Found problems: 85335
2003 Cono Sur Olympiad, 3
Let $ABC$ be an acute triangle such that $\angle{B}=60$. The circle with diameter $AC$ intersects the internal angle bisectors of $A$ and $C$ at the points $M$ and $N$, respectively $(M\neq{A},$ $N\neq{C})$. The internal bisector of $\angle{B}$ intersects $MN$ and $AC$ at the points $R$ and $S$, respectively. Prove that $BR\leq{RS}$.
2004 Tournament Of Towns, 4
A circle with the center $I$ is entirely inside of a circle with center $O$. Consider all possible chords $AB$ of the larger circle which are tangent to the smaller one. Find the locus of the centers of the circles circumscribed about the triangle $AIB$.
2009 China Northern MO, 7
Let $\lfloor m \rfloor$ be the largest integer smaller than $m$ . Assume $x,y \in \mathbb{R+}$ ,
For all positive integer $n$ , $\lfloor x \lfloor ny \rfloor \rfloor =n-1$ .
Prove : $xy=1$ , $y$ is an irrational number larger than $ 1 $ .
1994 All-Russian Olympiad Regional Round, 10.1
We have seven equal pails with water, filled to one half, one third, one quarter, one fifth, one eighth, one ninth, and one tenth, respectively. We are allowed to pour water from one pail into another until the first pail empties or the second
one fills to the brim. Can we obtain a pail that is filled to
(a) one twelfth,
(b) one sixth
after several such steps?
2020 Canadian Mathematical Olympiad Qualification, 1
Show that for all integers $a \ge 1$,$ \lfloor \sqrt{a}+\sqrt{a+1}+\sqrt{a+2}\rfloor = \lfloor \sqrt{9a+8}\rfloor$
2018 Oral Moscow Geometry Olympiad, 6
Cut each of the equilateral triangles with sides $2$ and $3$ into three parts and construct an equilateral triangle from all received parts.
2022 Oral Moscow Geometry Olympiad, 5
Circle $\omega$ is tangent to the interior of the circle $\Omega$ at the point C. Chord $AB$ of circle $\Omega$ is tangent to $\omega$. Chords $CF$ and $BG$ of circle $\Omega$ intersect at point $E$ lying on $\omega$. Prove that the circumcircle of triangle $CGE$ is tangent to straight line $AF$.
(I. Kukharchuk)
2010 Dutch IMO TST, 1
Consider sequences $a_1, a_2, a_3,...$ of positive integers. Determine the smallest possible value of $a_{2010}$ if
(i) $a_n < a_{n+1}$ for all $n\ge 1$,
(ii) $a_i + a_l > a_j + a_k$ for all quadruples $ (i, j, k, l)$ which satisfy $1 \le i < j \le k < l$.
2005 China Western Mathematical Olympiad, 8
For $n$ people, if it is known that
(a) there exist two people knowing each other among any three people, and
(b) there exist two people not knowing each other among any four people.
Find the maximum of $n$.
Here, we assume that if $A$ knows $B$, then $B$ knows $A$.
2020 BMT Fall, 4
Let $\varphi$ be the positive solution to the equation $$x^2=x+1.$$ For $n\ge 0$, let $a_n$ be the unique integer such that $\varphi^n-a_n\varphi$ is also an integer. Compute $$\sum_{n=0}^{10}a_n.$$
2006 Vietnam Team Selection Test, 1
Prove that for all real numbers $x,y,z \in [1,2]$ the following inequality always holds:
\[ (x+y+z)(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})\geq 6(\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}). \]
When does the equality occur?
2019-2020 Fall SDPC, 4
Let $\triangle{ABC}$ be an acute, scalene triangle with orthocenter $H$, and let $AH$ meet the circumcircle of $\triangle{ABC}$ at a point $D \neq A$. Points $E$ and $F$ are chosen on $AC$ and $AB$ such that $DE \perp AC$ and $DF \perp AB$. Show that $BE$, $CF$, and the line through $H$ parallel to $EF$ concur.
2022 AMC 8 -, 9
A cup of boiling water ($212^{\circ}\text{F}$) is placed to cool in a room whose temperature remains constant at $68^{\circ}\text{F}$. Suppose the difference between the water temperature and the room temperature is halved every $5$ minutes. What is the water temperature, in degrees Fahrenheit, after $15$ minutes?
$\textbf{(A)} ~77\qquad\textbf{(B)} ~86\qquad\textbf{(C)} ~92\qquad\textbf{(D)} ~98\qquad\textbf{(E)} ~104\qquad$
2007 Pre-Preparation Course Examination, 6
Let $a,b$ be two positive integers and $b^2+a-1|a^2+b-1$. Prove that $b^2+a-1$ has at least two prime divisors.
PEN E Problems, 25
Prove that $\ln n \geq k\ln 2$, where $n$ is a natural number and $k$ is the number of distinct primes that divide $n$.
