Olympiad problems

2000 Belarus Team Selection Test, 4.1

Find all functions $f ,g,h : R\to R$ such that $f(x+y^3)+g(x^3+y) = h(xy)$ for all $x,y \in R$

2007 QEDMO 4th, 13

Tags: combinatorics , function

Let $n$ and $k$ be integers such that $0\leq k\leq n$. Prove that $\sum_{u=0}^{k}\binom{n+u-1}{u}\binom{n}{k-2u}=\binom{n+k-1}{k}$. Note that we use the following conventions: $\binom{r}{0}=1$ for every integer $r$; $\binom{u}{v}=0$ if $u$ is a nonnegative integer and $v$ is an integer satisfying $v<0$ or $v>u$. Darij

2010 Today's Calculation Of Integral, 563

Tags: integration , calculus , function , quadratic , calculus computations

Determine the pair of constant numbers $ a,\ b,\ c$ such that for a quadratic function $ f(x) \equal{} x^2 \plus{} ax \plus{} b$, the following equation is identity with respect to $ x$. \[ f(x \plus{} 1) \equal{} c\int_0^1 (3x^2 \plus{} 4xt)f'(t)dt\] .

2010 IberoAmerican, 1

Tags: combinatorics , analytic geometry , function , vector

There are ten coins a line, which are indistinguishable. It is known that two of them are false and have consecutive positions on the line. For each set of positions, you may ask how many false coins it contains. Is it possible to identify the false coins by making only two of those questions, without knowing the answer to the first question before making the second?

2007 Nicolae Păun, 2

Tags: function , linear algebra , commutative algebra , matrix

For a given natural number, $ n\ge 2, $ consider two matrices $ A,B\in\mathcal{M}_n(\mathbb{C}) $ that commute and such that $ A $ is invertible and that the function $ M:\mathbb{C}\longrightarrow\mathbb{C} ,M(x)=\det (A+xB) $ is bounded above or below. Prove that $ B^n=0. $ [i]Sorin Rădulescu[/i] and [i]Ion Savu[/i]

2013 ISI Entrance Examination, 3

Tags: real analysis , inequalities , function

Let $f:\mathbb R\to\mathbb R$ satisfy \[|f(x+y)-f(x-y)-y|\leq y^2\] For all $(x,y)\in\mathbb R^2.$ Show that $f(x)=\frac x2+c$ where $c$ is a constant.

2009 Today's Calculation Of Integral, 506

Tags: conic , integration , parabola , function , calculus computations , geometry , calculus

Let $ a,\ b$ be the real numbers such that $ 0\leq a\leq b\leq 1$. Find the minimum value of $ \int_0^1 |(x\minus{}a)(x\minus{}b)|\ dx$.

2017 Bulgaria EGMO TST, 1

Tags: algebra , number theory , function , algorithm , euclidean algorithm , functional equation , induction

Let $\mathbb{Q^+}$ denote the set of positive rational numbers. Determine all functions $f: \mathbb{Q^+} \to \mathbb{Q^+}$ that satisfy the conditions \[ f \left( \frac{x}{x+1}\right) = \frac{f(x)}{x+1} \qquad \text{and} \qquad f \left(\frac{1}{x}\right)=\frac{f(x)}{x^3}\] for all $x \in \mathbb{Q^+}.$

2002 AMC 12/AHSME, 20

Tags: function

Let $f$ be a real-valued function such that \[f(x)+2f\left(\dfrac{2002}x\right)=3x\] for all $x>0$. Find $f(2)$. $\textbf{(A) }1000\qquad\textbf{(B) }2000\qquad\textbf{(C) }3000\qquad\textbf{(D) }4000\qquad\textbf{(E) }6000$

2006 Stanford Mathematics Tournament, 2

Tags: calculus , derivative , function

Find the minimum value of $ 2x^2\plus{}2y^2\plus{}5z^2\minus{}2xy\minus{}4yz\minus{}4x\minus{}2z\plus{}15$ for real numbers $ x$, $ y$, $ z$.

2020 BMT Fall, 5

Tags: functional equation , function , algebra

Let $f:\mathbb{R}^+\to \mathbb{R}^+$ be a function such that for all $x,y \in \mathbb{R}+,\, f(x)f(y)=f(xy)+f\left(\frac{x}{y}\right)$, where $\mathbb{R}^+$ represents the positive real numbers. Given that $f(2)=3$, compute the last two digits of $f\left(2^{2^{2020}}\right)$.

