Olympiad problems

MathLinks Contest 2nd, 6.1

Determine the parity of the positive integer $N$, where $$N = \lfloor \frac{2002!}{2001 \cdot2003} \rfloor.$$

1987 Balkan MO, 2

Tags: function , algebra

Find all real numbers $x,y$ greater than $1$, satisfying the condition that the numbers $\sqrt{x-1}+\sqrt{y-1}$ and $\sqrt{x+1}+\sqrt{y+1}$ are nonconsecutive integers.

2011 Romanian Master of Mathematics, 5

Tags: vector , function , geometry , geometric transformation , reflection , combinatorial geometry , perpendicular bisector

For every $n\geq 3$, determine all the configurations of $n$ distinct points $X_1,X_2,\ldots,X_n$ in the plane, with the property that for any pair of distinct points $X_i$, $X_j$ there exists a permutation $\sigma$ of the integers $\{1,\ldots,n\}$, such that $\textrm{d}(X_i,X_k) = \textrm{d}(X_j,X_{\sigma(k)})$ for all $1\leq k \leq n$. (We write $\textrm{d}(X,Y)$ to denote the distance between points $X$ and $Y$.) [i](United Kingdom) Luke Betts[/i]

2022 USAMTS Problems, 2

Tags: function , functional equation , algebra

Let $Z^+$ denote the set of positive integers. Determine , with proof, if there exists a function $f:\mathbb{Z^+}\rightarrow\mathbb {Z^+}$ such that $f(f(f(f(f(n)))))$ = $2022n$ for all positive integers $n$.

1997 ITAMO, 2

Tags: odd , function , periodic , periodic function , algebra

Let a real function $f$ defined on the real numbers satisfy the following conditions: (i) $f(10+x) = f(10- x)$ (ii) $f(20+x) = - f(20- x)$ for all $x$. Prove that f is odd and periodic.

1990 IMO Longlists, 80

Tags: function , induction , algebra

Function $f(x, y): \mathbb N \times \mathbb N \to \mathbb Q$ satisfies the conditions: (i) $f(1, 1) =1$, (ii) $f(p + 1, q) + f(p, q + 1) = f(p, q)$ for all $p, q \in \mathbb N$, and (iii) $qf(p + 1, q) = pf(p, q + 1)$ for all $p, q \in \mathbb N$. Find $f(1990, 31).$

1974 Miklós Schweitzer, 5

Tags: function , integration , real analysis

Let $ \{f_n \}_{n=0}^{\infty}$ be a uniformly bounded sequence of real-valued measurable functions defined on $ [0,1]$ satisfying \[ \int_0^1 f_n^2=1.\] Further, let $ \{ c_n \}$ be a sequence of real numbers with \[ \sum_{n=0}^{\infty} c_n^2= +\infty.\] Prove that some re-arrangement of the series $ \sum_{n=0}^{\infty} c_nf_n$ is divergent on a set of positive measure. [i]J. Komlos[/i]

2007 Germany Team Selection Test, 1

Tags: inequalities , function , algebra

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]

1984 IMO Longlists, 52

Tags: trigonometry , function , geometry

Construct a scalene triangle such that \[a(\tan B - \tan C) = b(\tan A - \tan C)\]

2007 Moldova National Olympiad, 12.6

Tags: hyperbola , ellipse , trigonometry , function , algebra , conic

Show that the distance between a point on the hyperbola $xy=5$ and a point on the ellipse $x^{2}+6y^{2}=6$ is at least $\frac{9}{7}$.

2008 Serbia National Math Olympiad, 3

Tags: function , cauchy inequality , inequalities

Let $ a$, $ b$, $ c$ be positive real numbers such that $ a \plus{} b \plus{} c \equal{} 1$. Prove inequality: \[ \frac{1}{bc \plus{} a \plus{} \frac{1}{a}} \plus{} \frac{1}{ac \plus{} b \plus{} \frac{1}{b}} \plus{} \frac{1}{ab \plus{} c \plus{} \frac{1}{c}} \leqslant \frac{27}{31}.\]

1984 Tournament Of Towns, (063) O4

Tags: function , graph , combinatorics , combinatorial geometry , geometry , rectangle

Prove that, for any natural number $n$, the graph of any increasing function $f : [0,1] \to [0, 1]$ can be covered by $n$ rectangles each of area whose sides are parallel to the coordinate axes. Assume that a rectangle includes both its interior and boundary points. (a) Assume that $f(x)$ is continuous on $[0,1]$. (b) Do not assume that $f(x)$ is continuous on $[0,1]$. (A Andjans, Riga) PS. (a) for O Level, (b) for A Level

