Olympiad problems

1989 National High School Mathematics League, 10

A positive number, if its fractional part, integeral part, and itself are geometric series, then the number is________.

2013 Iran MO (3rd Round), 4

Tags: function , functional equation , algebra

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f(0) \in \mathbb Q$ and \[f(x+f(y)^2 ) = {f(x+y)}^2.\] (25 points)

2017 Harvard-MIT Mathematics Tournament, 10

Tags: function , modular arithmetic

Let $\mathbb{N}$ denote the natural numbers. Compute the number of functions $f:\mathbb{N}\rightarrow \{0, 1, \dots, 16\}$ such that $$f(x+17)=f(x)\qquad \text{and} \qquad f(x^2)\equiv f(x)^2+15 \pmod {17}$$ for all integers $x\ge 1$.

2011 Romania National Olympiad, 3

Tags: function , cyclic function , algebra

Let be three positive real numbers $ a,b,c. $ Show that the function $ f:\mathbb{R}\longrightarrow\mathbb{R} , $ $$ f(x)=\frac{a^x}{b^x+c^x} +\frac{b^x}{a^x+c^x} +\frac{c^x}{a^x+b^x} , $$ is nondecresing on the interval $ \left[ 0,\infty \right) $ and nonincreasing on the interval $ \left( -\infty ,0 \right] . $

2005 MOP Homework, 4

Tags: algebra , function

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $f(x^3)-f(y^3)=(x^2+xy+y^2)(f(x)-f(y))$.

2008 Iran Team Selection Test, 4

Tags: geometry , function , 3d geometry , tetrahedron , vector , inequalities

Let $ P_1,P_2,P_3,P_4$ be points on the unit sphere. Prove that $ \sum_{i\neq j}\frac1{|P_i\minus{}P_j|}$ takes its minimum value if and only if these four points are vertices of a regular pyramid.

2016 IFYM, Sozopol, 1

Tags: inequalities , triangle inequality , function , algebra

Find all functions $f: \mathbb{R}^+\rightarrow \mathbb{R}^+$ with the following property: $a,b,$ and $c$ are lengths of sides of a triangle, if and only if $f(a),f(b),$ and $f(c)$ are lengths of sides of a triangle.

2005 Alexandru Myller, 1

Tags: real analysis , integration , function

Let $f:[a,b]\to\mathbb R$ be a continous function with the property that there exists a constant $\lambda\in\mathbb R$ so that for every $x\in[a,b]$ there exists a $y\in[a,b]-\{x\}$ s.t. $\int_x^yf(x)dx=\lambda$. Prove that the function $f$ has at least two zeros in $(a,b)$. [i]Eugen Paltanea[/i]

2020 Romanian Master of Mathematics, 4

Tags: algebra , function

Let $\mathbb N$ be the set of all positive integers. A subset $A$ of $\mathbb N$ is [i]sum-free[/i] if, whenever $x$ and $y$ are (not necessarily distinct) members of $A$, their sum $x+y$ does not belong to $A$. Determine all surjective functions $f:\mathbb N\to\mathbb N$ such that, for each sum-free subset $A$ of $\mathbb N$, the image $\{f(a):a\in A\}$ is also sum-free. [i]Note: a function $f:\mathbb N\to\mathbb N$ is surjective if, for every positive integer $n$, there exists a positive integer $m$ such that $f(m)=n$.[/i]

2007 Today's Calculation Of Integral, 200

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

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

2021 Kosovo National Mathematical Olympiad, 2

Tags: algebra , function

Find all functions $f:\mathbb R\to\mathbb R$ so that the following relation holds for all $x, y\in\mathbb R$. $$f(f(x)f(y)-1) = xy - 1$$

1947 Putnam, B1

Tags: trigonometry , function

Let $f(x)$ be a function such that $f(1)=1$ and for $x \geq 1$ $$f'(x)= \frac{1}{x^2 +f(x)^{2}}.$$ Prove that $$\lim_{x\to \infty} f(x)$$ exists and is less than $1+ \frac{\pi}{4}.$

2013 Online Math Open Problems, 19

Tags: function , number theory , euler , geometric series , relatively prime

Let $\sigma(n)$ be the number of positive divisors of $n$, and let $\operatorname{rad} n$ be the product of the distinct prime divisors of $n$. By convention, $\operatorname{rad} 1 = 1$. Find the greatest integer not exceeding \[ 100\left(\sum_{n=1}^{\infty}\frac{\sigma(n)\sigma(n \operatorname{rad} n)}{n^2\sigma(\operatorname{rad} n)}\right)^{\frac{1}{3}}. \][i]Proposed by Michael Kural[/i]

