Found problems: 85335
1990 Tournament Of Towns, (267) 1
Given $$a=\dfrac{1}{2+\dfrac{1}{3+\dfrac{1}{...+\dfrac{...}{99}}}}, \,\,and\,\,\,
b=\dfrac{1}{2+\dfrac{1}{3+\dfrac{1}{...+\dfrac{...}{99+\dfrac{1}{100}}}}}$$
Prove that $$|a-b| <\frac{1}{99! 100!}$$
(G Galperin, Moscow)
2013 Stanford Mathematics Tournament, 2
How many alphabetic sequences (that is, sequences containing only letters from $a\cdots z$) of length $2013$ have letters in alphabetic order?
1950 Miklós Schweitzer, 6
Prove the following identity for determinants:
$ |c_{ik} \plus{} a_i \plus{} b_k \plus{} 1|_{i,k \equal{} 1,...,n} \plus{} |c_{ik}|_{i,k \equal{} 1,...,n} \equal{} |c_{ik} \plus{} a_i \plus{} b_k|_{i,k \equal{} 1,...,n} \plus{} |c_{ik} \plus{} 1|_{i,k \equal{} 1,...,n}$
2001 Saint Petersburg Mathematical Olympiad, 11.7
Rectangles $1\times20$, $1\times 19$, ..., $1\times 1$ were cut out of $20\times20$ table. Prove that at least 85 dominoes(1×2 rectangle) can be removed from the remainder.
Proposed by S. Berlov
1985 AMC 12/AHSME, 24
A non-zero digit is chosen in such a way that the probability of choosing digit $ d$ is $ \log_{10}(d\plus{}1) \minus{} \log_{10} d$. The probability that the digit $ 2$ is chosen is exactly $ \frac12$ the probability that the digit chosen is in the set
$ \textbf{(A)}\ \{2,3\} \qquad \textbf{(B)}\ \{3,4\} \qquad \textbf{(C)}\ \{4,5,6,7,8\} \qquad \textbf{(D)}\ \{5,6,7,8,9\} \qquad \textbf{(E)}\ \{4,5,6,7,8,9\}$
2015 Online Math Open Problems, 14
Let $ABCD$ be a square with side length $2015$. A disk with unit radius is packed neatly inside corner $A$ (i.e. tangent to both $\overline{AB}$ and $\overline{AD}$). Alice kicks the disk, which bounces off $\overline{CD}$, $\overline{BC}$, $\overline{AB}$, $\overline{DA}$, $\overline{DC}$ in that order, before landing neatly into corner $B$. What is the total distance the center of the disk travelled?
[i]Proposed by Evan Chen[/i]
2000 Czech And Slovak Olympiad IIIA, 6
Find all four-digit numbers $\overline{abcd}$ (in decimal system) such that $\overline{abcd}= (\overline{ac}+1).(\overline{bd} +1)$
MOAA Team Rounds, 2023.5
Angeline starts with a 6-digit number and she moves the last digit to the front. For example, if she originally had $100823$ she ends up with $310082$. Given that her new number is $4$ times her original number, find the smallest possible value of her original number.
[i]Proposed by Angeline Zhao[/i]
1998 Vietnam National Olympiad, 1
Let $a\geq 1$ be a real number. Put $x_{1}=a,x_{n+1}=1+\ln{(\frac{x_{n}^{2}}{1+\ln{x_{n}}})}(n=1,2,...)$. Prove that the sequence $\{x_{n}\}$ converges and find its limit.
2015 Puerto Rico Team Selection Test, 5
Each number of the set $\{1,2, 3,4,5,6, 7,8\}$ is colored red or blue, following the following rules:
(a) Number $4$ is colored red, and there is at least one blue number,
(b) if two numbers $x,y$ have different colors and $x + y \le 8$, so the number $x + y$ is colored blue,
(c) if two numbers $x,y$ have different colors and $x \cdot y \le 8$, then the number $x \cdot y$ is colored red.
Find all the possible ways to color this set.
2003 JHMMC 8, 15
Evaluate $\frac{100-99+98-97\cdots +4-3+2-1}{1-2+3-4\cdots +97-98+99-100}$.
2011 Switzerland - Final Round, 1
At a party, there are $2011$ people with a glass of fruit juice each sitting around a circular table. Once a second, they clink glasses obeying the following two rules:
(a) They do not clink glasses crosswise.
(b) At each point of time, everyone can clink glasses with at most one other person.
How many seconds pass at least until everyone clinked glasses with everybody else?
