Olympiad problems

2002 Canada National Olympiad, 5

Let $\mathbb N = \{0,1,2,\ldots\}$. Determine all functions $f: \mathbb N \to \mathbb N$ such that \[ xf(y) + yf(x) = (x+y) f(x^2+y^2) \] for all $x$ and $y$ in $\mathbb N$.

2006 Iran MO (3rd Round), 2

Tags: vector , modular arithmetic , combinatorics , linear algebra , function

Let $B$ be a subset of $\mathbb{Z}_{3}^{n}$ with the property that for every two distinct members $(a_{1},\ldots,a_{n})$ and $(b_{1},\ldots,b_{n})$ of $B$ there exist $1\leq i\leq n$ such that $a_{i}\equiv{b_{i}+1}\pmod{3}$. Prove that $|B| \leq 2^{n}$.

2015 AMC 10, 7

Tags: function

Consider the operation "minus the reciprocal of," defined by $a\diamond b=a-\frac{1}{b}$. What is $((1\diamond2)\diamond3)-(1\diamond(2\diamond3))$? $\textbf{(A) } -\dfrac{7}{30} \qquad\textbf{(B) } -\dfrac{1}{6} \qquad\textbf{(C) } 0 \qquad\textbf{(D) } \dfrac{1}{6} \qquad\textbf{(E) } \dfrac{7}{30} $

2014 Switzerland - Final Round, 3

Tags: induction , function , algebra

Find all such functions $f :\mathbb{R}\to \mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following holds : \[ f(x^2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y) \]

2015 District Olympiad, 4

Tags: function , algebra

Find the functions $ f:\mathbb{N}\longrightarrow\mathbb{N} $ that satisfy the following relation: $$ \gcd\left( x,f(y)\right)\cdot\text{lcm}\left(f(x), y\right) = \gcd (x,y)\cdot\text{lcm}\left( f(x), f(y)\right) ,\quad\forall x,y\in\mathbb{N} . $$

1993 Hungary-Israel Binational, 3

Tags: function , vector , trigonometry , inequalities , symmetry

Distinct points $A, B , C, D, E$ are given in this order on a semicircle with radius $1$. Prove that \[AB^{2}+BC^{2}+CD^{2}+DE^{2}+AB \cdot BC \cdot CD+BC \cdot CD \cdot DE < 4.\]

1973 Czech and Slovak Olympiad III A, 4

Tags: function , floor function , sum , sum of roots , algebra

For any integer $n\ge2$ evaluate the sum \[\sum_{k=1}^{n^2-1}\bigl\lfloor\sqrt k\bigr\rfloor.\]

2012 Iran MO (3rd Round), 1

Tags: algebra , induction , polynomial , modular arithmetic , function

Suppose $0<m_1<...<m_n$ and $m_i \equiv i (\mod 2)$. Prove that the following polynomial has at most $n$ real roots. ($\forall 1\le i \le n: a_i \in \mathbb R$). \[a_0+a_1x^{m_1}+a_2x^{m_2}+...+a_nx^{m_n}.\]

2006 Purple Comet Problems, 16

Tags: function

$f(x)$ and $g(x)$ are linear functions such that for all $x$, $f(g(x)) = g(f(x)) = x$. If $f(0) = 4$ and $g(5) = 17$, compute $f(2006)$.

1997 Taiwan National Olympiad, 1

Tags: algebra , function

Let $a$ be rational and $b,c,d$ are real numbers, and let $f: \mathbb{R}\to [-1.1]$ be a function satisfying $f(x+a+b)-f(x+b)=c[x+2a+[x]-2[x+a]-[b]]+d$ for all $x$. Show that $f$ is periodic.

2004 Nicolae Coculescu, 3

Tags: function , group theory , morphism , endomorphism

Let be a finite group $ G $ having an endomorphism $ \eta $ that has exactly one fixed point. [b]a)[/b] Demonstrate that the function $ f:G\longrightarrow G $ defined as $ f(x)=x^{-1}\cdot\eta (x) $ is bijective. [b]b)[/b] Show that $ G $ is commutative if the composition of the function $ f $ from [b]a)[/b] with itself is the identity function.

Gheorghe Țițeica 2025, P3

Tags: function , real analysis , polynomial

Let $\mathcal{P}_n$ be the set of all real monic polynomial functions of degree $n$. Prove that for any $a<b$, $$\inf_{P\in\mathcal{P}_n}\int_a^b |P(x)|\, dx >0.$$ [i]Cristi Săvescu[/i]

2007 Moldova National Olympiad, 12.3

Tags: inequalities , logarithm , function , algebra

For $a,b \in [1;\infty)$ show that \[ab\leq e^{a-1}+b\ln b\]

2012 Math Prize For Girls Problems, 19

Tags: derivative , induction , calculus , function

Define $L(x) = x - \frac{x^2}{2}$ for every real number $x$. If $n$ is a positive integer, define $a_n$ by \[ a_n = L \Bigl( L \Bigl( L \Bigl( \cdots L \Bigl( \frac{17}{n} \Bigr) \cdots \Bigr) \Bigr) \Bigr), \] where there are $n$ iterations of $L$. For example, \[ a_4 = L \Bigl( L \Bigl( L \Bigl( L \Bigl( \frac{17}{4} \Bigr) \Bigr) \Bigr) \Bigr). \] As $n$ approaches infinity, what value does $n a_n$ approach?

