Olympiad problems

2012 China Team Selection Test, 1

Given an integer $n\ge 4$. $S=\{1,2,\ldots,n\}$. $A,B$ are two subsets of $S$ such that for every pair of $(a,b),a\in A,b\in B, ab+1$ is a perfect square. Prove that \[\min \{|A|,|B|\}\le\log _2n.\]

2018 China Team Selection Test, 4

Tags: function , algebra , functional equation , fe

Functions $f,g:\mathbb{Z}\to\mathbb{Z}$ satisfy $$f(g(x)+y)=g(f(y)+x)$$ for any integers $x,y$. If $f$ is bounded, prove that $g$ is periodic.

2019 Harvard-MIT Mathematics Tournament, 4

Tags: hmmt , algebra , function

Let $\mathbb{N}$ be the set of positive integers, and let $f: \mathbb{N} \to \mathbb{N}$ be a function satisfying [list] [*] $f(1) = 1$, [*] for $n \in \mathbb{N}$, $f(2n) = 2f(n)$ and $f(2n+1) = 2f(n) - 1$. [/list] Determine the sum of all positive integer solutions to $f(x) = 19$ that do not exceed 2019.

1963 Putnam, B3

Tags: function , algebra , functional equation

Find every twice-differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfies the functional equation $$ f(x)^2 -f(y)^2 =f(x+y)f(x-y)$$ for all $x,y \in \mathbb{R}. $

2019 Silk Road, 4

Tags: combinatorics , sum , floor function , even , algebra , function

The sequence $ \{a_n \} $ is defined as follows: $ a_0 = 1 $ and $ {a_n} = \sum \limits_ {k = 1} ^ {[\sqrt n]} {{a_ {n - {k ^ 2 }}}} $ for $ n \ge 1. $ Prove that among $ a_1, a_2, \ldots, a_ {10 ^ 6} $ there are at least $500$ even numbers. (Here, $ [x] $ is the largest integer not exceeding $ x $.)

1997 AIME Problems, 12

Tags: function , limit , quadratic , linear algebra

The function $f$ defined by $\displaystyle f(x)= \frac{ax+b}{cx+d}$. where $a,b,c$ and $d$ are nonzero real numbers, has the properties $f(19)=19, f(97)=97$ and $f(f(x))=x$ for all values except $\displaystyle \frac{-d}{c}$. Find the unique number that is not in the range of $f$.

2009 BMO TST, 1

Tags: function , algebra

Given the equation $x^4-x^3-1=0$ [b](a)[/b] Find the number of its real roots. [b](b)[/b] We denote by $S$ the sum of the real roots and by $P$ their product. Prove that $P< - \frac{11}{10}$ and $S> \frac {6}{11}$.

2014 Thailand TSTST, 1

Tags: function , algebra

Find all functions $f: {\mathbb{R^\plus{}}}\to{\mathbb{R^\plus{}}}$ such that \[ f(1\plus{}xf(y))\equal{}yf(x\plus{}y)\] for all $x,y\in\mathbb{R^\plus{}}$.

2019 BAMO, 4

Tags: function , functional equation , algebra

Let $S$ be a finite set of nonzero real numbers, and let $f : S\to S$ be a function with the following property: for each $x \in S$, either $f ( f (x)) = x+ f (x)$ or $f ( f (x)) = \frac{x+ f (x)}{2}$. Prove that $f (x) = x$ for all $x \in S$.

2010 Miklós Schweitzer, 8

Tags: topology , function , harmonic functions

Let $ D \subset \mathbb {R} ^ {2} $ be a finite Lebesgue measure of a connected open set and $ u: D \rightarrow \mathbb {R} $ a harmonic function. Show that it is either a constant $ u $ or for almost every $ p \in D $ $$ f ^ {\prime} (t) = (\operatorname {grad} u) (f (t)), \quad f (0) = p $$has no initial value problem(differentiable everywhere) solution to $ f:[0,\infty) \rightarrow D $.

2010 China Team Selection Test, 2

Tags: polynomial , function , complex number , algebra

Let $A=\{a_1,a_2,\cdots,a_{2010}\}$ and $B=\{b_1,b_2,\cdots,b_{2010}\}$ be two sets of complex numbers. Suppose \[\sum_{1\leq i<j\leq 2010} (a_i+a_j)^k=\sum_{1\leq i<j\leq 2010}(b_i+b_j)^k\] holds for every $k=1,2,\cdots, 2010$. Prove that $A=B$.

2009 Hong Kong TST, 5

Tags: inequalities , quadratic , function , algebra

Let $ a,b,c$ be the three sides of a triangle. Determine all possible values of $ \frac {a^2 \plus{} b^2 \plus{} c^2}{ab \plus{} bc \plus{} ca}$

1974 AMC 12/AHSME, 12

Tags: function

If $ g(x)\equal{}1\minus{}x^2$ and $ f(g(x)) \equal{} \frac{1\minus{}x^2}{x^2}$ when $ x\neq0$, then $ f(1/2)$ equals $ \textbf{(A)}\ 3/4 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ \sqrt2/2 \qquad \textbf{(E)}\ \sqrt2$

MathLinks Contest 7th, 4.3

Tags: inequalities , function , parameterization , algebra , polynomial , vieta , cauchy inequality

Let $ a,b,c$ be positive real numbers such that $ ab\plus{}bc\plus{}ca\equal{}3$. Prove that \[ \frac 1{1\plus{}a^2(b\plus{}c)} \plus{} \frac 1{1\plus{}b^2(c\plus{}a)} \plus{} \frac 1 {1\plus{}c^2(a\plus{}b) } \leq \frac 3 {1\plus{}2abc} .\]

1989 National High School Mathematics League, 9

Tags: function , geometry

Functions $f_0(x)=|x|,f_1(x)=|f_0(x)-1|,f_2(x)=|f_1(x)-2|$. Area of the closed part between the figure of $f_2(x)$ and $x$-axis is________.