[i](Swiss Mathematical Olympiad 2011, Final round, problem 1)[/i]
1935 Moscow Mathematical Olympiad, 019
a) How many distinct ways are there are there of painting the faces of a cube six different colors?
(Colorations are considered distinct if they do not coincide when the cube is rotated.)
b)* How many distinct ways are there are there of painting the faces of a dodecahedron $12$ different colors?
(Colorations are considered distinct if they do not coincide when the cube is rotated.)
2007 Tournament Of Towns, 7
Nancy shuffles a deck of $52$ cards and spreads the cards out in a circle face up, leaving one spot empty. Andy, who is in another room and does not see the cards, names a card. If this card is adjacent to the empty spot, Nancy moves the card to the empty spot, without telling Andy; otherwise nothing happens. Then Andy names another card and so on, as many times as he likes, until he says "stop."
[list][b](a)[/b] Can Andy guarantee that after he says "stop," no card is in its initial spot?
[b](b)[/b] Can Andy guarantee that after he says "stop," the Queen of Spades is not adjacent to
the empty spot?[/list]
2023 Middle European Mathematical Olympiad, 2
If $a, b, c, d>0$ and $abcd=1$, show that $$\frac{ab+1}{a+1}+\frac{bc+1}{b+1}+\frac{cd+1}{c+1}+\frac{da+1}{d+1} \geq 4. $$ When does equality hold?
2016 Regional Olympiad of Mexico Northeast, 1
Determine if there is any triple of nonnegative integers, not necessarily different, $(a, b, c)$ such that:
$$a^3 + b^3 + c^3 = 2016$$
2014 AMC 8, 22
A $2$-digit number is such that the product of the digits plus the sum of the digits is equal to the number. What is the units digit of the number?
$\textbf{(A) }1\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }7\qquad \textbf{(E) }9$
1998 Tournament Of Towns, 5
A "labyrinth" is an $8 \times 8$ chessboard with walls between some neighboring squares. If a rook can traverse the entire board without jumping over the walls, the labyrinth is "good" ; otherwise it is "bad" . Are there more good labyrinths or bad labyrinths?
(A Shapovalov)
1997 Romania National Olympiad, 1
Let $n_1 = \overline{abcabc}$ and $n_2= \overline{d00d}$ be numbers represented in the decimal system, with $a\ne 0$ and $d \ne 0$.
a) Prove that $\sqrt{n_1}$ cannot be an integer.
b) Find all positive integers $n_1$ and $n_2$ such that $\sqrt{n_1+n_2}$ is an integer number.
c) From all the pairs $(n_1,n_2)$ such that $\sqrt{n_1 n_2}$ is an integer find those for which $\sqrt{n_1 n_2}$ has the greatest possible value
2013 ELMO Shortlist, 2
Let $ABC$ be a scalene triangle with circumcircle $\Gamma$, and let $D$,$E$,$F$ be the points where its incircle meets $BC$, $AC$, $AB$ respectively. Let the circumcircles of $\triangle AEF$, $\triangle BFD$, and $\triangle CDE$ meet $\Gamma$ a second time at $X,Y,Z$ respectively. Prove that the perpendiculars from $A,B,C$ to $AX,BY,CZ$ respectively are concurrent.
[i]Proposed by Michael Kural[/i]
1997 Iran MO (3rd Round), 2
Show that for any arbitrary triangle $ABC$, we have
\[\sin\left(\frac{A}{2}\right) \cdot \sin\left(\frac{B}{2}\right) \cdot \sin\left(\frac{C}{2}\right) \leq \frac{abc}{(a+b)(b+c)(c+a)}.\]
2013 Romania Team Selection Test, 3
Given an integer $n\geq 2$, determine all non-constant polynomials $f$ with complex coefficients satisfying the condition
\[1+f(X^n+1)=f(X)^n.\]
2016 BMT Spring, 5
Find
$$\frac{\tan 1^o}{1 + \tan 1^o }+\frac{\tan 2^o}{1 + \tan 2^o } + ... + \frac{\tan 89^o}{1 + \tan 89^o}$$
2011 Oral Moscow Geometry Olympiad, 6
One triangle lies inside another. Prove that at least one of the two smallest sides (out of six) is the side of the inner triangle.
2023 Yasinsky Geometry Olympiad, 2
In triangle $ABC$, the difference between angles $B$ and $C$ is equal to $90^o$, and $AL$ is the angle bisector of triangle $ABC$. The bisector of the exterior angle $A$ of the triangle $ABC$ intersects the line $BC$ at the point $F$. Prove that $AL = AF$.
(Alexander Dzyunyak)