Olympiad problems

2002 Iran MO (3rd Round), 4

Tags: geometry , algebra , polynomial , function , inequalities , geometric transformation

$a_{n}$ ($n$ is integer) is a sequence from positive reals that \[a_{n}\geq \frac{a_{n+2}+a_{n+1}+a_{n-1}+a_{n-2}}4\] Prove $a_{n}$ is constant.

1979 Austrian-Polish Competition, 4

Tags: functional equation , algebra , function

Determine all functions $f : N_0 \to R$ satisfying $f (x+y)+ f (x-y)= f (3x)$ for all $x,y$.

2024 Israel TST, P2

Tags: combinatorics , function , tree , average

A positive integer $N$ is given. Panda builds a tree on $N$ vertices, and writes a real number on each vertex, so that $1$ plus the number written on each vertex is greater or equal to the average of the numbers written on the neighboring vertices. Let the maximum number written be $M$ and the minimal number written $m$. Mink then gives Panda $M-m$ kilograms of bamboo. What is the maximum amount of bamboo Panda can get?

2022 Turkey MO (2nd round), 2

Tags: number theory , function

For positive integers $k$ and $n$, we know $k \geq n!$. Prove that $ \phi (k) \geq (n-1)!$

2001 Miklós Schweitzer, 6

Tags: real analysis , inequalities , functional inequality , function

Let $I\subset \mathbb R$ be a non-empty open interval, $\varepsilon\geq 0$ and $f\colon I\rightarrow\mathbb R$ a function satisfying the $$f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)+\varepsilon t(1-t)|x-y|$$ inequality for all $x,y\in I$ and $t\in [0,1]$. Prove that there exists a convex $g\colon I\rightarrow\mathbb R$ function, such that the function $l :=f-g$ has the $\varepsilon$-Lipschitz property, that is $$|l(x)-l(y)|\leq \varepsilon|x-y|\text{ for all }x,y\in I$$

2014 Online Math Open Problems, 24

Tags: function , inequalities , vector , floor function

Let $\mathcal P$ denote the set of planes in three-dimensional space with positive $x$, $y$, and $z$ intercepts summing to one. A point $(x,y,z)$ with $\min \{x,y,z\} > 0$ lies on exactly one plane in $\mathcal P$. What is the maximum possible integer value of $\left(\frac{1}{4} x^2 + 2y^2 + 16z^2\right)^{-1}$? [i]Proposed by Sammy Luo[/i]

1998 All-Russian Olympiad, 2

Tags: geometry , function , trig identities , trigonometry , inequalities , rectangle

Two polygons are given on the plane. Assume that the distance between any two vertices of the same polygon is at most 1, and that the distance between any two vertices of different polygons is at least $ 1/\sqrt{2}$. Prove that these two polygons have no common interior points. By the way, can two sides of a polygon intersect?

1991 Arnold's Trivium, 8

Tags: function

How many maxima, minima, and saddle points does the function $x^4 + y^4 + z^4 + u^4 + v^4$ have on the surface $x+ ... +v = 0$, $x^2+ ... + v^2 = 1$, $x^3 + ... + v^3 = C$?

2012 USAMO, 3

Tags: function , modular arithmetic , floor function

Determine which integers $n > 1$ have the property that there exists an infinite sequence $a_1, a_2, a_3, \ldots$ of nonzero integers such that the equality \[a_k+2a_{2k}+\ldots+na_{nk}=0\]holds for every positive integer $k$.

2011 Romania Team Selection Test, 1

Tags: strong induction , induction , algebra , function

Given a positive integer number $k$, define the function $f$ on the set of all positive integer numbers to itself by \[f(n)=\begin{cases}1, &\text{if }n\le k+1\\ f(f(n-1))+f(n-f(n-1)), &\text{if }n>k+1\end{cases}\] Show that the preimage of every positive integer number under $f$ is a finite non-empty set of consecutive positive integers.

2009 Moldova Team Selection Test, 2

Tags: function , algebra

[color=darkred]Determine all functions $ f : [0; \plus{} \infty) \rightarrow [0; \plus{} \infty)$, such that \[ f(x \plus{} y \minus{} z) \plus{} f(2\sqrt {xz}) \plus{} f(2\sqrt {yz}) \equal{} f(x \plus{} y \plus{} z)\] for all $ x,y,z \in [0; \plus{} \infty)$, for which $ x \plus{} y\ge z$.[/color]

1985 IMO Longlists, 65

Tags: function , algebra , ceiling function , floor function

Define the functions $f, F : \mathbb N \to \mathbb N$, by \[f(n)=\left[ \frac{3-\sqrt 5}{2} n \right] , F(k) =\min \{n \in \mathbb N|f^k(n) > 0 \},\] where $f^k = f \circ \cdots \circ f$ is $f$ iterated $n$ times. Prove that $F(k + 2) = 3F(k + 1) - F(k)$ for all $k \in \mathbb N.$

2007 Bundeswettbewerb Mathematik, 4

Tags: function , algebra , floor function

Let $a$ be a positive integer. How many non-negative integer solutions x does the equation $\lfloor \frac{x}{a}\rfloor = \lfloor \frac{x}{a+1}\rfloor$ have? $\lfloor ~ \rfloor$ ---> [url=http://en.wikipedia.org/wiki/Floor_function]Floor Function[/url].