2005 USA Team Selection Test, 2

Tags: geometry , circumcircle , parallelogram , trigonometry , inequalities , function , geometric transformation

Let $A_{1}A_{2}A_{3}$ be an acute triangle, and let $O$ and $H$ be its circumcenter and orthocenter, respectively. For $1\leq i \leq 3$, points $P_{i}$ and $Q_{i}$ lie on lines $OA_{i}$ and $A_{i+1}A_{i+2}$ (where $A_{i+3}=A_{i}$), respectively, such that $OP_{i}HQ_{i}$ is a parallelogram. Prove that \[\frac{OQ_{1}}{OP_{1}}+\frac{OQ_{2}}{OP_{2}}+\frac{OQ_{3}}{OP_{3}}\geq 3.\]

MathLinks Contest 7th, 5.3

Tags: inequalities , symmetry , function , calculus , derivative , parameterization , logarithm

If $ a\geq b\geq c\geq d > 0$ such that $ abcd\equal{}1$, then prove that \[ \frac 1{1\plus{}a} \plus{} \frac 1{1\plus{}b} \plus{} \frac 1{1\plus{}c} \geq \frac {3}{1\plus{}\sqrt[3]{abc}}.\]

2005 Today's Calculation Of Integral, 76

Tags: calculus , integration , function , limit , inequalities , calculus computations

The function $f_n (x)\ (n=1,2,\cdots)$ is defined as follows. \[f_1 (x)=x,\ f_{n+1}(x)=2x^{n+1}-x^n+\frac{1}{2}\int_0^1 f_n(t)\ dt\ \ (n=1,2,\cdots)\] Evaluate \[\lim_{n\to\infty} f_n \left(1+\frac{1}{2n}\right)\]

2016 China Team Selection Test, 6

Tags: algebra , function

Find all functions $f: \mathbb R^+ \rightarrow \mathbb R^+$ satisfying the following condition: for any three distinct real numbers $a,b,c$, a triangle can be formed with side lengths $a,b,c$, if and only if a triangle can be formed with side lengths $f(a),f(b),f(c)$.

2025 Korea - Final Round, P2

Tags: algebra , function , functional equation

Let $\mathbb{R}$ be the set of real numbers. Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ that satisfy the following condition. Here, $f^{100}(x)$ is the function obtained by composing $f(x)$ $100$ times, that is, $(\underbrace{f \circ f \circ \cdots \circ f}_{100 \ \text{times}})(x).$ [b](Condition)[/b] For all $x, y \in \mathbb{R}$, $$f(x + f^{100}(y)) = x + y \ \ \ \text{or} \ \ \ f(f^{100}(x) + y) = x + y$$

2010 AMC 12/AHSME, 22

Tags: trigonometry , function , inequalities , trig identities , law of cosines

Let $ ABCD$ be a cyclic quadrilateral. The side lengths of $ ABCD$ are distinct integers less than $ 15$ such that $ BC\cdot CD\equal{}AB\cdot DA$. What is the largest possible value of $ BD$? $ \textbf{(A)}\ \sqrt{\frac{325}{2}} \qquad \textbf{(B)}\ \sqrt{185} \qquad \textbf{(C)}\ \sqrt{\frac{389}{2}} \qquad \textbf{(D)}\ \sqrt{\frac{425}{2}} \qquad \textbf{(E)}\ \sqrt{\frac{533}{2}}$

2018 Taiwan TST Round 1, 1

Tags: function , algebra

Find all functions $ f: \mathbb{R} \to \mathbb{R} $ such that $$ f\left(f\left(x\right)+y\right) = f\left(x^2-y\right)+4\left(y-2\right)\left(f\left(x\right)+2\right) $$ holds for all $ x, y \in \mathbb{R} $

2023 AMC 12/AHSME, 22

Tags: function

A real-valued function $f$ has the property that for all real numbers $a$ and $b,$ $$f(a + b) + f(a - b) = 2f(a) f(b).$$ Which one of the following cannot be the value of $f(1)?$ $ \textbf{(A) } 0 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } -1 \qquad \textbf{(D) } 2 \qquad \textbf{(E) } -2$

2003 Gheorghe Vranceanu, 2

Tags: function , real analysis , limit point

Let be a real number $ a $ and a function $ f:[a,\infty )\longrightarrow\mathbb{R} $ that is continuous at $ a. $ Prove that $ f $ is primitivable on $ (a,\infty ) $ if and only if $ f $ is primitivable on $ [a,\infty ) . $

2009 Today's Calculation Of Integral, 506

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

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$.