Found problems: 30
2004 IMO Shortlist, 2
Let $a_0$, $a_1$, $a_2$, ... be an infinite sequence of real numbers satisfying the equation $a_n=\left|a_{n+1}-a_{n+2}\right|$ for all $n\geq 0$, where $a_0$ and $a_1$ are two different positive reals.
Can this sequence $a_0$, $a_1$, $a_2$, ... be bounded?
[i]Proposed by Mihai Bălună, Romania[/i]
1962 Putnam, B3
Let $S$ be a convex region in the euclidean plane containing the origin. Assume that every ray from the origin has at least one point outside $S$. Prove that $S$ is bounded.
2005 Morocco TST, 3
Let $a_1,a_2,\ldots$ be an infinite sequence of real numbers, for which there exists a real number $c$ with $0\leq a_i\leq c$ for all $i$, such that \[\left\lvert a_i-a_j \right\rvert\geq \frac{1}{i+j} \quad \text{for all }i,\ j \text{ with } i \neq j. \] Prove that $c\geq1$.
2024 Mexican University Math Olympiad, 6
Let \( p \) be a monic polynomial with all distinct real roots. Show that there exists \( K \) such that
\[
(p(x)^2)'' \leq K(p'(x))^2.
\]
2003 IMO Shortlist, 3
Consider pairs of the sequences of positive real numbers \[a_1\geq a_2\geq a_3\geq\cdots,\qquad b_1\geq b_2\geq b_3\geq\cdots\] and the sums \[A_n = a_1 + \cdots + a_n,\quad B_n = b_1 + \cdots + b_n;\qquad n = 1,2,\ldots.\] For any pair define $c_n = \min\{a_i,b_i\}$ and $C_n = c_1 + \cdots + c_n$, $n=1,2,\ldots$.
(1) Does there exist a pair $(a_i)_{i\geq 1}$, $(b_i)_{i\geq 1}$ such that the sequences $(A_n)_{n\geq 1}$ and $(B_n)_{n\geq 1}$ are unbounded while the sequence $(C_n)_{n\geq 1}$ is bounded?
(2) Does the answer to question (1) change by assuming additionally that $b_i = 1/i$, $i=1,2,\ldots$?
Justify your answer.
2020 Swedish Mathematical Competition, 3
Determine all bounded functions $f: R \to R$, such that $f (f (x) + y) = f (x) + f (y)$, for all real $x, y$.
1998 Moldova Team Selection Test, 10
Let $P(x)$ denote the product of all (decimal) digits of a natural number $x$. For any positive integer $x_1$, define the sequence $(x_n)$ recursively by $x_{n+1} = x_n + P(x_n)$. Prove or disprove that the sequence $(x_n)$ is necessarily bounded.
2001 Nordic, 2
Let ${f}$ be a bounded real function defined for all real numbers and satisfying for all real numbers ${x}$ the condition ${ f \Big(x+\frac{1}{3}\Big) + f \Big(x+\frac{1}{2}\Big)=f(x)+ f \Big(x+\frac{5}{6}\Big)}$ . Show that ${f}$ is periodic.
2015 China Northern MO, 8
The sequence $\{a_n\}$ is defined as follows: $a_1$ is a positive rational number, $a_n= \frac{p_n}{q_n}$, ($n= 1,2,…$) is a positive integer, where $p_n$ and $q_n$ are positive integers that are relatively prime, then $a_{n+1} = \frac{p_n^2+2015}{p_nq_n}$ Is there a$_1>2015$, making the sequence $\{a_n\}$ a bounded sequence? Justify your conclusion.
2004 Germany Team Selection Test, 1
Consider pairs of the sequences of positive real numbers \[a_1\geq a_2\geq a_3\geq\cdots,\qquad b_1\geq b_2\geq b_3\geq\cdots\] and the sums \[A_n = a_1 + \cdots + a_n,\quad B_n = b_1 + \cdots + b_n;\qquad n = 1,2,\ldots.\] For any pair define $c_n = \min\{a_i,b_i\}$ and $C_n = c_1 + \cdots + c_n$, $n=1,2,\ldots$.
(1) Does there exist a pair $(a_i)_{i\geq 1}$, $(b_i)_{i\geq 1}$ such that the sequences $(A_n)_{n\geq 1}$ and $(B_n)_{n\geq 1}$ are unbounded while the sequence $(C_n)_{n\geq 1}$ is bounded?
(2) Does the answer to question (1) change by assuming additionally that $b_i = 1/i$, $i=1,2,\ldots$?
Justify your answer.
2019 Kosovo Team Selection Test, 2
Determine all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that for every $x,y \in \mathbb{R}$
$$f(x^{4}-y^{4})+4f(xy)^{2}=f(x^{4}+y^{4})$$
2015 Postal Coaching, 2
Prove that there exists a real number $C > 1$ with the following property.
Whenever $n > 1$ and $a_0 < a_1 < a_2 <\cdots < a_n$ are positive integers such that $\frac{1}{a_0},\frac{1}{a_1} \cdots \frac{1}{a_n}$ form an arithmetic progression, then $a_0 > C^n$.
1992 Putnam, B5
Let $D_n$ denote the value of the $(n -1) \times (n - 1)$ determinant
$$ \begin{pmatrix}
3 & 1 &1 & \ldots & 1\\
1 & 4 &1 & \ldots & 1\\
1 & 1 & 5 & \ldots & 1\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
1 & 1 & 1 & \ldots & n+1
\end{pmatrix}.$$
Is the set $\left\{ \frac{D_n }{n!} \, | \, n \geq 2\right\}$ bounded?