1970 AMC 12/AHSME, 16

Tags: function

If $F(n)$ is a function such that $F(1)=F(2)=F(3)=1$, and such that $F(n+1)=\dfrac{F(n)\cdot F(n-1)+1}{F(n-2)}$ for $n\ge 3$, then $F(6)$ is equal to $\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }7\qquad\textbf{(D) }11\qquad \textbf{(E) }26$

2024 OMpD, 4

Tags: real analysis , function

Let $ n $ be a positive integer. Determine the largest possible value of $ k $ with the following property: there exists a bijective function $ \phi: [0, 1] \to [0, 1]^k $ and a constant $ C > 0 $ such that, for all $ x, y \in [0, 1] $, \[ \| \phi(x) - \phi(y) \| \leq C \| x - y \|^k. \] Note: $ \| \cdot \| $ denotes the Euclidean norm, that is, $ \| (a_1, \ldots, a_n) \| = \sqrt{a_1^2 + \cdots + a_n^2} $.

2024 PErA, P5

Tags: functional equation , algebra , function

Find all functions $f\colon \mathbb{R}^+ \to \mathbb{R}^+$ such that \[ f(xf(x)+y^2) = x^2+yf(y) \] for any positive reals $x,y$.

2012 Kyrgyzstan National Olympiad, 4

Tags: functional equation , algebra , function

Find all functions $ f:\mathbb{R}\to\mathbb{R} $ such that $ f(f(x)^2+f(y)) = xf(x)+y $,$ \forall x,y\in R $.

1988 IMO Shortlist, 2

Tags: polynomial , algebra , generating functions , binomial coefficient , function

Let $ n$ be a positive integer. Find the number of odd coefficients of the polynomial \[ u_n(x) \equal{} (x^2 \plus{} x \plus{} 1)^n. \]

1963 Putnam, A5

Tags: trigonometry , function

i) Prove that if a function $f$ is continuous on the closed interval $[0, \pi]$ and $$ \int_{0}^{\pi} f(t) \cos t \; dt= \int_{0}^{\pi} f(t) \sin t \; dt=0,$$ then there exist points $0 < \alpha < \beta < \pi$ such that $f(\alpha) =f(\beta) =0.$ ii) Let $R$ be a bounded, convex, and open region in the Euclidean plane. Prove with the help of i) that the centroid of $R$ bisects at least three different chords of the boundary of $ R.$

2014-2015 SDML (High School), 3

Tags: functional equation , algebra , function

Suppose a non-identically zero function $f$ satisfies $f\left(x\right)f\left(y\right)=f\left(\sqrt{x^2+y^2}\right)$ for all $x$ and $y$. Compute $$f\left(1\right)-f\left(0\right)-f\left(-1\right).$$

2004 IMO Shortlist, 6

Tags: function , algebra , calculus , functional equation

Find all functions $f:\mathbb{R} \to \mathbb{R}$ satisfying the equation \[ f(x^2+y^2+2f(xy)) = (f(x+y))^2. \] for all $x,y \in \mathbb{R}$.

2010 Harvard-MIT Mathematics Tournament, 2

Tags: derivative , function , calculus

Let $f$ be a function such that $f(0)=1$, $f^\prime (0)=2$, and \[f^{\prime\prime}(t)=4f^\prime(t)-3f(t)+1\] for all $t$. Compute the $4$th derivative of $f$, evaluated at $0$.

1950 Miklós Schweitzer, 7

Tags: logarithm , calculus , function , euler , real analysis , integration

Examine the behavior of the expression $ \sum_{\nu\equal{}1}^{n\minus{}1}\frac{\log(n\minus{}\nu)}{\nu}\minus{}\log^2 n$ as $ n\rightarrow \infty$

2014 Dutch IMO TST, 1

Tags: algebra , function

Let $ f:\mathbb{Z}_{>0}\rightarrow\mathbb{R} $ be a function such that for all $n > 1$ there is a prime divisor $p$ of $n$ such that \[ f(n)=f\left(\frac{n}{p}\right)-f(p). \] Furthermore, it is given that $ f(2^{2014})+f(3^{2015})+f(5^{2016})=2013 $. Determine $ f(2014^2)+f(2015^3)+f(2016^5) $.

2013 Putnam, 5

Tags: matrix , algebra , linear algebra , function , induction , probability

Let $X=\{1,2,\dots,n\},$ and let $k\in X.$ Show that there are exactly $k\cdot n^{n-1}$ functions $f:X\to X$ such that for every $x\in X$ there is a $j\ge 0$ such that $f^{(j)}(x)\le k.$ [Here $f^{(j)}$ denotes the $j$th iterate of $f,$ so that $f^{(0)}(x)=x$ and $f^{(j+1)}(x)=f\left(f^{(j)}(x)\right).$]