2003 Romania National Olympiad, 3

Tags: limit , function , continuity , real analysis

Let be two functions $ f,g:\mathbb{R}_{\ge 0 }\longrightarrow\mathbb{R} $ having that properties that $ f $ is continuous, $ g $ is nondecreasing and unbounded, and for any sequence of rational numbers $ \left( x_n \right)_{n\ge 1} $ that diverges to $ \infty , $ we have $$ 1=\lim_{n\to\infty } f\left( x_n \right) g\left( x_n \right) . $$ Prove that $1=\lim_{x\to\infty } f\left( x \right) g\left( x \right) . $ [i]Radu Gologan[/i]

2015 Postal Coaching, Problem 6

Tags: number theory , function

Let $k \in \mathbb{N}$, let $x_k$ denote the nearest integer to $\sqrt k$. Show that for each $m \in \mathbb {N}$, $$\sum_{k=1}^{m} \frac{1}{x_k} = f(m)+ \frac{m}{f(m)+1}$$, where $f(m)$ is the integer part of $\frac{\sqrt{4m-3}-1}{2}$

2012 Balkan MO Shortlist, A6

Tags: algebra , function , domain , inequalities

Let $k$ be a positive integer. Find the maximum value of \[a^{3k-1}b+b^{3k-1}c+c^{3k-1}a+k^2a^kb^kc^k,\] where $a$, $b$, $c$ are non-negative reals such that $a+b+c=3k$.

2007 Today's Calculation Of Integral, 200

Tags: calculus , integration , trigonometry , function , abstract algebra , algebra , polynomial

Evaluate the following definite integral. \[\int_{0}^{\pi}\frac{\cos nx}{2-\cos x}dx\ (n=0,\ 1,\ 2,\ \cdots)\]

2011 Singapore MO Open, 5

Tags: modular arithmetic , algebra , function , domain , number theory

Find all pairs of positive integers $(m,n)$ such that \[m+n-\frac{3mn}{m+n}=\frac{2011}{3}.\]

1965 Swedish Mathematical Competition, 4

Tags: function , recurrence relation , sequence , algebra

Find constants $A > B$ such that $\frac{f\left( \frac{1}{1+2x}\right) }{f(x)}$ is independent of $x$, where $f(x) = \frac{1 + Ax}{1 + Bx}$ for all real $x \ne - \frac{1}{B}$. Put $a_0 = 1$, $a_{n+1} = \frac{1}{1 + 2a_n}$. Find an expression for an by considering $f(a_0), f(a_1), ...$.

2011 Morocco National Olympiad, 3

Tags: function , search , algebra

Find all functions $f : \mathbb{R} \to \mathbb{R} $ which verify the relation \[(x-2)f(y)+f(y+2f(x))= f(x+yf(x)), \qquad \forall x,y \in \mathbb R.\]

1995 Israel Mathematical Olympiad, 8

Tags: functional equation , functional , function , algebra

A real number $\alpha$ is given. Find all functions $f : R^+ \to R^+$ satisfying $\alpha x^2f\left(\frac{1}{x}\right) +f(x) =\frac{x}{x+1}$ for all $x > 0$.

2014 Bulgaria National Olympiad, 2

Tags: function , induction , limit , geometric series , algebra

Find all functions $f: \mathbb{Q}^+ \to \mathbb{R}^+ $ with the property: \[f(xy)=f(x+y)(f(x)+f(y)) \,,\, \forall x,y \in \mathbb{Q}^+\] [i]Proposed by Nikolay Nikolov[/i]

1982 Miklós Schweitzer, 9

Tags: topology , geometry , 3d geometry , function

Suppose that $ K$ is a compact Hausdorff space and $ K\equal{} \cup_{n\equal{}0}^{\infty}A_n$, where $ A_n$ is metrizable and $ A_n \subset A_m$ for $ n<m$. Prove that $ K$ is metrizable. [i]Z. Balogh[/i]

1992 Putnam, A1

Tags: function , algebra

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$.

2013 Saudi Arabia IMO TST, 2

Tags: increasing , function , inequalities , algebra

Let $S = f\{0.1. 2.3,...\}$ be the set of the non-negative integers. Find all strictly increasing functions $f : S \to S$ such that $n + f(f(n)) \le 2f(n)$ for every $n$ in $S$

2010 AMC 10, 4

Tags: function

For a real number $ x$, define $ \heartsuit (x)$ to be the average of $ x$ and $ x^2$. What is $ \heartsuit(1) \plus{} \heartsuit(2) \plus{}\heartsuit(3)$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 20$