Found problems: 4776
2013 Federal Competition For Advanced Students, Part 2, 4
For a positive integer $n$, let $a_1, a_2, \ldots a_n$ be nonnegative real numbers such that for all real numbers $x_1>x_2>\ldots>x_n>0$ with $x_1+x_2+\ldots+x_n<1$, the inequality $\sum_{k=1}^na_kx_k^3<1$ holds. Show that \[na_1+(n-1)a_2+\ldots+(n-j+1)a_j+\ldots+a_n\leqslant\frac{n^2(n+1)^2}{4}.\]
2000 Harvard-MIT Mathematics Tournament, 44
A function $f:\mathbb{Z}\implies\mathbb{Z}$ satisfies
$f(x+4)-f(x)=8x+20$
$f(x^2-1)=(f(x)-x)^2+x^2-2$
Find $f(0)$ and $f(1)$.
2012 Online Math Open Problems, 22
Let $c_1,c_2,\ldots,c_{6030}$ be 6030 real numbers. Suppose that for any 6030 real numbers $a_1,a_2,\ldots,a_{6030}$, there exist 6030 real numbers $\{b_1,b_2,\ldots,b_{6030}\}$ such that \[a_n = \sum_{k=1}^{n} b_{\gcd(k,n)}\] and \[b_n = \sum_{d\mid n} c_d a_{n/d}\] for $n=1,2,\ldots,6030$. Find $c_{6030}$.
[i]Victor Wang.[/i]
2015 AMC 10, 7
Consider the operation "minus the reciprocal of," defined by $a\diamond b=a-\frac{1}{b}$. What is $((1\diamond2)\diamond3)-(1\diamond(2\diamond3))$?
$\textbf{(A) } -\dfrac{7}{30}
\qquad\textbf{(B) } -\dfrac{1}{6}
\qquad\textbf{(C) } 0
\qquad\textbf{(D) } \dfrac{1}{6}
\qquad\textbf{(E) } \dfrac{7}{30}
$
1992 National High School Mathematics League, 6
$f(x)$ is a function defined on $\mathbb{R}$, satisfying: $f(10+x)=f(10-x),f(20-x)=-f(20+x)$. Then, $f(x)$ is
$\text{(A)}$even function, but not periodic function
$\text{(B)}$even function, and periodic function
$\text{(C)}$odd function, but not periodic function
$\text{(D)}$odd function, and periodic function
2007 Gheorghe Vranceanu, 1
Let $ M $ denote the set of the primitives of a function $ f:\mathbb{R}\longrightarrow\mathbb{R} . $
[b]ii)[/b] Show that $ M $ along with the operation $ *:M^2\longrightarrow M $ defined as $ F*G=F+G(2007) $ form a commutative group.
[b]iii)[/b] Show that $ M $ is isomorphic with the additive group of real numbers.
1981 National High School Mathematics League, 3
Let $\alpha$ be a real number and $\alpha\neq\frac{k\pi}{2} , k\in\mathbb{Z}$,
$$T=\frac{\sin\alpha+\tan\alpha}{\cos\alpha+\cot\alpha}$$.
$\text{(A)}$$T$ is negative.
$\text{(B)}$$T$ is nonnegative.
$\text{(C)}$$T$ is positive.
$\text{(D)}$$T$ can be either positive or negative.
