Olympiad problems

2022-IMOC, N6

Find all integer coefficient polynomial $P(x)$ such that for all positive integer $x$, we have $$\tau(P(x))\geq\tau(x)$$Where $\tau(n)$ denotes the number of divisors of $n$. Define $\tau(0)=\infty$. Note: you can use this conclusion. For all $\epsilon\geq0$, there exists a positive constant $C_\epsilon$ such that for all positive integer $n$, the $n$th smallest prime is at most $C_\epsilon n^{1+\epsilon}$. [i]Proposed by USJL[/i]

1978 IMO Longlists, 35

Tags: algebra , function

A sequence $(a_n)_0^N$ of real numbers is called concave if $2a_n\ge a_{n-1} + a_{n+1}$ for all integers $n, 1 \le n \le N - 1$. $(a)$ Prove that there exists a constant $C >0$ such that \[\left(\displaystyle\sum_{n=0}^{N}a_n\right)^2\ge C(N - 1)\displaystyle\sum_{n=0}^{N}a_n^2\:\:\:\:\:(1)\] for all concave positive sequences $(a_n)^N_0$ $(b)$ Prove that $(1)$ holds with $C = \frac{3}{4}$ and that this constant is best possible.

1975 Miklós Schweitzer, 7

Tags: calculus , real analysis , analytic geometry , function , derivative , graphing lines , slope

Let $ a<a'<b<b'$ be real numbers and let the real function $ f$ be continuous on the interval $ [a,b']$ and differentiable in its interior. Prove that there exist $ c \in (a,b), c'\in (a',b')$ such that \[ f(b)\minus{}f(a)\equal{}f'(c)(b\minus{}a),\] \[ f(b')\minus{}f(a')\equal{}f'(c')(b'\minus{}a'),\] and $ c<c'$. [i]B. Szokefalvi Nagy[/i]

2013 Iran Team Selection Test, 6

Tags: geometry , function

Points $A, B, C$ and $D$ lie on line $l$ in this order. Two circular arcs $C_1$ and $C_2$, which both lie on one side of line $l$, pass through points $A$ and $B$ and two circular arcs $C_3$ and $C_4$ pass through points $C$ and $D$ such that $C_1$ is tangent to $C_3$ and $C_2$ is tangent to $C_4$. Prove that the common external tangent of $C_2$ and $C_3$ and the common external tangent of $C_1$ and $C_4$ meet each other on line $l$. [i]Proposed by Ali Khezeli[/i]

1998 Greece National Olympiad, 4

Tags: floor function , function , algebra , induction

Let a function $g:\mathbb{N}_0\to\mathbb{N}_0$ satisfy $g(0)=0$ and $g(n)=n-g(g(n-1))$ for all $n\ge 1$. Prove that: a) $g(k)\ge g(k-1)$ for any positive integer $k$. b) There is no $k$ such that $g(k-1)=g(k)=g(k+1)$.

2012 Turkey Team Selection Test, 1

Tags: function , algebra , integration , calculus , limit

Let $S_r(n)=1^r+2^r+\cdots+n^r$ where $n$ is a positive integer and $r$ is a rational number. If $S_a(n)=(S_b(n))^c$ for all positive integers $n$ where $a, b$ are positive rationals and $c$ is positive integer then we call $(a,b,c)$ as [i]nice triple.[/i] Find all nice triples.