2008 ISI B.Stat Entrance Exam, 3

Tags: calculus , function , derivative , geometry , 3d geometry , calculus computations

Study the derivatives of the function \[y=\sqrt{x^3-4x}\] and sketch its graph on the real line.

2014 AMC 12/AHSME, 21

Tags: logarithm , function , algebra , floor function , domain

For every real number $x$, let $\lfloor x\rfloor$ denote the greatest integer not exceeding $x$, and let \[f(x)=\lfloor x\rfloor(2014^{x-\lfloor x\rfloor}-1).\] The set of all numbers $x$ such that $1\leq x<2014$ and $f(x)\leq 1$ is a union of disjoint intervals. What is the sum of the lengths of those intervals? $\textbf{(A) }1\qquad \textbf{(B) }\dfrac{\log 2015}{\log 2014}\qquad \textbf{(C) }\dfrac{\log 2014}{\log 2013}\qquad \textbf{(D) }\dfrac{2014}{2013}\qquad \textbf{(E) }2014^{\frac1{2014}}\qquad$

1982 USAMO, 3

Tags: function , geometry , trigonometry , perimeter

If a point $A_1$ is in the interior of an equilateral triangle $ABC$ and point $A_2$ is in the interior of $\triangle{A_1BC}$, prove that \[\operatorname{I.Q.} (A_1BC) > \operatorname{I.Q.} (A_2BC),\] where the [i]isoperrimetric quotient[/i] of a figure $F$ is defined by \[\operatorname{I.Q.}(F) = \frac{\operatorname{Area}(F)}{[\operatorname{Perimeter}(F)]^2}.\]

2008 Junior Balkan MO, 1

Tags: function , polynomial , inequalities , algebra

Find all real numbers $ a,b,c,d$ such that \[ \left\{\begin{array}{cc}a \plus{} b \plus{} c \plus{} d \equal{} 20, \\ ab \plus{} ac \plus{} ad \plus{} bc \plus{} bd \plus{} cd \equal{} 150. \end{array} \right.\]

2003 Alexandru Myller, 4

Tags: function , integration , calculus

[b]a)[/b] Prove that the function $ 1\le t\mapsto\int_{1}^t\frac{\sin x}{x^n} dx $ has an horizontal asymptote, for any natural number $ n. $ [b]b)[/b] Calculate $ \lim_{n\to\infty }\lim_{t\to\infty }\int_{1}^t\frac{\sin x}{x^n} . $ [i]Mihai Piticari[/i]

1995 Cono Sur Olympiad, 2

Tags: ratio , function , geometry

The semicircle with centre $O$ and the diameter $AC$ is divided in two arcs $AB$ and $BC$ with ratio $1: 3$. $M$ is the midpoint of the radium $OC$. Let $T$ be the point of arc $BC$ such that the area of the cuadrylateral $OBTM$ is maximum. Find such area in fuction of the radium.

2003 AMC 12-AHSME, 19

Tags: function , geometry , analytic geometry , conic , parabola , geometric transformation , reflection

A parabola with equation $ y \equal{} ax^2 \plus{} bx \plus{} c$ is reﬂected about the $ x$-axis. The parabola and its reﬂection are translated horizontally ﬁve units in opposite directions to become the graphs of $ y \equal{} f(x)$ and $ y \equal{} g(x)$, respectively. Which of the following describes the graph of $ y \equal{} (f \plus{} g)(x)$? $ \textbf{(A)}\ \text{a parabola tangent to the }x\text{ \minus{} axis}$ $ \textbf{(B)}\ \text{a parabola not tangent to the }x\text{ \minus{} axis} \qquad \textbf{(C)}\ \text{a horizontal line}$ $ \textbf{(D)}\ \text{a non \minus{} horizontal line} \qquad \textbf{(E)}\ \text{the graph of a cubic function}$

1997 IMC, 6

Tags: function , trigonometry , real analysis

Let $f: [0,1]\rightarrow \mathbb{R}$ continuous. We say that $f$ crosses the axis at $x$ if $f(x)=0$ but $\exists y,z \in [x-\epsilon,x+\epsilon]: f(y)<0<f(z)$ for any $\epsilon$. (a) Give an example of a function that crosses the axis infinitely often. (b) Can a continuous function cross the axis uncountably often?

2014 PUMaC Algebra B, 6

Tags: algebra , ceiling function , function , floor function

There is a sequence with $a(2)=0$, $a(3)=1$ and $a(n)=a\left(\left\lfloor\dfrac n2\right\rfloor\right)+a\left(\left\lceil\dfrac n2\right\rceil\right)$ for $n\geq 4$. Find $a(2014)$. [Note that $\left\lfloor\dfrac n2\right\rfloor$ and $\left\lceil\dfrac n2\right\rceil$ denote the floor function (largest integer $\leq\tfrac n2$) and the ceiling function (smallest integer $\geq\tfrac n2$), respectively.]

1989 IMO Longlists, 6

Tags: function , geometry , analytic geometry

Let $ E$ be the set of all triangles whose only points with integer coordinates (in the Cartesian coordinate system in space), in its interior or on its sides, are its three vertices, and let $ f$ be the function of area of a triangle. Determine the set of values $ f(E)$ of $ f.$

2008 IMC, 1

Tags: function , functional equation , algebra

Find all continuous functions $f: \mathbb{R}\to \mathbb{R}$ such that \[ f(x)-f(y)\in \mathbb{Q}\quad \text{ for all }\quad x-y\in\mathbb{Q} \]