Found problems: 85335
2005 Gheorghe Vranceanu, 4
Let be a sequence of real numbers $ \left( x_n \right)_{n\geqslant 0} $ with $ x_0\neq 0,1 $ and defined as $ x_{n+1}=x_n+x_n^{-1/x_0} . $
[b]a)[/b] Show that the sequence $ \left( x_n\cdot n^{-\frac{x_0}{1+x_0}} \right)_{n\geqslant 0} $ is convergent.
[b]b)[/b] Prove that $ \inf_{x_0\neq 0,1} \lim_{n\to\infty } x_n\cdot n^{-\frac{x_0}{1+x_0}} =1. $
IV Soros Olympiad 1997 - 98 (Russia), 9.1
Solve the equation $$2(x-6)=\dfrac{x^2}{(1+\sqrt{x+1})^2}$$
2019 Simurgh, 1
Show that there exists a $10 \times 10$ table of distinct natural numbers such that if $R_i$ is equal to the multiplication of numbers of row $i$ and $S_i$ is equal to multiplication of numbers of column $i$, then numbers $R_1$, $R_2$, ... , $R_{10}$ make a nontrivial arithmetic sequence and numbers $S_1$, $S_2$, ... , $S_{10}$ also make a nontrivial arithmetic sequence.
(A nontrivial arithmetic sequence is an arithmetic sequence with common difference between terms not equal to $0$).
2018 Mexico National Olympiad, 5
Let $n\geq 5$ an integer and consider a regular $n$-gon. Initially, Nacho is situated in one of the vertices of the $n$-gon, in which he puts a flag. He will start moving clockwise. First, he moves one position and puts another flag, then, two positions and puts another flag, etcetera, until he finally moves $n-1$ positions and puts a flag, in such a way that he puts $n$ flags in total. ¿For which values of $n$, Nacho will have put a flag in each of the $n$ vertices?
2003 Putnam, 1
Let $n$ be a fixed positive integer. How many ways are there to write $n$ as a sum of positive integers,
\[n = a_1 + a_2 + \cdots a_k\]
with $k$ an arbitrary positive integer and $a_1 \le a_2 \le \cdots \le a_k \le a_1 + 1$? For example, with $n = 4$, there are four ways: $4$, $2 + 2$, $1 + 1 + 2$, $1 + 1 + 1 + 1$.
2019 Jozsef Wildt International Math Competition, W. 3
Compute $$\int \limits_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\frac{\cos x+1-x^2}{(1+x\sin x)\sqrt{1-x^2}}dx$$
1989 Greece Junior Math Olympiad, 1
Let $A$ be the sum of three consecutive integers and $B$ be the sum of the exact three consecutive integers. Is it possible to have $AB=33333$ ?
2014 BMT Spring, 19
Evaluate the integral $\int_0^{\pi/2} \sqrt{\tan \theta} d\theta$.
2006 Princeton University Math Competition, 3
Find the minimum value of $x^2+2x+ \frac{24}{x}$ for $x > 0$.
2005 ITAMO, 3
Two circles $\gamma_1, \gamma_2$ in a plane, with centers $A$ and $B$ respectively, intersect at $C$ and $D$. Suppose that the circumcircle of $ABC$ intersects $\gamma_1$ in $E$ and $\gamma_2$ in $F$, where the arc $EF$ not containing $C$ lies outside $\gamma_1$ and $\gamma_2$. Prove that this arc $EF$ is bisected by the line $CD$.
2015 Balkan MO Shortlist, N1
Let $d$ be an even positive integer.
John writes the numbers $1^2 ,3^2 ,\ldots,(2n-1)^2 $ on the blackboard and then chooses three of them, let them be ${a_1}, {a_2}, {a_3}$, erases them and writes the number $1+ \displaystyle\sum_{1\le i<j\leq 3} |{a_i} -{a_j}|$
He continues until two numbers remain written on on the blackboard.
Prove that the sum of squares of those two numbers is different than the numbers $1^2 ,3^2 ,\ldots,(2n-1)^2$.
(Albania)
1998 Iran MO (3rd Round), 1
Find all functions $f: \mathbb N \to \mathbb N$ such that for all positive integers $m,n$,
[b](i)[/b] $mf(f(m))=\left( f(m) \right)^2$,
[b](ii)[/b] If $\gcd(m,n)=d$, then $f(mn) \cdot f(d)=d \cdot f(m) \cdot f(n)$,
[b](iii)[/b] $f(m)=m$ if and only if $m=1$.
