Olympiad problems

2007 Today's Calculation Of Integral, 216

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

Let $ a_{n}$ is a positive number such that $ \int_{0}^{a_{n}}\frac{e^{x}\minus{}1}{1\plus{}e^{x}}\ dx \equal{}\ln n$. Find $ \lim_{n\to\infty}(a_{n}\minus{}\ln n)$.

2013 Waseda University Entrance Examination, 3

Tags: function , calculus , calculus computations , logarithm

Let $f(x)=\frac 12e^{2x}+2e^x+x$. Answer the following questions. (1) For a real number $t$, set $g(x)=tx-f(x).$ When $x$ moves in the range of all real numbers, find the range of $t$ for which $g(x)$ has maximum value, then for the range of $t$, find the maximum value of $g(x)$ and the value of $x$ which gives the maximum value. (2) Denote by $m(t)$ the maximum value found in $(1)$. Let $a$ be a constant, consider a function of $t$, $h(t)=at-m(t)$. When $t$ moves in the range of $t$ found in $(1)$, find the maximum value of $h(t)$.

2018 Costa Rica - Final Round, F3

Tags: function , algebra

Consider a function $f: R \to R$ that fulfills the following two properties: $f$ is periodic of period $5$ (that is, for all $x\in R$, $f (x + 5) = f (x)$), and by restricting $f$ to the interval $[-2,3]$, $f$ coincides to $x^2$. Determine the value of $f(2018).$

1999 AIME Problems, 9

Tags: function , complex number , number theory , relatively prime

A function $f$ is defined on the complex numbers by $f(z)=(a+bi)z,$ where $a$ and $b$ are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given that $|a+bi|=8$ and that $b^2=m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

2011 Northern Summer Camp Of Mathematics, 2

Tags: function , number theory

Find all functions $f: \mathbb N \cup \{0\} \to \mathbb N\cup \{0\}$ such that $f(1)>0$ and \[f(m^2+3n^2)=(f(m))^2 + 3(f(n))^2 \quad \forall m,n \in \mathbb N\cup \{0\}.\]

STEMS 2023 Math Cat A, 4

Tags: function , computational , algebra

Let $f : \mathbb{N} \to \mathbb{N}$ be a function such that the following conditions hold: $\qquad\ (1) \; f(1) = 1.$ $\qquad\ (2) \; \dfrac{(x + y)}{2} < f(x + y) \le f(x) + f(y) \; \forall \; x, y \in \mathbb{N}.$ $\qquad\ (3) \; f(4n + 1) < 2f(2n + 1) \; \forall \; n \ge 0.$ $\qquad\ (4) \; f(4n + 3) \le 2f(2n + 1) \; \forall \; n \ge 0.$ Find the sum of all possible values of $f(2023)$.

2007 IMC, 3

Tags: function , limit

Let $ C$ be a nonempty closed bounded subset of the real line and $ f: C\to C$ be a nondecreasing continuous function. Show that there exists a point $ p\in C$ such that $ f(p) \equal{} p$. (A set is closed if its complement is a union of open intervals. A function $ g$ is nondecreasing if $ g(x)\le g(y)$ for all $ x\le y$.)

2013 Putnam, 3

Tags: algebra , polynomial , function , integration

Suppose that the real numbers $a_0,a_1,\dots,a_n$ and $x,$ with $0<x<1,$ satisfy \[\frac{a_0}{1-x}+\frac{a_1}{1-x^2}+\cdots+\frac{a_n}{1-x^{n+1}}=0.\] Prove that there exists a real number $y$ with $0<y<1$ such that \[a_0+a_1y+\cdots+a_ny^n=0.\]

1954 Putnam, B5

Tags: function , limit , differentiability

Let $f(x)$ be a real-valued function, defined for $-1<x<1$ for which $f'(0)$ exists. Let $(a_n) , (b_n)$ be two sequences such that $-1 <a_n <0 <b_n <1$ for all $n$ and $\lim_{n \to \infty } a_n = 0 =\lim_{n \to \infty} b_n.$ Prove that $$ \lim_{n \to \infty} \frac{ f(b_n )- f(a_n ) }{b_n -a_n} =f'(0).$$

2013 ELMO Shortlist, 5

Tags: function , calculus , derivative , 3-variable inequality , logarithm , inequalities

Let $a,b,c$ be positive reals satisfying $a+b+c = \sqrt[7]{a} + \sqrt[7]{b} + \sqrt[7]{c}$. Prove that $a^a b^b c^c \ge 1$. [i]Proposed by Evan Chen[/i]

Today's calculation of integrals, 849

Tags: calculus , integration , function , calculus computations

Evaluate $\int_1^{e^2} \frac{(2x^2+2x+1)e^{x}}{\sqrt{x}}\ dx.$

1984 Czech And Slovak Olympiad IIIA, 6

Tags: algebra , functional equation , function

Let f be a function from the set Z of all integers into itself, that satisfies the condition for all $m \in Z$, $$f(f(m)) =-m. \ \ (1)$$ Then: (a) $f$ is a mutually unique mapping, i.e. a simple mapping of the set $Z$ onto the set $Z$ , (b) for all $m \in Z$ holds that $f(-m) = -f(m)$ , (c) $f(m) = 0$ if and only if $m = 0$ . Prove these statements and construct an example of a mapping f that satisfies condition (1).

2004 Harvard-MIT Mathematics Tournament, 7

Tags: function

If $x$, $y$, $k$ are positive reals such that \[3=k^2\left(\dfrac{x^2}{y^2}+\dfrac{y^2}{x^2}\right)+k\left(\dfrac{x}{y}+\dfrac{y}{x}\right),\] find the maximum possible value of $k$.

