Olympiad problems

2001 Czech And Slovak Olympiad IIIA, 6

Let be given natural numbers $a_1,a_2,...,a_n$ and a function $f : Z \to R$ such that $f(x) = 1$ for all integers $x < 0$ and $f(x) = 1- f(x-a_1)f(x-a_2)... f(x-a_n)$ for all integers $x \ge 0$. Prove that there exist natural numbers $s$ and $t$ such that for all integers $x > s$ it holds that $f(x+t) = f(x)$.

1974 All Soviet Union Mathematical Olympiad, 203

Tags: algebra , function , inequalities

Given a function $f(x)$ on the segment $0\le x\le 1$. For all $x, f(x)\ge 0, f(1)=1$. For all the couples of $(x_1,x_2)$ such, that all the arguments are in the segment $$f(x_1+x_2)\ge f(x_1)+f(x_2).$$ a) Prove that for all $x$ holds $f(x) \le 2x$. b) Is the inequality $f(x) \le 1.9x$ valid?

2001 National High School Mathematics League, 11

Tags: function

The range of function $y=x+\sqrt{x^2-3x+2}(x\in\mathbb{R})$ is________.

2015 Chile TST Ibero, 1

Tags: algebra , function

Determine the number of functions $f: \mathbb{N} \to \mathbb{N}$ and $g: \mathbb{N} \to \mathbb{N}$ such that for all $n \in \mathbb{N}$: \[ f(g(n)) = n + 2015, \] \[ g(f(n)) = n^2 + 2015. \]

2004 Baltic Way, 8

Tags: logarithm , polynomial , number theory , algebra , function , prime number , inequalities

Let $f\left(x\right)$ be a non-constant polynomial with integer coefficients, and let $u$ be an arbitrary positive integer. Prove that there is an integer $n$ such that $f\left(n\right)$ has at least $u$ distinct prime factors and $f\left(n\right) \neq 0$.

2010 Today's Calculation Of Integral, 576

Tags: rotation , function , logarithm , geometric transformation , calculus , integration , geometry

For a function $ f(x)\equal{}(\ln x)^2\plus{}2\ln x$, let $ C$ be the curve $ y\equal{}f(x)$. Denote $ A(a,\ f(a)),\ B(b,\ f(b))\ (a<b)$ the points of tangency of two tangents drawn from the origin $ O$ to $ C$ and the curve $ C$. Answer the following questions. (1) Examine the increase and decrease, extremal value and inflection point , then draw the approximate garph of the curve $ C$. (2) Find the values of $ a,\ b$. (3) Find the volume by a rotation of the figure bounded by the part from the point $ A$ to the point $ B$ and line segments $ OA,\ OB$ around the $ y$-axis.

1990 IMO Longlists, 29

Tags: function , algebra , floor function

Function $f(n), n \in \mathbb N$, is defined as follows: Let $\frac{(2n)!}{n!(n+1000)!} = \frac{A(n)}{B(n)}$ , where $A(n), B(n)$ are coprime positive integers; if $B(n) = 1$, then $f(n) = 1$; if $B(n) \neq 1$, then $f(n)$ is the largest prime factor of $B(n)$. Prove that the values of $f(n)$ are finite, and find the maximum value of $f(n).$

2021 ISI Entrance Examination, 4

Tags: function , calculus

Let $g:(0,\infty) \rightarrow (0,\infty)$ be a differentiable function whose derivative is continuous, and such that $g(g(x)) = x$ for all $x> 0$. If $g$ is not the identity function, prove that $g$ must be strictly decreasing.

2024 European Mathematical Cup, 4

Tags: algebra , function , functional equation

Find all functions $ f: \mathbb{R}^{+} \to \mathbb{R}^{+}$ such that $f(x+yf(x)) = xf(1+y)$ for all x, y positive reals.

1994 Korea National Olympiad, Problem 1

Tags: algebra , function

Let $ S$ be the set of nonnegative integers. Find all functions $ f,g,h: S\rightarrow S$ such that $ f(m\plus{}n)\equal{}g(m)\plus{}h(n),$ for all $ m,n\in S$, and $ g(1)\equal{}h(1)\equal{}1$.

2012 Today's Calculation Of Integral, 784

Tags: limit , integration , logarithm , function , calculus , analytic geometry , geometry

Define for positive integer $n$, a function $f_n(x)=\frac{\ln x}{x^n}\ (x>0).$ In the coordinate plane, denote by $S_n$ the area of the figure enclosed by $y=f_n(x)\ (x\leq t)$, the $x$-axis and the line $x=t$ and denote by $T_n$ the area of the rectagle with four vertices $(1,\ 0),\ (t,\ 0),\ (t,\ f_n(t))$ and $(1,\ f_n(t))$. (1) Find the local maximum $f_n(x)$. (2) When $t$ moves in the range of $t>1$, find the value of $t$ for which $T_n(t)-S_n(t)$ is maximized. (3) Find $S_1(t)$ and $S_n(t)\ (n\geq 2)$. (4) For each $n\geq 2$, prove that there exists the only $t>1$ such that $T_n(t)=S_n(t)$. Note that you may use $\lim_{x\to\infty} \frac{\ln x}{x}=0.$

2006 Federal Competition For Advanced Students, Part 1, 1

Tags: function , number theory

Let $ n$ be a non-negative integer, which ends written in decimal notation on exactly $ k$ zeros, but which is bigger than $ 10^k$. For a $ n$ is only $ k\equal{}k(n)\geq2$ known. In how many different ways (as a function of $ k\equal{}k(n)\geq2$) can $ n$ be written as difference of two squares of non-negative integers at least?