1950 AMC 12/AHSME, 41

Tags: function

The least value of the function $ ax^2\plus{}bx\plus{}c$ with $a>0$ is: $\textbf{(A)}\ -\dfrac{b}{a} \qquad \textbf{(B)}\ -\dfrac{b}{2a} \qquad \textbf{(C)}\ b^2-4ac \qquad \textbf{(D)}\ \dfrac{4ac-b^2}{4a}\qquad \textbf{(E)}\ \text{None of these}$

1988 IMO Longlists, 3

Tags: polynomial , function , algebra , generating functions , binomial coefficient

Let $ n$ be a positive integer. Find the number of odd coefficients of the polynomial \[ u_n(x) \equal{} (x^2 \plus{} x \plus{} 1)^n. \]

2002 All-Russian Olympiad, 4

Tags: algorithm , induction , function , combinatorics , graph theory

There are 2002 towns in a kingdom. Some of the towns are connected by roads in such a manner that, if all roads from one city closed, one can still travel between any two cities. Every year, the kingdom chooses a non-self-intersecting cycle of roads, founds a new town, connects it by roads with each city from the chosen cycle, and closes all the roads from the original cycle. After several years, no non-self-intersecting cycles remained. Prove that at that moment there are at least 2002 towns, exactly one road going out from each of them.

1990 Irish Math Olympiad, 3

Tags: function , induction

Determine whether there exists a function $ f: \mathbb{N}\longrightarrow \mathbb{N}$ such that $ f(n)\equal{}f(f(n\minus{}1))\plus{}f(f(n\plus{}1))$ for all natural numbers $ n\ge 2$.

2014 Contests, 1

Tags: function , number theory

Let $f(x)$ is such function, that $f(x)=1$ for integer $x$ and $f(x)=0$ for non integer $x$. Build such function using only variable $x$, integer numbers, and operations $+,-,*,/,[.]$(plus, minus, multiply,divide and integer part)

2005 IberoAmerican Olympiad For University Students, 4

Tags: function , trigonometry , analytic geometry , geometry , conic , hyperbola , asymptote

A variable tangent $t$ to the circle $C_1$, of radius $r_1$, intersects the circle $C_2$, of radius $r_2$ in $A$ and $B$. The tangents to $C_2$ through $A$ and $B$ intersect in $P$. Find, as a function of $r_1$ and $r_2$, the distance between the centers of $C_1$ and $C_2$ such that the locus of $P$ when $t$ varies is contained in an equilateral hyperbola. [b]Note[/b]: A hyperbola is said to be [i]equilateral[/i] if its asymptotes are perpendicular.

1970 AMC 12/AHSME, 25

Tags: function

For every real number $x$, let $[x]$ be the greatest integer less than or equal to $x$. If the postal rate for first class mail is six cents for every ounce or portion thereof, then the cost in cents of first-class postage on a letter weighing $W$ ounces is always $\textbf{(A) }6W\qquad\textbf{(B) }6[W]\qquad\textbf{(C) }6([W]-1)\qquad\textbf{(D) }6([W]+1)\qquad \textbf{(E) }-6[-W]$

1974 Putnam, B4

Tags: function , continuous

A function $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ is said to be [i]continuous in each variable separately [/i] if, for each fixed value $y_0$ of $y$, the function $f(x, y_0)$ is contnuous in the usual sense as a function in $x,$ and similarly $f(x_0 , y)$ is continuous as a function of $y$ for each fixed $x_0$. Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be continuous in each variable separately. Show that there exists a sequence of continuous functions $g_n: \mathbb{R}^{2} \rightarrow \mathbb{R}$ such that $$f(x,y) =\lim_{n\to \infty}g_{n}(x,y)$$ for all $(x,y)\in \mathbb{R}^{2}.$

2012 Romanian Masters In Mathematics, 3

Tags: algebra , combinatorics , function , additive combinatorics

Each positive integer is coloured red or blue. A function $f$ from the set of positive integers to itself has the following two properties: (a) if $x\le y$, then $f(x)\le f(y)$; and (b) if $x,y$ and $z$ are (not necessarily distinct) positive integers of the same colour and $x+y=z$, then $f(x)+f(y)=f(z)$. Prove that there exists a positive number $a$ such that $f(x)\le ax$ for all positive integers $x$. [i](United Kingdom) Ben Elliott[/i]

2007 Italy TST, 3

Tags: functional equation , algebra , function , polynomial

Find all $f: R \longrightarrow R$ such that \[f(xy+f(x))=xf(y)+f(x)\] for every pair of real numbers $x,y$.

