Found problems: 15925
2015 BMT Spring, 4
Let $\{a_n\}$ be a sequence of real numbers with $a_1=-1$, $a_2=2$ and for all $n\ge3$,
$$a_{n+1}-a_n-a_{n+2}=0.$$
Find $a_1+a_2+a_3+\ldots+a_{2015}$.
2014 IMO Shortlist, A2
Define the function $f:(0,1)\to (0,1)$ by \[\displaystyle f(x) = \left\{ \begin{array}{lr} x+\frac 12 & \text{if}\ \ x < \frac 12\\ x^2 & \text{if}\ \ x \ge \frac 12 \end{array} \right.\] Let $a$ and $b$ be two real numbers such that $0 < a < b < 1$. We define the sequences $a_n$ and $b_n$ by $a_0 = a, b_0 = b$, and $a_n = f( a_{n -1})$, $b_n = f (b_{n -1} )$ for $n > 0$. Show that there exists a positive integer $n$ such that \[(a_n - a_{n-1})(b_n-b_{n-1})<0.\]
[i]Proposed by Denmark[/i]
2017 India PRMO, 7
Find the number of positive integers $n$, such that $\sqrt{n} + \sqrt{n + 1} < 11$.
2010 IberoAmerican Olympiad For University Students, 7
(a) Prove that, for any positive integers $m\le \ell$ given, there is a positive integer $n$ and positive integers $x_1,\cdots,x_n,y_1,\cdots,y_n$ such that the equality \[ \sum_{i=1}^nx_i^k=\sum_{i=1}^ny_i^k\] holds for every $k=1,2,\cdots,m-1,m+1,\cdots,\ell$, but does not hold for $k=m$.
(b) Prove that there is a solution of the problem, where all numbers $x_1,\cdots,x_n,y_1,\cdots,y_n$ are distinct.
[i]Proposed by Ilya Bogdanov and Géza Kós.[/i]
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 \}$.
2002 Bundeswettbewerb Mathematik, 2
We consider the sequences strictely increasing $(a_0,a_1,...)$ of naturals which have the following property :
For every natural $n$, there is exactly one representation of $n$ as $a_i+2a_j+4a_k$, where $i,j,k$ can be equal.
Prove that there is exactly a such sequence and find $a_{2002}$
1977 IMO Longlists, 2
Find all functions $f : \mathbb{N}\rightarrow \mathbb{N}$ satisfying following condition:
\[f(n+1)>f(f(n)), \quad \forall n \in \mathbb{N}.\]
2025 VJIMC, 2
Determine all real numbers $x>1$ such that
\[ \left\lfloor\frac{n+1}{x}\right\rfloor = n - \left\lfloor \frac{n}{x} \right\rfloor + \left \lfloor \frac{\left \lfloor \frac{n}{x} \right\rfloor}{x}\right \rfloor - \left \lfloor \frac{\left \lfloor \frac{\left\lfloor \frac{n}{x} \right\rfloor}{x} \right\rfloor}{x}\right \rfloor + \cdots \]
for any positive integer $n$.
2019 India PRMO, 8
How many positive integers $n$ are there such that $3 \leq n \leq 100$ and $x^{2^{n}} + x + 1$ is divisible by $x^2 + x + 1$?
2018 China Team Selection Test, 6
Suppose $a_i, b_i, c_i, i=1,2,\cdots ,n$, are $3n$ real numbers in the interval $\left [ 0,1 \right ].$ Define $$S=\left \{ \left ( i,j,k \right ) |\, a_i+b_j+c_k<1 \right \}, \; \; T=\left \{ \left ( i,j,k \right ) |\, a_i+b_j+c_k>2 \right \}.$$ Now we know that $\left | S \right |\ge 2018,\, \left | T \right |\ge 2018.$ Try to find the minimal possible value of $n$.
2022 LMT Spring, 6
For all $y$, define cubic $f_y (x)$ such that $f_y (0) = y$, $f_y (1) = y +12$, $f_y (2) = 3y^2$, $f_y (3) = 2y +4$. For all $y$, $f_y(4)$ can be expressed in the form $ay^2 +by +c$ where $a,b,c$ are integers. Find $a +b +c$.