1998 Spain Mathematical Olympiad, 2

Tags: algebra , function , induction

Find all strictly increasing functions $f:\mathbb{N}\rightarrow\mathbb{N}$ that satisfy \[f(n+f(n))=2f(n)\quad\text{for all}\ n\in\mathbb{N} \]

1998 Romania Team Selection Test, 4

Tags: combinatorics , function , trigonometry , analytic geometry

Consider in the plane a finite set of segments such that the sum of their lengths is less than $\sqrt{2}$. Prove that there exists an infinite unit square grid covering the plane such that the lines defining the grid do not intersect any of the segments. [i]Vasile Pop[/i]

1957 AMC 12/AHSME, 34

Tags: parameterization , graphing lines , slope , function , algebra , analytic geometry , inequalities

The points that satisfy the system $ x \plus{} y \equal{} 1,\, x^2 \plus{} y^2 < 25,$ constitute the following set: $ \textbf{(A)}\ \text{only two points} \qquad \\ \textbf{(B)}\ \text{an arc of a circle}\qquad \\ \textbf{(C)}\ \text{a straight line segment not including the end\minus{}points}\qquad \\ \textbf{(D)}\ \text{a straight line segment including the end\minus{}points}\qquad \\ \textbf{(E)}\ \text{a single point}$

1991 Romania Team Selection Test, 4

Tags: combinatorial geometry , function , geometry

Let $S$ be the set of all polygonal areas in a plane. Prove that there is a function $f : S \to (0,1)$ which satisfies $f(S_1 \cup S_2) = f(S_1)+ f(S_2)$ for any $S_1,S_2 \in S$ which have common points only on their borders

2024 Philippine Math Olympiad, P4

Tags: combinatorics , number theory , function

Let $n$ be a positive integer. Suppose for any $\mathcal{S} \subseteq \{1, 2, \cdots, n\}$, $f(\mathcal{S})$ is the set containing all positive integers at most $n$ that have an odd number of factors in $\mathcal{S}$. How many subsets of $\{1, 2, \cdots, n\}$ can be turned into $\{1\}$ after finitely many (possibly zero) applications of $f$?

2019 District Olympiad, 2

Tags: function , integrability , calculus

Let $n$ be a positive integer and $f:[0,1] \to \mathbb{R}$ be an integrable function. Prove that there exists a point $c \in \left[0,1- \frac{1}{n} \right],$ such that [center] $ \int\limits_c^{c+\frac{1}{n}}f(x)\mathrm{d}x=0$ or $\int\limits_0^c f(x) \mathrm{d}x=\int\limits_{c+\frac{1}{n}}^1f(x)\mathrm{d}x.$ [/center]

2011 India IMO Training Camp, 3

Tags: function , number theory

Let $\{a_0,a_1,\ldots\}$ and $\{b_0,b_1,\ldots\}$ be two infinite sequences of integers such that \[(a_{n}-a_{n-1})(a_n-a_{n-2}) +(b_n-b_{n-1})(b_n-b_{n-2})=0\] for all integers $n\geq 2$. Prove that there exists a positive integer $k$ such that \[a_{k+2011}=a_{k+2011^{2011}}.\]

2022 District Olympiad, P1

Tags: darboux s property , function

Let $f,g:\mathbb{R}\to\mathbb{R}$ be functions which satisfy \[\inf_{x>a}f(x)=g(a)\text{ and }\sup_{x<a}g(x)=f(a),\]for all $a\in\mathbb{R}.$ Given that $f$ has Darboux's Property (intermediate value property), show that functions $f$ and $g$ are continuous and equal to each other. [i]Mathematical Gazette [/i]

2005 Korea National Olympiad, 4

Tags: function , derivative , calculus , functional equation , induction , algebra

Find all $f: \mathbb R \to\mathbb R$ such that for all real numbers $x$, $f(x) \geq 0$ and for all real numbers $x$ and $y$, \[ f(x+y)+f(x-y)-2f(x)-2y^2=0. \]

2006 ISI B.Stat Entrance Exam, 10

Tags: ratio , function , probability , geometric sequence , induction , algebra

Consider a function $f$ on nonnegative integers such that $f(0)=1, f(1)=0$ and $f(n)+f(n-1)=nf(n-1)+(n-1)f(n-2)$ for $n \ge 2$. Show that \[\frac{f(n)}{n!}=\sum_{k=0}^n \frac{(-1)^k}{k!}\]

2014 Romania National Olympiad, 1

Tags: function , algebra

Find all continuous functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that satisfy: $ \text{(i)}\text{id}+f $ is nondecreasing $ \text{(ii)} $ There is a natural number $ m $ such that $ \text{id}+f+f^2\cdots +f^m $ is nonincreasing. Here, $ \text{id} $ represents the identity function, and ^ denotes functional power.

2007 Germany Team Selection Test, 1

Tags: function , algebra , inequalities

Prove the inequality: \[\sum_{i < j}{\frac {a_{i}a_{j}}{a_{i} \plus{} a_{j}}}\leq \frac {n}{2(a_{1} \plus{} a_{2} \plus{}\cdots \plus{} a_{n})}\cdot \sum_{i < j}{a_{i}a_{j}}\] for positive reals $ a_{1},a_{2},\ldots,a_{n}$. [i]Proposed by Dusan Dukic, Serbia[/i]

1990 USAMO, 1

Tags: function , geometry , 3d geometry , tetrahedron

A certain state issues license plates consisting of six digits (from 0 to 9). The state requires that any two license plates differ in at least two places. (For instance, the numbers 027592 and 020592 cannot both be used.) Determine, with proof, the maximum number of distinct license plates that the state can use.