Found problems: 15925
2018 Balkan MO, 2
Let $q$ be a positive rational number. Two ants are initially at the same point $X$ in the plane. In the $n$-th minute $(n = 1,2,...)$ each of them chooses whether to walk due north, east, south or west and then walks the distance of $q^n$ metres. After a whole number of minutes, they are at the same point in the plane (not necessarily $X$), but have not taken exactly the same route within that time. Determine all possible values of $q$.
Proposed by Jeremy King, UK
2023 Estonia Team Selection Test, 5
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 $a_{2022}\leq 1$.
2022 Bulgaria National Olympiad, 4
Let $n\geq 4$ be a positive integer and $x_{1},x_{2},\ldots ,x_{n},x_{n+1},x_{n+2}$ be real numbers such that $x_{n+1}=x_{1}$ and $x_{n+2}=x_{2}$. If there exists an $a>0$ such that
\[x_{i}^2=a+x_{i+1}x_{i+2}\quad\forall 1\leq i\leq n\]
then prove that at least $2$ of the numbers $x_{1},x_{2},\ldots ,x_{n}$ are negative.
2014 ELMO Shortlist, 3
Let $t$ and $n$ be fixed integers each at least $2$. Find the largest positive integer $m$ for which there exists a polynomial $P$, of degree $n$ and with rational coefficients, such that the following property holds: exactly one of \[ \frac{P(k)}{t^k} \text{ and } \frac{P(k)}{t^{k+1}} \] is an integer for each $k = 0,1, ..., m$.
[i]Proposed by Michael Kural[/i]
2005 India IMO Training Camp, 3
For real numbers $a,b,c,d$ not all equal to $0$ , define a real function $f(x) = a +b\cos{2x} + c\sin{5x} +d \cos{8x}$. Suppose $f(t) = 4a$ for some real $t$. prove that there exist a real number $s$ s.t. $f(s)<0$
2017 China Team Selection Test, 2
Let $x>1$ ,$n$ be positive integer. Prove that$$\sum_{k=1}^{n}\frac{\{kx \}}{[kx]}<\sum_{k=1}^{n}\frac{1}{2k-1}$$
Where $[kx ]$ be the integer part of $kx$ ,$\{kx \}$ be the decimal part of $kx$.
2017 Czech And Slovak Olympiad III A, 3
Find all functions $f: R \to R$ such that for all real numbers $x, y$ holds $f(y - xy) = f(x)y + (x - 1)^2 f(y)$
2017 Hanoi Open Mathematics Competitions, 8
Determine all real solutions $x, y, z$ of the following system of equations: $\begin{cases}
x^3 - 3x = 4 - y \\
2y^3 - 6y = 6 - z \\
3z^3 - 9z = 8 - x\end{cases}$
1984 Poland - Second Round, 3
The given sequences are $ (x_1, x_2, \ldots, x_n) $, $ (y_1, y_2, \ldots, y_n) $ with positive terms. Prove that there exists a permutation $ p $ of the set $ \{1, 2, \ldots, n\} $ such that for every real $ t $ the sequence
$$ (x_{p(1)}+ty_{p(1)}, x_{p(2)}+ty_{p(2)}, \ldots, x_{p(n)}+ty_{p(n) })$$ has the following property: there is a number $ k $ such that $ 1 \leq k \leq n $ and all non-zero terms of the sequence with indices less than $ k $ are of the same sign and all non-zero terms of the sequence with indices not less than $ k $ are the same sign.
2024 IFYM, Sozopol, 5
The function $f: A \rightarrow A$ is such that $f(x) \leq x^2 \mbox{ and } f(x+y) \leq f(x) + f(y) + 2xy$ for any $x, y \in A$.
a) If $A = \mathbb{R}$, find all functions satisfying the conditions.
b) If $A = \mathbb{R}^{-}$, prove that there are infinitely many functions satisfying the conditions.
[i](With $\mathbb{R}^{-}$ we denote the set of negative real numbers.)[/i]
2015 Tuymaada Olympiad, 3
$P(x,y)$ is polynomial with real coefficients and $P(x+2y,x+y)=P(x,y)$. Prove that exists polynomial $Q(t)$ such that $P(x,y)=Q((x^2-2y^2)^2)$
[i]A. Golovanov[/i]
1981 IMO Shortlist, 9
A sequence $(a_n)$ is defined by means of the recursion
\[a_1 = 1, a_{n+1} = \frac{1 + 4a_n +\sqrt{1+ 24a_n}}{16}.\]
Find an explicit formula for $a_n.$
2020 USA IMO Team Selection Test, 1
Choose positive integers $b_1, b_2, \dotsc$ satisfying
\[1=\frac{b_1}{1^2} > \frac{b_2}{2^2} > \frac{b_3}{3^2} > \frac{b_4}{4^2} > \dotsb\]
and let $r$ denote the largest real number satisfying $\tfrac{b_n}{n^2} \geq r$ for all positive integers $n$. What are the possible values of $r$ across all possible choices of the sequence $(b_n)$?