2016 Saint Petersburg Mathematical Olympiad, 1
In the sequence of integers $(a_n)$, the sum $a_m + a_n$ is divided by $m + n$ with any different $m$ and $n$. Prove that $a_n$ is a multiple of $n$ for any $n$.
KoMaL A Problems 2020/2021, A. 796
Let $ABCD$ be a cyclic quadrilateral. Let lines $AB$ and $CD$ intersect in $P,$ and lines $BC$ and $DA$ intersect in $Q.$ The feet of the perpendiculars from $P$ to $BC$ and $DA$ are $K$ and $L,$ and the feet of the perpendiculars from $Q$ to $AB$ and $CD$ are $M$ and $N.$ The midpoint of diagonal $AC$ is $F.$
Prove that the circumcircles of triangles $FKN$ and $FLM,$ and the line $PQ$ are concurrent.
[i]Based on a problem by Ádám Péter Balogh, Szeged[/i]
2005 India IMO Training Camp, 2
Given real numbers $a,\alpha,\beta, \sigma \ and \ \varrho$ s.t. $\sigma, \varrho > 0$ and $\sigma \varrho = \frac{1}{16}$, prove that there exist integers $x$ and $y$ s.t.
\[ - \sigma \leq (x+\alpha_(ax + y + \beta ) \leq \varrho \]
1990 Spain Mathematical Olympiad, 1
Prove that $\sqrt{x}+\sqrt{y}+\sqrt{xy}$ is equal to$ \sqrt{x}+\sqrt{y+xy+2y\sqrt{x}}$
and compare the numbers $\sqrt{3}+\sqrt{10+2\sqrt{3}}$ and $\sqrt{5+\sqrt{22}}+\sqrt{8-
\sqrt{22}+2\sqrt{15-3\sqrt{22}}}$.
2022 Novosibirsk Oral Olympiad in Geometry, 7
Altitudes $AA_1$ and $CC_1$ of an acute-angled triangle $ABC$ intersect at point $H$. A straight line passing through point $H$ parallel to line $A_1C_1$ intersects the circumscribed circles of triangles $AHC_1$ and $CHA_1$ at points $X$ and $Y$, respectively. Prove that points $X$ and $Y$ are equidistant from the midpoint of segment $BH$.
1990 IMO Longlists, 33
Let S be a 1990-element set and P be a set of 100-ary sequences $(a_1,a_2,...,a_{100})$ ,where $a_i's$ are distinct elements of S.An ordered pair (x,y) of elements of S is said to [i]appear[/i] in $(a_1,a_2,...,a_{100})$ if $x=a_i$ and $y=a_j$ for some i,j with $1\leq i<j\leq 100$.Assume that every ordered pair (x,y) of elements of S appears in at most one member in P.Show that $|P|\leq 800$.
2024 LMT Fall, 10
David starts at the point $A$ and goes up and right along the grid lines to point $B$. At each of the points $C$, $D$, and $E$ there is a bully. Find the number of paths David can take which make him encounter exactly one bully.
[asy]
size(150);
draw((0,0)--(4,0)--(4,3)--(0,3)--cycle);
draw((0,1)--(4,1));
draw((0,2)--(4,2));
draw((1,0)--(1,3));
draw((2,0)--(2,3));
draw((3,0)--(3,3));
dot((0,0)); label("A", (0,0), W);
dot((4,3)); label("B", (4,3), E);
dot((1,1.5)); label("C", (1,1.5), W);
dot((2,0.5)); label("D", (2,0.5), W);
dot((2.5,2)); label("E", (2.5,2), N);
[/asy]
1962 Putnam, B4
The euclidean plane is divided into regions by drawing a finite number of circles. Show that it is possible to color each of these regions either red or blue in such a way that no two adjacent regions have the same color.
2020 CHKMO, 1
Given that ${a_n}$ and ${b_n}$ are two sequences of integers defined by
\begin{align*}
a_1=1, a_2=10, a_{n+1}=2a_n+3a_{n-1} & ~~~\text{for }n=2,3,4,\ldots, \\
b_1=1, b_2=8, b_{n+1}=3b_n+4b_{n-1} & ~~~\text{for }n=2,3,4,\ldots.
\end{align*}
Prove that, besides the number $1$, no two numbers in the sequences are identical.
1965 Miklós Schweitzer, 7
Prove that any uncountable subset of the Euclidean $ n$-space contains an countable subset with the property that the distances between different pairs of points are different (that is, for any points $ P_1 \not\equal{} P_2$ and $ Q_1\not\equal{} Q_2$ of this subset, $ \overline{P_1P_2}\equal{}\overline{Q_1Q_2}$ implies either $ P_1\equal{}Q_1$ and $ P_2\equal{}Q_2$, or $ P_1\equal{}Q_2$ and $ P_2\equal{}Q_1$). Show that a similar statement is not valid if the Euclidean $ n$-space is replaced with a (separable) Hilbert space.