1997 Putnam, 4

Tags: function

Let $G$ be group with identity $e$ and $\phi :G\to G$ be a function such that : \[ \phi(g_1)\cdot \phi(g_2)\cdot \phi(g_3)=\phi(h_1)\cdot \phi(h_2)\cdot \phi(h_3) \] Whenever $g_1\cdot g_2\cdot g_3=e=h_1\cdot h_2\cdot h_3$ Show there exists $a\in G$ such that $\psi(x)=a\phi(x)$ is a homomorphism. (that is $\psi(x\cdot y)=\psi (x)\cdot \psi(y)$ for all $x,y\in G$ )

1999 National Olympiad First Round, 28

Tags: trigonometry , function

Find the number of functions defined on positive real numbers such that $ f\left(1\right) \equal{} 1$ and for every $ x,y\in \Re$, $ f\left(x^{2} y^{2} \right) \equal{} f\left(x^{4} \plus{} y^{4} \right)$. $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{Infinitely many}$

2001 Romania National Olympiad, 2

Tags: number theory , function , geometry , analytic geometry

For every rational number $m>0$ we consider the function $f_m:\mathbb{R}\rightarrow\mathbb{R},f_m(x)=\frac{1}{m}x+m$. Denote by $G_m$ the graph of the function $f_m$. Let $p,q,r$ be positive rational numbers. a) Show that if $p$ and $q$ are distinct then $G_p\cap G_q$ is non-empty. b) Show that if $G_p\cap G_q$ is a point with integer coordinates, then $p$ and $q$ are integer numbers. c) Show that if $p,q,r$ are consecutive natural numbers, then the area of the triangle determined by intersections of $G_p,G_q$ and $G_r$ is equal to $1$.

2024 Turkey EGMO TST, 2

Tags: divisibility , functional equation , function , number theory

Find all functions $f:\mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+}$ such that the conditions $\quad a) \quad a-b \mid f(a)-f(b)$ for all $a\neq b$ and $a,b \in \mathbb{Z}^{+}$ $\quad b) \quad f(\varphi(a))=\varphi(f(a))$ for all $a \in \mathbb{Z}^{+}$ where $\varphi$ is the Euler's totient function. holds

2010 Romania National Olympiad, 3

Tags: function , real analysis

Let $f:\mathbb{R}\rightarrow [0,\infty)$. Prove that $f(x+y)\ge (y+1)f(x),\ (\forall)x\in \mathbb{R}$ if and only if the function $g:\mathbb{R}\rightarrow [0,\infty),\ g(x)=e^{-x}f(x),\ (\forall)x\in \mathbb{R}$ is increasing.

2010 Romanian Master of Mathematics, 2

Tags: algebra , cauchy inequality , absolute value , inequalities , function

For each positive integer $n$, find the largest real number $C_n$ with the following property. Given any $n$ real-valued functions $f_1(x), f_2(x), \cdots, f_n(x)$ defined on the closed interval $0 \le x \le 1$, one can find numbers $x_1, x_2, \cdots x_n$, such that $0 \le x_i \le 1$ satisfying \[|f_1(x_1)+f_2(x_2)+\cdots f_n(x_n)-x_1x_2\cdots x_n| \ge C_n\] [i]Marko Radovanović, Serbia[/i]

1993 Tournament Of Towns, (377) 5

Tags: algebra , functional equation , functional , function

Does there exist a piecewise linear function $f$ defined on the segment [$-1,1]$ (including the ends) such that $f(f(x)) = -x$ for all x? (A function is called piecewise linear if its graph is the union of a finite set of points and intervals; it may be discontinuous).

2006 MOP Homework, 4

Tags: system of equations , algebra , inequalities , function , polynomial

Let $n$ be a positive integer. Solve the system of equations \begin{align*}x_{1}+2x_{2}+\cdots+nx_{n}&= \frac{n(n+1)}{2}\\ x_{1}+x_{2}^{2}+\cdots+x_{n}^{n}&= n\end{align*} for $n$-tuples $(x_{1},x_{2},\ldots,x_{n})$ of nonnegative real numbers.

1995 AIME Problems, 14

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

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 $m+n+d.$

1969 Miklós Schweitzer, 3

Tags: real analysis , abstract algebra , group theory , function

Let $ f(x)$ be a nonzero, bounded, real function on an Abelian group $ G$, $ g_1,...,g_k$ are given elements of $ G$ and $ \lambda_1,...,\lambda_k$ are real numbers. Prove that if \[ \sum_{i=1}^k \lambda_i f(g_ix) \geq 0\] holds for all $ x \in G$, then \[ \sum_{i=1}^k \lambda_i \geq 0.\] [i]A. Mate[/i]

2018 China Team Selection Test, 6

Tags: function , functional equation , algebra , divisibility , iteration

Let $M,a,b,r$ be non-negative integers with $a,r\ge 2$, and suppose there exists a function $f:\mathbb{Z}\rightarrow\mathbb{Z}$ satisfying the following conditions: (1) For all $n\in \mathbb{Z}$, $f^{(r)}(n)=an+b$ where $f^{(r)}$ denotes the composition of $r$ copies of $f$ (2) For all $n\ge M$, $f(n)\ge 0$ (3) For all $n>m>M$, $n-m|f(n)-f(m)$ Show that $a$ is a perfect $r$-th power.