2006 Moldova National Olympiad, 12.2
Let $a, b, n \in \mathbb{N}$, with $a, b \geq 2.$ Also, let $I_{1}(n)=\int_{0}^{1} \left \lfloor{a^n x} \right \rfloor dx $ and $I_{2} (n) = \int_{0}^{1} \left \lfloor{b^n x} \right \rfloor dx.$ Find $\lim_{n \to \infty} \dfrac{I_1(n)}{I_{2}(n)}.$
2018 Romania National Olympiad, 3
Let $f:[a,b] \to \mathbb{R}$ be an integrable function and $(a_n) \subset \mathbb{R}$ such that $a_n \to 0.$
$\textbf{a) }$ If $A= \{m \cdot a_n \mid m,n \in \mathbb{N}^* \},$ prove that every open interval of strictly positive real numbers contains elements from $A.$
$\textbf{b) }$ If, for any $n \in \mathbb{N}^*$ and for any $x,y \in [a,b]$ with $|x-y|=a_n,$ the inequality $\left| \int_x^yf(t)dt \right| \leq |x-y|$ is true, prove that $$\left| \int_x^y f(t)dt \right| \leq |x-y|, \: \forall x,y \in [a,b]$$
[i]Nicolae Bourbacut[/i]
2006 South africa National Olympiad, 6
Consider the function $f$ defined by
\[f(n)=\frac{1}{n}\left (\left \lfloor\frac{n}{1}\right \rfloor+\left \lfloor\frac{n}{2}\right \rfloor+\cdots+\left \lfloor\frac{n}{n}\right \rfloor \right )\]
for all positive integers $n$. (Here $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.) Prove that
(a) $f(n+1)>f(n)$ for infinitely many $n$.
(b) $f(n+1)<f(n)$ for infinitely many $n$.
2023 USA IMOTST, 1
Let $\lfloor \bullet \rfloor$ denote the floor function. For nonnegative integers $a$ and $b$, their [i]bitwise xor[/i], denoted $a \oplus b$, is the unique nonnegative integer such that $$ \left \lfloor \frac{a}{2^k} \right \rfloor+ \left\lfloor\frac{b}{2^k} \right\rfloor - \left\lfloor \frac{a\oplus b}{2^k}\right\rfloor$$ is even for every $k \ge 0$. Find all positive integers $a$ such that for any integers $x>y\ge 0$, we have \[ x\oplus ax \neq y \oplus ay. \]
[i]Carl Schildkraut[/i]
2016 Puerto Rico Team Selection Test, 6
$N$ denotes the set of all natural numbers. Define a function $T: N \to N$ such that $T (2k) = k$ and $T (2k + 1) = 2k + 2$. We write $T^2 (n) = T (T (n))$ and in general $T^k (n) = T^{k-1} (T (n))$ for all $k> 1$.
(a) Prove that for every $n \in N$, there exists $k$ such that $T^k (n) = 1$.
(b) For $k \in N$, $c_k$ denotes the number of elements in the set $\{n: T^k (n) = 1\}$.
Prove that $c_{k + 2} = c_{k + 1} + c_k$, for $1 \le k$.
2007 IberoAmerican, 1
Given an integer $ m$, define the sequence $ \left\{a_{n}\right\}$ as follows:
\[ a_{1}\equal{}\frac{m}{2},\ a_{n\plus{}1}\equal{}a_{n}\left\lceil a_{n}\right\rceil,\textnormal{ if }n\geq 1\]
Find all values of $ m$ for which $ a_{2007}$ is the first integer appearing in the sequence.
Note: For a real number $ x$, $ \left\lceil x\right\rceil$ is defined as the smallest integer greater or equal to $ x$. For example, $ \left\lceil\pi\right\rceil\equal{}4$, $ \left\lceil 2007\right\rceil\equal{}2007$.
2008 China Team Selection Test, 5
For two given positive integers $ m,n > 1$, let $ a_{ij} (i = 1,2,\cdots,n, \; j = 1,2,\cdots,m)$ be nonnegative real numbers, not all zero, find the maximum and the minimum values of $ f$, where
\[ f = \frac {n\sum_{i = 1}^{n}(\sum_{j = 1}^{m}a_{ij})^2 + m\sum_{j = 1}^{m}(\sum_{i= 1}^{n}a_{ij})^2}{(\sum_{i = 1}^{n}\sum_{j = 1}^{m}a_{ij})^2 + mn\sum_{i = 1}^{n}\sum_{j=1}^{m}a_{ij}^2}. \]
2017 Bosnia and Herzegovina Team Selection Test, Problem 2
Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
1994 IMO, 5
Let $ S$ be the set of all real numbers strictly greater than −1. Find all functions $ f: S \to S$ satisfying the two conditions:
(a) $ f(x \plus{} f(y) \plus{} xf(y)) \equal{} y \plus{} f(x) \plus{} yf(x)$ for all $ x, y$ in $ S$;
(b) $ \frac {f(x)}{x}$ is strictly increasing on each of the two intervals $ \minus{} 1 < x < 0$ and $ 0 < x$.