[i]Carl Schildkraut and Milan Haiman[/i]
2022 Saudi Arabia JBMO TST, 2
Consider non-negative real numbers $a, b, c$ satisfying the condition $a^2 + b^2 + c^2 = 2$ . Find the maximum value of the following expression $$P=\frac{\sqrt{b^2+c^2}}{3-a}+\frac{\sqrt{c^2+a^2}}{3-b}+a+b-2022c$$
2007 Nicolae Coculescu, 1
Calculate $ \left\lfloor \frac{(a^2+b^2+c^2)(a+b+c)}{a^3+b^3+c^3} \right\rfloor , $ where $ a,b,c $ are the lengths of the side of a triangle.
[i]Costel Anghel[/i]
1940 Putnam, A1
Prove that if $f(x)$ is a polynomial with integer coefficients and there exists an integer $k$ such that none of $f(1),\ldots,f(k)$ is divisible by $k$, then $f(x)$ has no integral root.
2010 IFYM, Sozopol, 7
Prove the following equality:
$4 sin\frac{2\pi }{7}-tg \frac{\pi }{7}=\sqrt{7}$
1972 All Soviet Union Mathematical Olympiad, 169
Let $x,y$ be positive numbers, $s$ -- the least of $$\{ x, (y+ 1/x), 1/y\}$$ What is the greatest possible value of $s$? To what $x$ and $y$ does it correspond?
2021 APMO, 1
Prove that for each real number $r>2$, there are exactly two or three positive real numbers $x$ satisfying the equation $x^2=r\lfloor x \rfloor$.
2012 Pre-Preparation Course Examination, 5
Suppose that for the linear transformation $T:V \longrightarrow V$ where $V$ is a vector space, there is no trivial subspace $W\subset V$ such that $T(W)\subseteq W$. Prove that for every polynomial $p(x)$, the transformation $p(T)$ is invertible or zero.
1993 Kurschak Competition, 3
Let $n$ be a fixed positive integer. Compute over $\mathbb{R}$ the minimum of the following polynomial:
\[f(x)=\sum_{t=0}^{2n}(2n+1-t)x^t.\]
2012 Stars of Mathematics, 3
For all triplets $a,b,c$ of (pairwise) distinct real numbers, prove the inequality
$$ \left | \dfrac {a+b} {a-b} \right | + \left | \dfrac {b+c} {b-c} \right | + \left | \dfrac {c+a} {c-a} \right | \geq 2$$
and determine all cases of equality.
Prove that if we also impose $a,b,c \geq 0$, then
$$ \left | \dfrac {a+b} {a-b} \right | + \left | \dfrac {b+c} {b-c} \right | + \left | \dfrac {c+a} {c-a} \right | > 3,$$
with the value $3$ being the best constant possible.
([i]Dan Schwarz[/i])
2025 CMIMC Algebra/NT, 10
Let $a_n$ be a recursively defined sequence with $a_0=2024$ and $a_{n+1}=a_n^3+5a_n^2+10a_n+6$ for $n\ge 0.$ Determine the value of $$\sum_{n=0}^{\infty} \frac{2^n(a_n+1)}{a_n^2+3a_n+4}.$$
2015 Harvard-MIT Mathematics Tournament, 8
Find the number of ordered pairs of integers $(a,b)\in\{1,2,\ldots,35\}^2$ (not necessarily distinct) such that $ax+b$ is a "quadratic residue modulo $x^2+1$ and $35$", i.e. there exists a polynomial $f(x)$ with integer coefficients such that either of the following $\textit{equivalent}$ conditions holds:
[list]
[*] there exist polynomials $P$, $Q$ with integer coefficients such that $f(x)^2-(ax+b)=(x^2+1)P(x)+35Q(x)$;
[*] or more conceptually, the remainder when (the polynomial) $f(x)^2-(ax+b)$ is divided by (the polynomial) $x^2+1$ is a polynomial with integer coefficients all divisible by $35$.
[/list]
2000 Tournament Of Towns, 4
Let $a_1 , a_2 , ..., a_n$ be non-zero integers that satisfy the equation
$$a_1 +\dfrac{1}{a_2+\dfrac{1}{a_3+ ... \dfrac{1}{a_n+\dfrac{1}{x}} } } = x$$
for all values of $x$ for which the lefthand side of the equation makes sense.
(a) Prove that $n$ is even.
(b) What is the smallest n for which such numbers $a_1 , a_2 , ..., a_n$ exist?
(M Skopenko)