2015 Romania Team Selection Tests, 2
Let $(a_n)_{n \geq 0}$ and $(b_n)_{n \geq 0}$ be sequences of real numbers such that $ a_0>\frac{1}{2}$ , $a_{n+1} \geq a_n$ and $b_{n+1}=a_n(b_n+b_{n+2})$ for all non-negative integers $n$ . Show that the sequence $(b_n)_{n \geq 0}$ is bounded .
1991 IMO, 3
An infinite sequence $ \,x_{0},x_{1},x_{2},\ldots \,$ of real numbers is said to be [b]bounded[/b] if there is a constant $ \,C\,$ such that $ \, \vert x_{i} \vert \leq C\,$ for every $ \,i\geq 0$. Given any real number $ \,a > 1,\,$ construct a bounded infinite sequence $ x_{0},x_{1},x_{2},\ldots \,$ such that
\[ \vert x_{i} \minus{} x_{j} \vert \vert i \minus{} j \vert^{a}\geq 1
\]
for every pair of distinct nonnegative integers $ i, j$.
2005 Taiwan TST Round 3, 1
Let $a_0$, $a_1$, $a_2$, ... be an infinite sequence of real numbers satisfying the equation $a_n=\left|a_{n+1}-a_{n+2}\right|$ for all $n\geq 0$, where $a_0$ and $a_1$ are two different positive reals.
Can this sequence $a_0$, $a_1$, $a_2$, ... be bounded?
[i]Proposed by Mihai Bălună, Romania[/i]
2024 Singapore Senior Math Olympiad, Q5
Let $a_1,a_2,\dots$ be a sequence of positive numbers satisfying, for any positive integers $k,l,m,n$ such that $k+n=m+l$, $$\frac{a_k+a_n}{1+a_ka_n}=\frac{a_m+a_l}{1+a_ma_l}.$$Show that there exist positive numbers $b,c$ so that $b\le a_n\le c$ for any positive integer $n$.
2007 IMO Shortlist, 5
Let $ c > 2,$ and let $ a(1), a(2), \ldots$ be a sequence of nonnegative real numbers such that
\[ a(m \plus{} n) \leq 2 \cdot a(m) \plus{} 2 \cdot a(n) \text{ for all } m,n \geq 1,
\]
and $ a\left(2^k \right) \leq \frac {1}{(k \plus{} 1)^c} \text{ for all } k \geq 0.$ Prove that the sequence $ a(n)$ is bounded.
[i]Author: Vjekoslav Kovač, Croatia[/i]
2025 All-Russian Olympiad, 10.6
What is the smallest value of \( k \) such that for any polynomial \( f(x) \) of degree $100$ with real coefficients, there exists a polynomial \( g(x) \) of degree at most \( k \) with real coefficients such that the graphs of \( y = f(x) \) and \( y = g(x) \) intersect at exactly $100$ points? \\
1965 Swedish Mathematical Competition, 5
Let $S$ be the set of all real polynomials $f(x) = ax^3 + bx^2 + cx + d$ such that $|f(x)| \le 1$ for all $ -1 \le x \le 1$. Show that the set of possible $|a|$ for $f$ in $S$ is bounded above and find the smallest upper bound.
2024 Olimphíada, 3
A sequence of positive real numbers $a_1, a_2, \dots$ is called $\textit{phine}$ if it satisfies
$$a_{n+2}=\frac{a_{n+1}+a_{n-1}}{a_n},$$
for all $n\geq2$. Is there a $\textit{phine}$ sequence such that, for every real number $r$, there is some $n$ for which $a_n>r$?
1982 Austrian-Polish Competition, 4
Let $P(x)$ denote the product of all (decimal) digits of a natural number $x$. For any positive integer $x_1$, define the sequence $(x_n)$ recursively by $x_{n+1} = x_n + P(x_n)$. Prove or disprove that the sequence $(x_n)$ is necessarily bounded.
1991 IMO Shortlist, 28
An infinite sequence $ \,x_{0},x_{1},x_{2},\ldots \,$ of real numbers is said to be [b]bounded[/b] if there is a constant $ \,C\,$ such that $ \, \vert x_{i} \vert \leq C\,$ for every $ \,i\geq 0$. Given any real number $ \,a > 1,\,$ construct a bounded infinite sequence $ x_{0},x_{1},x_{2},\ldots \,$ such that
\[ \vert x_{i} \minus{} x_{j} \vert \vert i \minus{} j \vert^{a}\geq 1
\]
for every pair of distinct nonnegative integers $ i, j$.
2005 Germany Team Selection Test, 1
Let $a_0$, $a_1$, $a_2$, ... be an infinite sequence of real numbers satisfying the equation $a_n=\left|a_{n+1}-a_{n+2}\right|$ for all $n\geq 0$, where $a_0$ and $a_1$ are two different positive reals.
Can this sequence $a_0$, $a_1$, $a_2$, ... be bounded?
[i]Proposed by Mihai Bălună, Romania[/i]
1974 Czech and Slovak Olympiad III A, 4
Let $\mathcal M$ be the set of all polynomial functions $f$ of degree at most 3 such that \[\forall x\in[-1,1]:\ |f(x)|\le 1.\] Denote $a$ the (possibly zero) coefficient of $f$ at $x^3.$ Show that there is a positive number $k$ such that \[\forall f\in\mathcal M:\ |a|\le k\] and find the least $k$ with this property.