1986 IMO Longlists, 5
Let $ABC$ and $DEF$ be acute-angled triangles. Write $d = EF, e = FD, f = DE.$ Show that there exists a point $P$ in the interior of $ABC$ for which the value of the expression $X=d \cdot AP +e \cdot BP +f \cdot CP$ attains a minimum.
1967 Putnam, A3
Consider polynomial functions $ax^2 -bx +c$ with integer coefficients which have two distinct zeros in the open interval $(0,1).$ Exhibit with proof the least positive integer value of $a$ for which such a polynomial exists.
1993 USAMO, 4
Let $\, a,b \,$ be odd positive integers. Define the sequence $\, (f_n ) \,$ by putting $\, f_1 = a,$ $f_2 = b, \,$ and by letting $\, f_n \,$ for $\, n \geq 3 \,$ be the greatest odd divisor of $\, f_{n-1} + f_{n-2}$. Show that $\, f_n \,$ is constant for $\, n \,$ sufficiently large and determine the eventual value as a function of $\, a \,$ and $\, b$.
2014 AMC 12/AHSME, 24
Let $f_0(x)=x+|x-100|-|x+100|$, and for $n\geq 1$, let $f_n(x)=|f_{n-1}(x)|-1$. For how many values of $x$ is $f_{100}(x)=0$?
$\textbf{(A) }299\qquad
\textbf{(B) }300\qquad
\textbf{(C) }301\qquad
\textbf{(D) }302\qquad
\textbf{(E) }303\qquad$
2021 China Second Round A1, 2
Find a necessary and sufficient condition of $a,b,n\in\mathbb{N^*}$ such that for $S=\{a+bt\mid t=0,1,2,\cdots,n-1\}$, there exists a one-to-one mapping $f: S\to S$ such that for all $x\in S$, $\gcd(x,f(x))=1$.
2020 Iran MO (3rd Round), 3
Find all functions $f$ from positive integers to themselves, such that the followings hold.
$1)$.for each positive integer $n$ we have $f(n)<f(n+1)<f(n)+2020$.
$2)$.for each positive integer $n$ we have $S(f(n))=f(S(n))$ where $S(n)$ is the sum of digits of $n$ in base $10$ representation.
2002 All-Russian Olympiad, 4
There are 2002 towns in a kingdom. Some of the towns are connected by roads in such a manner that, if all roads from one city closed, one can still travel between any two cities. Every year, the kingdom chooses a non-self-intersecting cycle of roads, founds a new town, connects it by roads with each city from the chosen cycle, and closes all the roads from the original cycle. After several years, no non-self-intersecting cycles remained. Prove that at that moment there are at least 2002 towns, exactly one road going out from each of them.
2010 Federal Competition For Advanced Students, Part 1, 2
For a positive integer $n$, we define the function $f_n(x)=\sum_{k=1}^n |x-k|$ for all real numbers $x$. For any two-digit number $n$ (in decimal representation), determine the set of solutions $\mathbb{L}_n$ of the inequality $f_n(x)<41$.
[i](41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 2)[/i]
1984 National High School Mathematics League, 8
Lengths of five edges of a tetrahedron are $1$, while the last one is $x$. Its volume is $F(x)$. On its domain of definition, we have
$\text{(A)}$ $F(x)$ is an increasing function, it has no maximum value.
$\text{(B)}$ $F(x)$ is an increasing function, it has maximum value.
$\text{(C)}$ $F(x)$ is not an increasing function, it has no maximum value.
$\text{(D)}$ $F(x)$ is an increasing function, it has maximum value.