2014 Contests, 1

Tags: inequalities , algebra , polynomial , function , domain

Let $a$, $b$, $c$, $d$ be real numbers such that $b-d \ge 5$ and all zeros $x_1, x_2, x_3,$ and $x_4$ of the polynomial $P(x)=x^4+ax^3+bx^2+cx+d$ are real. Find the smallest value the product $(x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1)$ can take.

2012 ELMO Shortlist, 6

Tags: number theory , least common multiple , inequalities , induction , function , pigeonhole principle , logarithm

Consider a directed graph $G$ with $n$ vertices, where $1$-cycles and $2$-cycles are permitted. For any set $S$ of vertices, let $N^{+}(S)$ denote the out-neighborhood of $S$ (i.e. set of successors of $S$), and define $(N^{+})^k(S)=N^{+}((N^{+})^{k-1}(S))$ for $k\ge2$. For fixed $n$, let $f(n)$ denote the maximum possible number of distinct sets of vertices in $\{(N^{+})^k(X)\}_{k=1}^{\infty}$, where $X$ is some subset of $V(G)$. Show that there exists $n>2012$ such that $f(n)<1.0001^n$. [i]Linus Hamilton.[/i]

2014 Cezar Ivănescu, 2

Tags: antiderivative , function , real analysis , darboux , derivability , primitivability

[b]a)[/b] Give an example of function $ f:\mathbb{R}\longrightarrow\mathbb{R}_{>0 } $ that admits a primitive $ F:\mathbb{R}\longrightarrow\mathbb{R}_{>0 } $ having the property that $ F^e $ is a primitive of $ f^e. $ [b]b)[/b] Prove that there is no derivable function $ g:\mathbb{R}\longrightarrow\mathbb{R} $ that has a primitive $ G:\mathbb{R}\longrightarrow\mathbb{R} $ such that $ e^G $ is a primitive of $ e^g. $

2023 Belarus Team Selection Test, 1.3

Tags: number theory , prime factorization , function , prime number , prime

Let $Q$ be a set of prime numbers, not necessarily finite. For a positive integer $n$ consider its prime factorization: define $p(n)$ to be the sum of all the exponents and $q(n)$ to be the sum of the exponents corresponding only to primes in $Q$. A positive integer $n$ is called [i]special[/i] if $p(n)+p(n+1)$ and $q(n)+q(n+1)$ are both even integers. Prove that there is a constant $c>0$ independent of the set $Q$ such that for any positive integer $N>100$, the number of special integers in $[1,N]$ is at least $cN$. (For example, if $Q=\{3,7\}$, then $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)

1999 AMC 12/AHSME, 18

Tags: function , trigonometry

How many zeros does $ f(x) \equal{} \cos(\log(x)))$ have on the interval $ 0 < x < 1$? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ \text{infinitely many}$

2010 Contests, 1

Tags: function , functional equation , algebra

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all $x, y\in\mathbb{R}$, we have \[f(x+y)+f(x)f(y)=f(xy)+(y+1)f(x)+(x+1)f(y).\]

2006 ISI B.Math Entrance Exam, 4

Tags: calculus , derivative , function , induction , calculus computations

Let $f:\mathbb{R} \to \mathbb{R}$ be a function that is a function that is differentiable $n+1$ times for some positive integer $n$ . The $i^{th}$ derivative of $f$ is denoted by $f^{(i)}$ . Suppose- $f(1)=f(0)=f^{(1)}(0)=...=f^{(n)}(0)=0$. Prove that $f^{(n+1)}(x)=0$ for some $x \in (0,1)$

2012 Grigore Moisil Intercounty, 4

Tags: function , continuum , real analysis

A real continuous function has the property that its evaluation at any point is nilpotent under composition with itself. Prove that this function is $ 0. $ [i]Vasile Pop[/i]

2007 Today's Calculation Of Integral, 197

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

Let $|a|<\frac{\pi}{2}.$ Evaluate the following definite integral. \[\int_{0}^{\frac{\pi}{2}}\frac{dx}{\{\sin (a+x)+\cos x\}^{2}}\]

1998 Moldova Team Selection Test, 5

Tags: function

Let $A=\{a_1,a_2,\ldots,a_n\}$ be a set with $a_1<a_2\ldots<a_n$ and $B=\{b_1,b_2,\ldots,b_n\}$ be a set with $b_1<b_2\ldots<b_n$. Show that for every bijective function $f:A\rightarrow B$ the following relation takes place $$\max_{1\leq i\leq n} |a_i-f(a_i)| \geq \max_{1\leq i\leq n} |a_i-b_i|.$$

2008 Singapore Team Selection Test, 2

Tags: function , inequalities

Let $ x_1, x_2,\ldots , x_n$ be positive real numbers such that $ x_1x_2\cdots x_n \equal{} 1$. Prove that \[\sum_{i \equal{} 1}^n \frac {1}{n \minus{} 1 \plus{} x_i}\le 1.\]

1965 AMC 12/AHSME, 23

Tags: inequalities , function

If we write $ |x^2 \minus{} 4| < N$ for all $ x$ such that $ |x \minus{} 2| < 0.01$, the smallest value we can use for $ N$ is: $ \textbf{(A)}\ .0301 \qquad \textbf{(B)}\ .0349 \qquad \textbf{(C)}\ .0399 \qquad \textbf{(D)}\ .0401 \qquad \textbf{(E)}\ .0499 \qquad$