Found problems: 4776
2009 Princeton University Math Competition, 6
Find the number of functions $f:\mathbb{Z}\mapsto\mathbb{Z}$ for which $f(h+k)+f(hk)=f(h)f(k)+1$, for all integers $h$ and $k$.
2004 Postal Coaching, 13
Find all functions $f,g : \mathbb{R} \times \mathbb{R} \mapsto \mathbb{R}^{+}$ such that
\[ ( \sum_{j=1}^{n}a_{j}b_{j})^2 \leq (\sum_{j=1}^{n} f({a_{j},b_{j}))(\sum_{j=1}^{n} g({a_{j},b_{j})) \leq (\sum_{j=1}^{n} (a_j)^2 )(\sum_{j=1}^{n} (b_j)^2 ) }}\] for any two sets $a_j$ and $b_j$ of real numbers.
2011 Putnam, A5
Let $F:\mathbb{R}^2\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ be twice continuously differentiable functions with the following properties:
• $F(u,u)=0$ for every $u\in\mathbb{R};$
• for every $x\in\mathbb{R},g(x)>0$ and $x^2g(x)\le 1;$
• for every $(u,v)\in\mathbb{R}^2,$ the vector $\nabla F(u,v)$ is either $\mathbf{0}$ or parallel to the vector $\langle g(u),-g(v)\rangle.$
Prove that there exists a constant $C$ such that for every $n\ge 2$ and any $x_1,\dots,x_{n+1}\in\mathbb{R},$ we have
\[\min_{i\ne j}|F(x_i,x_j)|\le\frac{C}{n}.\]
2024 Romania National Olympiad, 3
Find the functions $f: \mathbb{R} \to \mathbb{R}$ that satisfy $$(f(x)-y)f(x+f(y))=f(x^2)-yf(y),$$ for all real numbers $x$ and $y.$
2001 India National Olympiad, 6
Find all functions $f : \mathbb{R} \to\mathbb{R}$ such that $f(x +y) = f(x) f(y) f(xy)$ for all $x, y \in \mathbb{R}.$
1989 AIME Problems, 15
Point $P$ is inside $\triangle ABC$. Line segments $APD$, $BPE$, and $CPF$ are drawn with $D$ on $BC$, $E$ on $AC$, and $F$ on $AB$ (see the figure at right). Given that $AP=6$, $BP=9$, $PD=6$, $PE=3$, and $CF=20$, find the area of $\triangle ABC$.
[asy]
size(200);
pair A=origin, B=(7,0), C=(3.2,15), D=midpoint(B--C), F=(3,0), P=intersectionpoint(C--F, A--D), ex=B+40*dir(B--P), E=intersectionpoint(B--ex, A--C);
draw(A--B--C--A--D^^C--F^^B--E);
pair point=P;
label("$A$", A, dir(point--A));
label("$B$", B, dir(point--B));
label("$C$", C, dir(point--C));
label("$D$", D, dir(point--D));
label("$E$", E, dir(point--E));
label("$F$", F, dir(point--F));
label("$P$", P, dir(0));[/asy]
2012 Kazakhstan National Olympiad, 3
There are $n$ balls numbered from $1$ to $n$, and $2n-1$ boxes numbered from $1$ to $2n-1$. For each $i$, ball number $i$ can only be put in the boxes with numbers from $1$ to $2i-1$. Let $k$ be an integer from $1$ to $n$. In how many ways we can choose $k$ balls, $k$ boxes and put these balls in the selected boxes so that each box has exactly one ball?
2011 IMO Shortlist, 4
Determine all pairs $(f,g)$ of functions from the set of positive integers to itself that satisfy \[f^{g(n)+1}(n) + g^{f(n)}(n) = f(n+1) - g(n+1) + 1\] for every positive integer $n$. Here, $f^k(n)$ means $\underbrace{f(f(\ldots f)}_{k}(n) \ldots ))$.
[i]Proposed by Bojan Bašić, Serbia[/i]
2015 Indonesia MO Shortlist, A4
Determine all functions $f: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ such that
\[ f(x,y) + f(y,z) + f(z,x) = \max \{ x,y,z \} - \min \{ x,y,z \} \] for every $x,y,z \in \mathbb{R}$
and there exists some real $a$ such that $f(x,a) = f(a,x) $ for every $x \in \mathbb{R}$.
2001 AMC 12/AHSME, 13
The parabola with equation $ y \equal{} ax^2 \plus{} bx \plus{} c$ and vertex $ (h,k)$ is reflected about the line $ y \equal{} k$. This results in the parabola with equation $ y \equal{} dx^2 \plus{} ex \plus{} f$. Which of the following equals $ a \plus{} b \plus{} c \plus{} d \plus{} e \plus{} f$?
$ \textbf{(A)} \ 2b \qquad \textbf{(B)} \ 2c \qquad \textbf{(C)} \ 2a \plus{} 2b \qquad \textbf{(D)} \ 2h \qquad \textbf{(E)} \ 2k$
2017 Taiwan TST Round 2, 4
Find all integer $c\in\{0,1,...,2016\}$ such that the number of $f:\mathbb{Z}\rightarrow\{0,1,...,2016\}$ which satisfy the following condition is minimal:\\
(1) $f$ has periodic $2017$\\
(2) $f(f(x)+f(y)+1)-f(f(x)+f(y))\equiv c\pmod{2017}$\\
Proposed by William Chao
1993 Mexico National Olympiad, 4
$f(n,k)$ is defined by
(1) $f(n,0) = f(n,n) = 1$ and
(2) $f(n,k) = f(n-1,k-1) + f(n-1,k)$ for $0 < k < n$.