1992 Putnam, A1

Tags: algebra , function

Find all functions $ f : Z\rightarrow Z$ for which we have $ f (0) \equal{} 1$ and $ f ( f (n)) \equal{} f ( f (n\plus{}2)\plus{}2) \equal{} n$, for every natural number $ n$.

2019 LIMIT Category C, Problem 2

Tags: function

Which of the following are true? $\textbf{(A)}~\exists f:\mathbb N\to\mathbb Z\text{ onto and increasing}$ $\textbf{(B)}~\exists f:\mathbb Z\to\mathbb Q\text{ onto and increasing}$ $\textbf{(C)}~\exists f:\mathbb Q\to\mathbb Z\text{ onto and increasing and bounded}$ $\textbf{(D)}~\text{None of the above}$

2013 Purple Comet Problems, 25

Tags: function , number theory

In how many ways can you write $12$ as an ordered sum of integers where the smallest of those integers is equal to $2$? For example, $2+10$, $10+2$, and $3+2+2+5$ are three such ways.

2008 Estonia Team Selection Test, 3

Tags: function , calculus , inequalities , algebra

Let $ n$ be a positive integer, and let $ x$ and $ y$ be a positive real number such that $ x^n \plus{} y^n \equal{} 1.$ Prove that \[ \left(\sum^n_{k \equal{} 1} \frac {1 \plus{} x^{2k}}{1 \plus{} x^{4k}} \right) \cdot \left( \sum^n_{k \equal{} 1} \frac {1 \plus{} y^{2k}}{1 \plus{} y^{4k}} \right) < \frac {1}{(1 \minus{} x) \cdot (1 \minus{} y)}. \] [i]Author: Juhan Aru, Estonia[/i]

2016 Korea USCM, 5

Tags: function , limit , real analysis , lp norm

For $f(x) = \cos\left(\frac{3\sqrt{3}\pi}{8}(x-x^3 ) \right)$, find the value of $$\lim_{t\to\infty} \left( \int_0^1 f(x)^t dx \right)^\frac{1}{t} + \lim_{t\to-\infty} \left( \int_0^1 f(x)^t dx \right)^\frac{1}{t} $$

1963 Miklós Schweitzer, 6

Tags: function , limit , real analysis , integration , inequalities

Show that if $ f(x)$ is a real-valued, continuous function on the half-line $ 0\leq x < \infty$, and \[ \int_0^{\infty} f^2(x)dx <\infty\] then the function \[ g(x)\equal{}f(x)\minus{}2e^{\minus{}x}\int_0^x e^tf(t)dt\] satisfies \[ \int _0^{\infty}g^2(x)dx\equal{}\int_0^{\infty}f^2(x)dx.\] [B. Szokefalvi-Nagy]

1989 Romania Team Selection Test, 4

Tags: function , combinatorics

Let $r,n$ be positive integers. For a set $A$, let ${A \choose r}$ denote the family of all $r$-element subsets of $A$. Prove that if $A$ is infinite and $f : {A \choose r} \to {1,2,...,n}$ is any function, then there exists an infinite subset $B$ of $A$ such that $f(X) = f(Y)$ for all $X,Y \in {B \choose r}$.

1996 Moldova Team Selection Test, 9

Tags: function , inequalities

Let $x_1,x_2,...,x_n \in [0;1]$ prove that $x_1(1-x_2)+x_2(1-x_3)+...+x_{n-1}(1-x_n)+x_n(1-x_1) \leq [\frac{n}{2}]$

2016 Switzerland Team Selection Test, Problem 9

Tags: algebra , function , functional equation

Find all functions $f : \mathbb{R} \mapsto \mathbb{R} $ such that $$ \left(f(x)+y\right)\left(f(x-y)+1\right)=f\left(f(xf(x+1))-yf(y-1)\right)$$ for all $x,y \in \mathbb{R}$