Olympiad problems

2011 AMC 10, 25

Let $R$ be a square region and $n\ge4$ an integer. A point $X$ in the interior of $R$ is called $n\text{-}ray$ partitional if there are $n$ rays emanating from $X$ that divide $R$ into $n$ triangles of equal area. How many points are 100-ray partitional but not 60-ray partitional? $\textbf{(A)}\,1500 \qquad\textbf{(B)}\,1560 \qquad\textbf{(C)}\,2320 \qquad\textbf{(D)}\,2480 \qquad\textbf{(E)}\,2500$

2006 Iran Team Selection Test, 5

Tags: function , circumcircle , inequalities , trigonometry , geometry , incenter

Let $ABC$ be an acute angle triangle. Suppose that $D,E,F$ are the feet of perpendicluar lines from $A,B,C$ to $BC,CA,AB$. Let $P,Q,R$ be the feet of perpendicular lines from $A,B,C$ to $EF,FD,DE$. Prove that \[ 2(PQ+QR+RP)\geq DE+EF+FD \]

2001 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: floor function , function , sequence , radical , algebra

For every integer $n > 1$, the sequence $\left( {{S}_{n}} \right)$ is defined by ${{S}_{n}}=\left\lfloor {{2}^{n}}\underbrace{\sqrt{2+\sqrt{2+...+\sqrt{2}}}}_{n\ radicals} \right\rfloor $ where $\left\lfloor x \right\rfloor$ denotes the floor function of $x$. Prove that ${{S}_{2001}}=2\,{{S}_{2000}}+1$. .

2006 Romania National Olympiad, 1

Tags: algebra , induction , function

Let $\displaystyle M$ be a set composed of $\displaystyle n$ elements and let $\displaystyle \mathcal P (M)$ be its power set. Find all functions $\displaystyle f : \mathcal P (M) \to \{ 0,1,2,\ldots,n \}$ that have the properties (a) $\displaystyle f(A) \neq 0$, for $\displaystyle A \neq \phi$; (b) $\displaystyle f \left( A \cup B \right) = f \left( A \cap B \right) + f \left( A \Delta B \right)$, for all $\displaystyle A,B \in \mathcal P (M)$, where $\displaystyle A \Delta B = \left( A \cup B \right) \backslash \left( A \cap B \right)$.

Dumbest FE I ever created, 2.

Tags: algebra , functional equation , function

Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R} $ such that for all $ x, y \in \mathbb{R} $, \[ f(x + f(2y)) + f(x^2 - y) = f(f(x)) f(x + 1) + 2y - f(y). \]

2012 China Team Selection Test, 2

Tags: floor function , algebra , function , inequalities , pigeonhole principle , difference of squares , ceiling function

Prove that there exists a positive real number $C$ with the following property: for any integer $n\ge 2$ and any subset $X$ of the set $\{1,2,\ldots,n\}$ such that $|X|\ge 2$, there exist $x,y,z,w \in X$(not necessarily distinct) such that \[0<|xy-zw|<C\alpha ^{-4}\] where $\alpha =\frac{|X|}{n}$.

PEN J Problems, 11

Tags: function , inequalities , logarithm , floor function , modular arithmetic , number theory , induction

Prove that ${d((n^2 +1)}^2)$ does not become monotonic from any given point onwards.

2012 Math Prize for Girls Olympiad, 4

Tags: factorial , function , pigeonhole principle

Let $f$ be a function from the set of rational numbers to the set of real numbers. Suppose that for all rational numbers $r$ and $s$, the expression $f(r + s) - f(r) - f(s)$ is an integer. Prove that there is a positive integer $q$ and an integer $p$ such that \[ \Bigl\lvert f\Bigl(\frac{1}{q}\Bigr) - p \Bigr\rvert \le \frac{1}{2012} \, . \]

2006 International Zhautykov Olympiad, 2

Tags: trapezoid , parallelogram , analytic geometry , function , ratio , geometry , trigonometry

Let $ ABC$ be a triangle and $ K$ and $ L$ be two points on $ (AB)$, $ (AC)$ such that $ BK \equal{} CL$ and let $ P \equal{} CK\cap BL$. Let the parallel through $ P$ to the interior angle bisector of $ \angle BAC$ intersect $ AC$ in $ M$. Prove that $ CM \equal{} AB$.

2005 SNSB Admission, 2

Tags: measure theory , real analysis , function

Let $ \lambda $ be the Lebesgue measure in the plane, let $ u,v\in\mathbb{R}^2, $ let $ A\subset\mathbb{R}^2 $ such that $ \lambda (A)>0 $ and let be the function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ defined as $$ f(t)=\int_A \chi_{A+tu}\cdot\chi_{A+tv}\cdot d\lambda $$ [b]a)[/b] Show that $ f $ is continuous. [b]b)[/b] Prove that any Lebesgue measurable subset of the plane that has nonzero Lebesgue measure contains the vertices of an equilateral triangle.

1972 Miklós Schweitzer, 7

Tags: inequalities , advanced fields , function

Let $ f(x,y,z)$ be a nonnegative harmonic function in the unit ball of $ \mathbb{R}^3$ for which the inequality $ f(x_0,0,0) \leq \varepsilon^2$ holds for some $ 0\leq x_0 \leq 1$ and $ 0<\varepsilon<(1\minus{}x_0)^2$. Prove that $ f(x,y,z) \leq \varepsilon$ in the ball with center at the origin an radius $ (1\minus{}3\varepsilon^{1/4}).$ [i]P. Turan[/i]

1993 Hungary-Israel Binational, 3

Tags: abstract algebra , superior algebra , group theory , function

In the questions below: $G$ is a ﬁnite group; $H \leq G$ a subgroup of $G; |G : H |$ the index of $H$ in $G; |X |$ the number of elements of $X \subseteq G; Z (G)$ the center of $G; G'$ the commutator subgroup of $G; N_{G}(H )$ the normalizer of $H$ in $G; C_{G}(H )$ the centralizer of $H$ in $G$; and $S_{n}$ the $n$-th symmetric group. Show that every element of $S_{n}$ is a product of $2$-cycles.

