Olympiad problems

2002 USAMTS Problems, 2

Tags: function

The integer 72 is the first of three consecutive integers 72, 73, and 74, that can each be expressed as the sum of the squares of two positive integers. The integers 72, 288, and 800 are the first three members of an infinite increasing sequence of integers with the above property. Find a function that generates the sequence and give the next three members.

1995 AIME Problems, 14

Tags: geometry , trigonometry , asymptote , vector , function , area of a triangle , heron's formula

In a circle of radius 42, two chords of length 78 intersect at a point whose distance from the center is 18. The two chords divide the interior of the circle into four regions. Two of these regions are bordered by segments of unequal lenghts, and the area of either of them can be expressed uniquley in the form $m\pi-n\sqrt{d},$ where $m, n,$ and $d$ are positive integers and $d$ is not divisible by the square of any prime number. Find $m+n+d.$

2016 District Olympiad, 2

Tags: function , calculus , integration

Let $ f:\mathbb{R}\longrightarrow (0,\infty ) $ be a continuous and periodic function having a period of $ 2, $ and such that the integral $ \int_0^2 \frac{f(1+x)}{f(x)} dx $ exists. Show that $$ \int_0^2 \frac{f(1+x)}{f(x)} dx\ge 2, $$ with equality if and only if $ 1 $ is also a period of $ f. $

2010 Putnam, B4

Tags: algebra , polynomial , function , induction , rational function , recursion

Find all pairs of polynomials $p(x)$ and $q(x)$ with real coefficients for which \[p(x)q(x+1)-p(x+1)q(x)=1.\]

2007 District Olympiad, 1

Tags: function , algebra

We say that a function $f: \mathbb{N}\rightarrow\mathbb{N}$ has the $(\mathcal{P})$ property if, for any $y\in\mathbb{N}$, the equation $f(x)=y$ has exactly 3 solutions. a) Prove that there exist an infinity of functions with the $(\mathcal{P})$ property ; b) Find all monotonously functions with the $(\mathcal{P})$ property ; c) Do there exist monotonously functions $f: \mathbb{Q}\rightarrow\mathbb{Q}$ satisfying the $(\mathcal{P})$ property ?

2012 Indonesia TST, 1

Tags: function , induction , floor function , algebra

Suppose a function $f : \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ satisfies $f(f(n)) + f(n+1) = n+2$ for all positive integer $n$. Prove that $f(f(n)+n) = n+1$ for all positive integer $n$.

2012 Bogdan Stan, 2

Tags: function , find all functions , calculus , integration

Find the continuous functions $ f:\left[ 0,\frac{1}{3} \right] \longrightarrow (0,\infty ) $ that satisfy the functional relation $$ 54\int_0^{1/3} f(x)dx +32\int_0^{1/3} \frac{dx}{\sqrt{x+f(x)}} =21. $$ [i]Cristinel Mortici[/i]

2012 IMC, 1

Tags: function

For every positive integer $n$, let $p(n)$ denote the number of ways to express $n$ as a sum of positive integers. For instance, $p(4)=5$ because \[4=3+1=2+2=2+1+1=1+1+1.\] Also define $p(0)=1$. Prove that $p(n)-p(n-1)$ is the number of ways to express $n$ as a sum of integers each of which is strictly greater than 1. [i]Proposed by Fedor Duzhin, Nanyang Technological University.[/i]

2021 CIIM, 4

Tags: function , inequalities

Let $\mathbb{Z}^{+}$ be the set of positive integers. [b]a)[/b] Prove that there is only one function $f:\mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+}$, strictly increasing, such that $f(f(n))=2n+1$ for every $n\in \mathbb{Z}^{+}$. [b]b)[/b] For the function in [b]a[/b]. Prove that for every $n\in \mathbb{Z}^{+}$ $\frac{4n+1}{3}\leq f(n)\leq \frac{3n+1}{2}$ [b]c) [/b] Prove that in each inequality side of [b]b[/b] the equality can reach by infinite positive integers $n$.

2010 Romanian Masters In Mathematics, 4

Tags: polynomial , function , ceiling function , inequalities , algebra

Determine whether there exists a polynomial $f(x_1, x_2)$ with two variables, with integer coefficients, and two points $A=(a_1, a_2)$ and $B=(b_1, b_2)$ in the plane, satisfying the following conditions: (i) $A$ is an integer point (i.e $a_1$ and $a_2$ are integers); (ii) $|a_1-b_1|+|a_2-b_2|=2010$; (iii) $f(n_1, n_2)>f(a_1, a_2)$ for all integer points $(n_1, n_2)$ in the plane other than $A$; (iv) $f(x_1, x_2)>f(b_1, b_2)$ for all integer points $(x_1, x_2)$ in the plane other than $B$. [i]Massimo Gobbino, Italy[/i]

2006 Princeton University Math Competition, 5

Tags: calculus , integration , function , inequalities , floor function

Find the greatest integer less than the number $1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots+\frac{1}{\sqrt{1000000}}$

2010 Contests, 3

Tags: trigonometry , calculus , function , complex number , algebra , inequalities

Given complex numbers $a,b,c$, we have that $|az^2 + bz +c| \leq 1$ holds true for any complex number $z, |z| \leq 1$. Find the maximum value of $|bc|$.