2007 IMAR Test, 1
For real numbers $ x_{i}>1,1\leq i\leq n,n\geq 2,$ such that:
$ \frac{x_{i}^2}{x_{i}\minus{}1}\geq S\equal{}\displaystyle\sum^n_{j\equal{}1}x_{j},$ for all $ i\equal{}1,2\dots, n$
find, with proof, $ \sup S.$
2010 Iran MO (3rd Round), 7
[b]interesting function[/b]
$S$ is a set with $n$ elements and $P(S)$ is the set of all subsets of $S$ and
$f : P(S) \rightarrow \mathbb N$
is a function with these properties:
for every subset $A$ of $S$ we have $f(A)=f(S-A)$.
for every two subsets of $S$ like $A$ and $B$ we have
$max(f(A),f(B))\ge f(A\cup B)$
prove that number of natural numbers like $x$ such that there exists $A\subseteq S$ and $f(A)=x$ is less than $n$.
time allowed for this question was 1 hours and 30 minutes.
1984 IMO Shortlist, 6
Let $c$ be a positive integer. The sequence $\{f_n\}$ is defined as follows:
\[f_1 = 1, f_2 = c, f_{n+1} = 2f_n - f_{n-1} + 2 \quad (n \geq 2).\]
Show that for each $k \in \mathbb N$ there exists $r \in \mathbb N$ such that $f_kf_{k+1}= f_r.$
2018 European Mathematical Cup, 1
A partition of a positive integer is even if all its elements are even numbers. Similarly, a partition
is odd if all its elements are odd. Determine all positive integers $n$ such that the number of even partitions of
$n$ is equal to the number of odd partitions of $n$.
Remark: A partition of a positive integer $n$ is a non-decreasing sequence of positive integers whose sum of
elements equals $n$. For example, $(2; 3; 4), (1; 2; 2; 2; 2)$ and $(9) $ are partitions of $9.$
1971 Swedish Mathematical Competition, 4
Find
\[
\frac{65533^3 + 65534^3 + 65535^3 + 65536^3 + 65537^3 + 65538^3+ 65539^3}{32765\cdot 32766 + 32767\cdot 32768 + 32768\cdot 32769 + 32770\cdot 32771}
\]
2021 South East Mathematical Olympiad, 3
Let $p$ be an odd prime and $\{u_i\}_{i\ge 0}$be an integer sequence.
Let $v_n=\sum_{i=0}^{n} C_{n}^{i} p^iu_i$ where $C_n^i$ denotes the binomial coefficients.
If $v_n=0$ holds for infinitely many $n$ , prove that it holds for every positive integer $n$.
2009 Saint Petersburg Mathematical Olympiad, 6
$(x_n)$ is sequence, such that $x_{n+2}=|x_{n+1}|-x_n$. Prove, that it is periodic.
2012 Kyoto University Entry Examination, 3
When real numbers $x,\ y$ moves in the constraint with $x^2+xy+y^2=6.$
Find the range of $x^2y+xy^2-x^2-2xy-y^2+x+y.$
30 points
Russian TST 2016, P1
The positive numbers $a, b, c$ are such that $a^2<16bc, b^2<16ca$ and $c^2<16ab$. Prove that \[a^2+b^2+c^2<2(ab+bc+ca).\]
1999 Junior Balkan Team Selection Tests - Romania, 3
Consider the set $ \mathcal{M}=\left\{ \gcd(2n+3m+13,3n+5m+1,6n+6m-1) | m,n\in\mathbb{N} \right\} . $
Show that there is a natural $ k $ such that the set of its positive divisors is $ \mathcal{M} . $
[i]Dan Brânzei[/i]
2019 Latvia Baltic Way TST, 2
Let $\mathbb R$ be set of real numbers. Determine all functions $f:\mathbb R\to \mathbb R$ such that
$$f(y^2 - f(x)) = yf(x)^2+f(x^2y+y)$$
holds for all real numbers $x; y$
2019 Switzerland - Final Round, 3
Find all periodic sequences $x_1,x_2,\dots$ of strictly positive real numbers such that $\forall n \geq 1$ we have $$x_{n+2}=\frac{1}{2} \left( \frac{1}{x_{n+1}}+x_n \right)$$
1984 IMO Longlists, 40
Find one pair of positive integers $a,b$ such that $ab(a+b)$ is not divisible by $7$, but $(a+b)^7-a^7-b^7$ is divisible by $7^7$.
2012 ELMO Problems, 3
Let $f,g$ be polynomials with complex coefficients such that $\gcd(\deg f,\deg g)=1$. Suppose that there exist polynomials $P(x,y)$ and $Q(x,y)$ with complex coefficients such that $f(x)+g(y)=P(x,y)Q(x,y)$. Show that one of $P$ and $Q$ must be constant.
[i]Victor Wang.[/i]