How many times do we need to use (2) to find $f(3991,1993)$?
2020 USAMTS Problems, 5:
Let $n \geq 3$ be an integer. Let $f$ be a function from the set of all integers to itself with the following property: If the integers $a_1,a_2,\ldots,a_n$ form an arithmetic progression, then the numbers
$$f(a_1),f(a_2),\ldots,f(a_n)$$
form an arithmetic progression (possibly constant) in some order. Find all values for $n$ such that the only functions $f$ with this property are the functions of the form $f(x)=cx+d$, where $c$ and $d$ are integers.
2007 Putnam, 3
Let $ k$ be a positive integer. Suppose that the integers $ 1,2,3,\dots,3k \plus{} 1$ are written down in random order. What is the probability that at no time during this process, the sum of the integers that have been written up to that time is a positive integer divisible by $ 3$ ? Your answer should be in closed form, but may include factorials.
2009 Czech-Polish-Slovak Match, 1
Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f : \mathbb{R}^+\to\mathbb{R}^+$ that satisfy \[ \Big(1+yf(x)\Big)\Big(1-yf(x+y)\Big)=1\] for all $x,y\in\mathbb{R}^+$.
1979 IMO Longlists, 45
For any positive integer $n$, we denote by $F(n)$ the number of ways in which $n$ can be expressed as the sum of three different positive integers, without regard to order. Thus, since $10 = 7+2+1 = 6+3+1 = 5+4+1 = 5+3+2$, we have $F(10) = 4$. Show that $F(n)$ is even if $n \equiv 2$ or $4 \pmod 6$, but odd if $n$ is divisible by $6$.
2003 Purple Comet Problems, 13
Let $P(x)$ be a polynomial such that, when divided by $x - 2$, the remainder is $3$ and, when divided by $x - 3$, the remainder is $2$. If, when divided by $(x - 2)(x - 3)$, the remainder is $ax + b$, find $a^2 + b^2$.
2022 Taiwan TST Round 3, 4
Let $\mathcal{X}$ be the collection of all non-empty subsets (not necessarily finite) of the positive integer set $\mathbb{N}$. Determine all functions $f: \mathcal{X} \to \mathbb{R}^+$ satisfying the following properties:
(i) For all $S$, $T \in \mathcal{X}$ with $S\subseteq T$, there holds $f(T) \le f(S)$.
(ii) For all $S$, $T \in \mathcal{X}$, there hold
\[f(S) + f(T) \le f(S + T),\quad f(S)f(T) = f(S\cdot T), \]
where $S + T = \{s + t\mid s\in S, t\in T\}$ and $S \cdot T = \{s\cdot t\mid s\in S, t\in T\}$.
[i]Proposed by Li4, Untro368, and Ming Hsiao.[/i]
2023 Brazil Team Selection Test, 3
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 $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)
1993 USAMO, 3
Consider functions $\, f: [0,1] \rightarrow \mathbb{R} \,$ which satisfy
(i) $f(x) \geq 0 \,$ for all $\, x \,$ in $\, [0,1],$
(ii) $f(1) = 1,$
(iii) $f(x) + f(y) \leq f(x+y)\,$ whenever $\, x, \, y, \,$ and $\, x + y \,$ are all in $\, [0,1]$.
Find, with proof, the smallest constant $\, c \,$ such that
\[ f(x) \leq cx \]
for every function $\, f \,$ satisfying (i)-(iii) and every $\, x \,$ in $\, [0,1]$.
1985 AIME Problems, 10
How many of the first 1000 positive integers can be expressed in the form
\[ \lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor, \]
where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$?
2007 France Team Selection Test, 2
Find all functions $f: \mathbb{Z}\rightarrow\mathbb{Z}$ such that for all $x,y \in \mathbb{Z}$:
\[f(x-y+f(y))=f(x)+f(y).\]
2015 AMC 12/AHSME, 24
Rational numbers $a$ and $b$ are chosen at random among all rational numbers in the interval $[0,2)$ that can be written as fractions $\tfrac nd$ where $n$ and $d$ are integers with $1\leq d\leq 5$. What is the probability that \[(\cos(a\pi)+i\sin(b\pi))^4\] is a real number?
$\textbf{(A) }\dfrac3{50}\qquad\textbf{(B) }\dfrac4{25}\qquad\textbf{(C) }\dfrac{41}{200}\qquad\textbf{(D) }\dfrac6{25}\qquad\textbf{(E) }\dfrac{13}{50}$
2003 India National Olympiad, 3
Show that $8x^4 - 16x^3 + 16x^2 - 8x + k = 0$ has at least one real root for all real $k$. Find the sum of the non-real roots.
2014 USAMTS Problems, 3b:
A group of people is lined up in [i]almost-order[/i] if, whenever person $A$ is to the left of person $B$ in the line, $A$ is not more than $8$ centimeters taller than $B$. For example, five people with heights $160, 165, 170, 175$, and $180$ centimeters could line up in [i]almost-order[/i] with heights (from left-to-right) of $160, 170, 165, 180, 175$ centimeters.
(b) How many different ways are there to line up $20$ people in [i]almost-order[/i] if their heights are $120, 125, 130,$ $135,$ $140,$ $145,$ $150,$ $155,$ $160,$ $164, 165, 170, 175, 180, 185, 190, 195, 200, 205$, and $210$ centimeters? (Note that there is someone of height $164$ centimeters.)