2022 Romania National Olympiad, P1

Tags: function , integral , calculus

Let $\mathcal{F}$ be the set of functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(2x)=f(x)$ for all $x\in\mathbb{R}.$ [list=a] [*]Determine all functions $f\in\mathcal{F}$ which admit antiderivatives on $\mathbb{R}.$ [*]Give an example of a non-constant function $f\in\mathcal{F}$ which is integrable on any interval $[a,b]\subset\mathbb{R}$ and satisfies \[\int_a^bf(x) \ dx=0\]for all real numbers $a$ and $b.$ [/list][i]Mihai Piticari and Sorin Rădulescu[/i]

2014 Contests, 2

Tags: function , algebra

Find all functions $f:\mathbb{R}\backslash\{0\}\rightarrow\mathbb{R}$ for which $xf(xy) + f(-y) = xf(x)$ for all non-zero real numbers $x, y$.

2012 Kyoto University Entry Examination, 3

Tags: function , geometry , geometric transformation , rotation , algebra

When real numbers $x,\ y$ moves in the constraint with $x^2+xy+y^2=6.$ Find the range of $x^2y+xy^2-x^2-2xy-y^2+x+y.$ 30 points

2014 Taiwan TST Round 1, 1

Tags: function , integration , logarithm

Find all increasing functions $f$ from the nonnegative integers to the integers satisfying $f(2)=7$ and \[ f(mn) = f(m) + f(n) + f(m)f(n) \] for all nonnegative integers $m$ and $n$.

1986 Iran MO (2nd round), 2

Tags: function , calculus , derivative , limit , algebra

[b](a)[/b] Sketch the diagram of the function $f$ if \[f(x)=4x(1-|x|) , \quad |x| \leq 1.\] [b](b)[/b] Does there exist derivative of $f$ in the point $x=0 \ ?$ [b](c)[/b] Let $g$ be a function such that \[g(x)=\left\{\begin{array}{cc}\frac{f(x)}{x} \quad : x \neq 0\\ \text{ } \\ 4 \ \ \ \ \quad : x=0\end{array}\right.\] Is the function $g$ continuous in the point $x=0 \ ?$ [b](d)[/b] Sketch the diagram of $g.$

1958 AMC 12/AHSME, 46

Tags: function , calculus , derivative , algebra , domain

For values of $ x$ less than $ 1$ but greater than $ \minus{}4$, the expression \[ \frac{x^2 \minus{} 2x \plus{} 2}{2x \minus{} 2} \] has: $ \textbf{(A)}\ \text{no maximum or minimum value}\qquad \\ \textbf{(B)}\ \text{a minimum value of }{\plus{}1}\qquad \\ \textbf{(C)}\ \text{a maximum value of }{\plus{}1}\qquad \\ \textbf{(D)}\ \text{a minimum value of }{\minus{}1}\qquad \\ \textbf{(E)}\ \text{a maximum value of }{\minus{}1}$

2019 Teodor Topan, 2

Tags: function , real analysis , functional analysis , analytic geometry

Let $ I $ be a nondegenerate interval, and let $ F $ be a primitive of a function $ f:I\longrightarrow\mathbb{R} . $ Show that for any distinct $ a,b\in I, $ the tangents to the graph of $ F $ at the points $ (a,F(a)) ,(b,F(b)) $ are concurrent at a point whose abscisa is situated in the interval $ (a,b). $ [i]Nicolae Bourbăcuț[/i]

2010 Polish MO Finals, 2

Tags: function , algebra , polynomial , number theory

Positive rational number $a$ and $b$ satisfy the equality \[a^3 + 4a^2b = 4a^2 + b^4.\] Prove that the number $\sqrt{a}-1$ is a square of a rational number.

1986 Vietnam National Olympiad, 1

Tags: function , inequalities

Let $ \frac{1}{2}\le a_1, a_2, \ldots, a_n \le 5$ be given real numbers and let $ x_1, x_2, \ldots, x_n$ be real numbers satisfying $ 4x_i^2\minus{} 4a_ix_i \plus{} \left(a_i \minus{} 1\right)^2 \le 0$. Prove that \[ \sqrt{\sum_{i\equal{}1}^n\frac{x_i^2}{n}}\le\sum_{i\equal{}1}^n\frac{x_i}{n}\plus{}1\]

2018 Chile National Olympiad, 4

Tags: floor function , algebra , number theory , function

Find all postitive integers n such that $$\left\lfloor \frac{n}{2} \right\rfloor \cdot \left\lfloor \frac{n}{3} \right\rfloor \cdot \left\lfloor \frac{n}{4} \right\rfloor=n^2$$ where $\lfloor x \rfloor$ represents the largest integer less than the real number $x$.

2011 Today's Calculation Of Integral, 739

Tags: calculus , integration , function , trigonometry , derivative , algebra , functional equation

Find the function $f(x)$ such that : \[f(x)=\cos x+\int_0^{2\pi} f(y)\sin (x-y)\ dy\]

2005 Vietnam Team Selection Test, 3

Tags: function , geometry , geometric transformation , 3d geometry , induction , calculus , integration

Find all functions $f: \mathbb{Z} \mapsto \mathbb{Z}$ satisfying the condition: $f(x^3 +y^3 +z^3 )=f(x)^3+f(y)^3+f(z)^3.$

1996 AMC 12/AHSME, 12

Tags: function

A function $ f$ from the integers to the integers is defined as follows: \[ f(n) \equal{} \begin{cases} n \plus{} 3 & \text{if n is odd} \\ n/2 & \text{if n is even} \end{cases} \]Suppose $ k$ is odd and $ f(f(f(k))) \equal{} 27$. What is the sum of the digits of $ k$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 15$