2015 China Second Round Olympiad, 2
In isoceles $\triangle ABC$, $AB=AC$, $I$ is its incenter, $D$ is a point inside $\triangle ABC$ such that $I,B,C,D$ are concyclic. The line through $C$ parallel to $BD$ meets $AD$ at $E$. Prove that $CD^2=BD\cdot CE$.
2013 Cuba MO, 5
Let the real numbers be $a, b, c, d$ with $a \ge b$ and $c \ge d$. Prove that the equation $$(x + a) (x + d) + (x + b) (x + c) = 0$$ has real roots.
1996 Portugal MO, 1
Consider a square on the hypotenuse of a right triangle $[ABC]$ (right at $B$). Prove that the line segment that joins vertex $B$ with the center of the square makes $45^o$ angles with legs of the triangle.
2007 Vietnam National Olympiad, 3
Let B,C be fixed points and A be roving point. Let H, G be orthecentre and centroid of triagle ABC. Known midpoint of HG lies on BC, find locus of A
2020 Taiwan TST Round 2, 4
Alice and Bob are stuck in quarantine, so they decide to play a game. Bob will write down a polynomial $f(x)$ with the following properties:
(a) for any integer $n$, $f(n)$ is an integer;
(b) the degree of $f(x)$ is less than $187$.
Alice knows that $f(x)$ satisfies (a) and (b), but she does not know $f(x)$. In every turn, Alice picks a number $k$ from the set $\{1,2,\ldots,187\}$, and Bob will tell Alice the value of $f(k)$. Find the smallest positive integer $N$ so that Alice always knows for sure the parity of $f(0)$ within $N$ turns.
[i]Proposed by YaWNeeT[/i]
2013 Puerto Rico Team Selection Test, 2
How many 3-digit numbers have the property that the sum of their digits is even?
1999 AMC 12/AHSME, 12
What is the maximum number of points of intersection of the graphs of two different fourth degree polynomial functions $ y \equal{} p(x)$ and $ y \equal{} q(x)$, each with leading coefficient $ 1$?
$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 8$
2006 Bulgaria Team Selection Test, 1
[b]Problem 4.[/b] Let $k$ be the circumcircle of $\triangle ABC$, and $D$ the point on the arc $\overarc{AB},$ which do not pass through $C$. $I_A$ and $I_B$ are the centers of incircles of $\triangle ADC$ and $\triangle BDC$, respectively. Proove that the circumcircle of $\triangle I_AI_BC$ touches $k$ iff \[ \frac{AD}{BD}=\frac{AC+CD}{BC+CD}. \]
[i] Stoyan Atanasov[/i]
1948 Moscow Mathematical Olympiad, 146
Consider two triangular pyramids $ABCD$ and $A'BCD$, with a common base $BCD$, and such that $A'$ is inside $ABCD$. Prove that the sum of planar angles at vertex $A'$ of pyramid $A'BCD$ is greater than the sum of planar angles at vertex $A$ of pyramid $ABCD$.
2008 AMC 10, 9
A quadratic equation $ ax^2\minus{}2ax\plus{}b\equal{}0$ has two real solutions. What is the average of the solutions?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ \frac{b}{a} \qquad
\textbf{(D)}\ \frac{2b}{a} \qquad
\textbf{(E)}\ \sqrt{2b\minus{}a}$
2024 HMNT, 3
Points $K,A,L,C,I,T,E$ are such that triangles $CAT$ and $ELK$ are equilateral, share a center $I,$ and points $E,L,K$ lie on sides $CA, AT, TC$ respectively. If the area of triangle $CAT$ is double the area of triangle $ELK$ and $CI = 2,$ compute the minimum possible value of $CK.$
2005 iTest, 27
Find the sum of all non-zero digits that can repeat at the end of a perfect square. (For example, if $811$ were a perfect square, $1$ would be one of these non-zero digits.)
2017 Macedonia National Olympiad, Problem 1
Find all functions $f:\mathbb{N} \to \mathbb{N}$ such that for each natural integer $n>1$ and for all $x,y \in \mathbb{N}$ the following holds:
$$f(x+y) = f(x) + f(y) + \sum_{k=1}^{n-1} \binom{n}{k}x^{n-k}y^k$$