\dfrac{x^2}{y^2}+\dfrac{y^2}{x^2}\right)+k\left(\dfrac{x}{y}+\dfrac{y}{x}\right),\] find the maximum possible value of $k$.
2011 Nordic, 3
Find all functions $f$ such that
\[f(f(x) + y) = f(x^2-y) + 4yf(x)\]
for all real numbers $x$ and $y$.
2023 Bulgaria EGMO TST, 2
Determine all integers $k$ for which there exists a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}$ such that $f(2023) = 2024$ and $f(ab) = f(a) + f(b) + kf(\gcd(a,b))$ for all positive integers $a$ and $b$.
1997 Yugoslav Team Selection Test, Problem 1
Consider a regular $n$-gon $A_1A_2\ldots A_n$ with area $S$. Let us draw the lines $l_1,l_2,\ldots,l_n$ perpendicular to the plane of the $n$-gon at $A_1,A_2,\ldots,A_n$ respectively. Points $B_1,B_2,\ldots,B_n$ are selected on lines $l_1,l_2,\ldots,l_n$ respectively so that:
(i) $B_1,B_2,\ldots,B_n$ are all on the same side of the plane of the $n$-gon;
(ii) Points $B_1,B_2,\ldots,B_n$ lie on a single plane;
(iii) $A_1B_1=h_1,A_2B_2=h_2,\ldots,A_nB_n=h_n$.
Express the volume of polyhedron $A_1A_2\ldots A_nB_1B_2\ldots B_n$ as a function in $S,h_1,\ldots,h_n$.
2005 Hungary-Israel Binational, 2
Let $F_{n}$ be the $n-$ th Fibonacci number (where $F_{1}= F_{2}= 1$). Consider the functions
$f_{n}(x)=\parallel . . . \parallel |x|-F_{n}|-F_{n-1}|-...-F_{2}|-F_{1}|, g_{n}(x)=| . . . \parallel x-1|-1|-...-1|$ ($F_{1}+...+F_{n}$ one’s).
Show that $f_{n}(x) = g_{n}(x)$ for every real number $x.$
1995 China National Olympiad, 3
Let $n(n>1)$ be an odd. We define $x_k=(x^{(k)}_1,x^{(k)}_2,\cdots ,x^{(k)}_n)$ as follow:
$x_0=(x^{(0)}_1,x^{(0)}_2,\cdots ,x^{(0)}_n)=(1,0,\cdots ,0,1)$;
$ x^{(k)}_i =\begin{cases}0, \quad x^{(k-1)}_i=x^{(k-1)}_{i+1},\\ 1, \quad x^{(k-1)}_i\not= x^{(k-1)}_{i+1},\end{cases} $
$i=1,2,\cdots ,n$, where $x^{(k-1)}_{n+1}= x^{(k-1)}_1$.
Let $m$ be a positive integer satisfying $x_0=x_m$. Prove that $m$ is divisible by $n$.
PEN F Problems, 8
Find all polynomials $W$ with real coefficients possessing the following property: if $x+y$ is a rational number, then $W(x)+W(y)$ is rational.
2002 Moldova National Olympiad, 2
Let $ a,b,c\geq 0$ such that $ a\plus{}b\plus{}c\equal{}1$. Prove that:
$ a^2\plus{}b^2\plus{}c^2\geq 4(ab\plus{}bc\plus{}ca)\minus{}1$
2021 Iran Team Selection Test, 2
Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for any two positive integers $m,n$ we have :
$$f(n)+1400m^2|n^2+f(f(m))$$
1999 South africa National Olympiad, 5
Let $S$ be the set of all rational numbers whose denominators are powers of 3. Let $a$, $b$ and $c$ be given non-zero real numbers. Determine all real-valued functions $f$ that are defined for $x \in S$, satisfy \[ f(x) = af(3x) + bf(3x - 1) + cf(3x - 2) \textrm{ if }0 \leq x \leq 1, \] and are zero elsewhere.
2009 Harvard-MIT Mathematics Tournament, 7
Let $s(n)$ denote the number of $1$'s in the binary representation of $n$. Compute
\[
\frac{1}{255}\sum_{0\leq n<16}2^n(-1)^{s(n)}.
\]
2011 Math Prize for Girls Olympiad, 4
Let $M$ be a matrix with $r$ rows and $c$ columns. Each entry of $M$ is a nonnegative integer. Let $a$ be the average of all $rc$ entries of $M$. If $r > {(10 a + 10)}^c$, prove that $M$ has two identical rows.
2018 Peru IMO TST, 9
A sequence of real numbers $a_1,a_2,\ldots$ satisfies the relation
$$a_n=-\max_{i+j=n}(a_i+a_j)\qquad\text{for all}\quad n>2017.$$
Prove that the sequence is bounded, i.e., there is a constant $M$ such that $|a_n|\leq M$ for all positive integers $n$.
1967 Miklós Schweitzer, 5
Let $ f$ be a continuous function on the unit interval $ [0,1]$. Show that \[ \lim_{n \rightarrow \infty} \int_0^1... \int_0^1f(\frac{x_1+...+x_n}{n})dx_1...dx_n=f(\frac12)\] and \[ \lim_{n \rightarrow \infty} \int_0^1... \int_0^1f (\sqrt[n]{x_1...x_n})dx_1...dx_n=f(\frac1e).\]
2012 Federal Competition For Advanced Students, Part 2, 2
We define $N$ as the set of natural numbers $n<10^6$ with the following property:
There exists an integer exponent $k$ with $1\le k \le 43$, such that $2012|n^k-1$.
Find $|N|$.
2020 LIMIT Category 2, 6
Let $f(x)$ be a real-valued function satisfying $af(x)+bf(-x)=px^2+qx+r$. $a$ and $b$ are distinct real numbers and $p,q,r$ are non-zero real numbers. Then $f(x)=0$ will have real solutions when
(A)$\left(\frac{a+b}{a-b}\right)\leq\frac{q^2}{4pr}$
(B)$\left(\frac{a+b}{a-b}\right)\leq\frac{4pr}{q^2}$
(C)$\left(\frac{a+b}{a-b}\right)\geq\frac{q^2}{4pr}$
(D)$\left(\frac{a+b}{a-b}\right)\geq\frac{4pr}{q^2}$