2007 ISI B.Math Entrance Exam, 8

Tags: function , algebra

Let $P:\mathbb{R} \to \mathbb{R}$ be a continuous function such that $P(X)=X$ has no real solution. Prove that $P(P(X))=X$ has no real solution.

1967 IMO Longlists, 58

Tags: inequality , linear function , algebra , geometry , function

A linear binomial $l(z) = Az + B$ with complex coefficients $A$ and $B$ is given. It is known that the maximal value of $|l(z)|$ on the segment $-1 \leq x \leq 1$ $(y = 0)$ of the real line in the complex plane $z = x + iy$ is equal to $M.$ Prove that for every $z$ \[|l(z)| \leq M \rho,\] where $\rho$ is the sum of distances from the point $P=z$ to the points $Q_1: z = 1$ and $Q_3: z = -1.$

1992 Baltic Way, 11

Tags: continued fraction , function , algebra

Let $ Q^\plus{}$ denote the set of positive rational numbers. Show that there exists one and only one function $f: Q^\plus{}\to Q^\plus{}$ satisfying the following conditions: (i) If $ 0<q<1/2$ then $ f(q)\equal{}1\plus{}f(q/(1\minus{}2q))$, (ii) If $ 1<q\le2$ then $ f(q)\equal{}1\plus{}f(q\minus{}1)$, (iii) $ f(q)\cdot f(1/q)\equal{}1$ for all $ q\in Q^\plus{}$.

2010 District Olympiad, 3

Tags: function , real analysis , inequalities

Let $ f: \mathbb{R}\rightarrow \mathbb{R}$ a strictly increasing function such that $ f\circ f$ is continuos. Prove that $ f$ is continuos.

1992 IMO Longlists, 45

Tags: function , additive number theory , additive combinatorics , counting , number theory

Let $n$ be a positive integer. Prove that the number of ways to express $n$ as a sum of distinct positive integers (up to order) and the number of ways to express $n$ as a sum of odd positive integers (up to order) are the same.

2009 Today's Calculation Of Integral, 431

Tags: function , integration , calculus , trigonometry , calculus computations

Consider the function $ f(\theta) \equal{} \int_0^1 |\sqrt {1 \minus{} x^2} \minus{} \sin \theta|dx$ in the interval of $ 0\leq \theta \leq \frac {\pi}{2}$. (1) Find the maximum and minimum values of $ f(\theta)$. (2) Evaluate $ \int_0^{\frac {\pi}{2}} f(\theta)\ d\theta$.

2009 Indonesia MO, 2

Tags: function , inequalities

Find the lowest possible values from the function \[ f(x) \equal{} x^{2008} \minus{} 2x^{2007} \plus{} 3x^{2006} \minus{} 4x^{2005} \plus{} 5x^{2004} \minus{} \cdots \minus{} 2006x^3 \plus{} 2007x^2 \minus{} 2008x \plus{} 2009\] for any real numbers $ x$.

2015 IFYM, Sozopol, 1

Tags: algebra , function

Find all functions $\mathbb R^+\to\mathbb R^+$ such that \[(f(a)+f(b))(f(c)+f(d))=(a+b)(c+d), \quad \forall a,b,c,d\in\mathbb R^+; \quad abcd=1\]

2016 Switzerland Team Selection Test, Problem 2

Tags: function , polynomial , algebra

Find all polynomial functions with real coefficients for which $$(x-2)P(x+2)+(x+2)P(x-2)=2xP(x)$$ for all real $x$

2009 Federal Competition For Advanced Students, P2, 2

Tags: floor function , function , algebra

(i) For positive integers $a<b$, let $M(a,b)=\frac{\Sigma^{b}_{k=a}\sqrt{k^2+3k+3}}{b-a+1}$. Calculate $[M(a,b)]$ (ii) Calculate $N(a,b)=\frac{\Sigma^{b}_{k=a}[\sqrt{k^2+3k+3}]}{b-a+1}$.

1979 IMO Longlists, 21

Tags: floor function , function , algebra

Let $E$ be the set of all bijective mappings from $\mathbb R$ to $\mathbb R$ satisfying \[f(t) + f^{-1}(t) = 2t, \qquad \forall t \in \mathbb R,\] where $f^{-1}$ is the mapping inverse to $f$. Find all elements of $E$ that are monotonic mappings.

2010 Ukraine Team Selection Test, 3

Tags: function , algebra , functional equation

Find all functions $f$ from the set of real numbers into the set of real numbers which satisfy for all $x$, $y$ the identity \[ f\left(xf(x+y)\right) = f\left(yf(x)\right) +x^2\] [i]Proposed by Japan[/i]

2009 Today's Calculation Of Integral, 513

Tags: integration , calculus computations , function , trigonometry , calculus

Find the constants $ a,\ b,\ c$ such that a function $ f(x)\equal{}a\sin x\plus{}b\cos x\plus{}c$ satisfies the following equation for any real numbers $ x$. \[ 5\sin x\plus{}3\cos x\plus{}1\plus{}\int_0^{\frac{\pi}{2}} (\sin x\plus{}\cos t)f(t)\ dt\equal{}f(x).\]