2014 Uzbekistan National Olympiad, 2

Tags: limit , function , functional equation , algebra

Find all functions $f:R\rightarrow R$ such that \[ f(x^3)+f(y^3)=(x+y)(f(x^2)+f(y^2)-f(xy)) \] for all $x,y\in R$.

2023 District Olympiad, P3

Tags: function , continuity , real analysis

Let $f:[a,b]\to[a,b]$ be a continuous function. It is known that there exist $\alpha,\beta\in (a,b)$ such that $f(\alpha)=a$ and $f(\beta)=b$. Prove that the function $f\circ f$ has at least three fixed points.

PEN K Problems, 33

Tags: algebra , function , functional equation , strong induction , induction

Find all functions $f: \mathbb{Q}\to \mathbb{Q}$ such that for all $x,y,z \in \mathbb{Q}$: \[f(x+y+z)+f(x-y)+f(y-z)+f(z-x)=3f(x)+3f(y)+3f(z).\]

2004 IberoAmerican, 3

Tags: combinatorics , function , geometry , quadratic , parallelogram , induction

Given a set $ \mathcal{H}$ of points in the plane, $ P$ is called an "intersection point of $ \mathcal{H}$" if distinct points $ A,B,C,D$ exist in $ \mathcal{H}$ such that lines $ AB$ and $ CD$ are distinct and intersect in $ P$. Given a finite set $ \mathcal{A}_{0}$ of points in the plane, a sequence of sets is defined as follows: for any $ j\geq0$, $ \mathcal{A}_{j+1}$ is the union of $ \mathcal{A}_{j}$ and the intersection points of $ \mathcal{A}_{j}$. Prove that, if the union of all the sets in the sequence is finite, then $ \mathcal{A}_{i}=\mathcal{A}_{1}$ for any $ i\geq1$.

2015 International Zhautykov Olympiad, 3

Tags: polynomial , function , algebra

Find all functions $ f\colon \mathbb{R} \to \mathbb{R} $ such that $ f(x^3+y^3+xy)=x^2f(x)+y^2f(y)+f(xy) $, for all $ x,y \in \mathbb{R} $.

2015 Turkey Junior National Olympiad, 1

Tags: function

For a non-constant function $f:\mathbb{R}\to \mathbb{R}$ prove that there exist real numbers $x,y$ satisfying $f(x+y)<f(xy)$

2012 South africa National Olympiad, 4

Tags: function , number theory

Let $p$ and $k$ be positive integers such that $p$ is prime and $k>1$. Prove that there is at most one pair $(x,y)$ of positive integers such that $x^k+px=y^k$.

2007 Today's Calculation Of Integral, 169

Tags: function , calculus , calculus computations , integration

(1) Let $f(x)$ be the differentiable and increasing function such that $f(0)=0.$Prove that $\int_{0}^{1}f(x)f'(x)dx\geq \frac{1}{2}\left(\int_{0}^{1}f(x)dx\right)^{2}.$ (2) $g_{n}(x)=x^{2n+1}+a_{n}x+b_{n}\ (n=1,\ 2,\ 3,\ \cdots)$ satisfies $\int_{-1}^{1}(px+q)g_{n}(x)dx=0$ for all linear equations $px+q.$ Find $a_{n},\ b_{n}.$

2014 ELMO Shortlist, 4

Tags: function , number theory

Let $\mathbb N$ denote the set of positive integers, and for a function $f$, let $f^k(n)$ denote the function $f$ applied $k$ times. Call a function $f : \mathbb N \to \mathbb N$ [i]saturated[/i] if \[ f^{f^{f(n)}(n)}(n) = n \] for every positive integer $n$. Find all positive integers $m$ for which the following holds: every saturated function $f$ satisfies $f^{2014}(m) = m$. [i]Proposed by Evan Chen[/i]

MathLinks Contest 5th, 5.2

Tags: inequalities , algebra , function

Prove or disprove the existence of a function $f : S \to R$ such that for all $x \ne y \in S$ we have $|f(x) - f(y)| \ge \frac{1}{x^2 + y^2}$, in each of the cases: a) $S = R$ b) $S = Q$.

1995 IMO Shortlist, 5

Tags: arithmetic sequence , recurrence relation , algebra , function

For positive integers $ n,$ the numbers $ f(n)$ are defined inductively as follows: $ f(1) \equal{} 1,$ and for every positive integer $ n,$ $ f(n\plus{}1)$ is the greatest integer $ m$ such that there is an arithmetic progression of positive integers $ a_1 < a_2 < \ldots < a_m \equal{} n$ for which \[ f(a_1) \equal{} f(a_2) \equal{} \ldots \equal{} f(a_m).\] Prove that there are positive integers $ a$ and $ b$ such that $ f(an\plus{}b) \equal{} n\plus{}2$ for every positive integer $ n.$

2015 India Regional MathematicaI Olympiad, 6

Tags: algebra , function , floor function

For how many integer values of $m$, (i) $1\le m \le 5000$ (ii) $[\sqrt{m}] =[\sqrt{m+125}]$ Note: $[x]$ is the greatest integer function