Found problems: 85335
2011 Kyrgyzstan National Olympiad, 7
Given that $g(n) = \frac{1}{{2 + \frac{1}{{3 + \frac{1}{{... + \frac{1}{{n - 1}}}}}}}}$ and $k(n) = \frac{1}{{2 + \frac{1}{{3 + \frac{1}{{... + \frac{1}{{n - 1 + \frac{1}{n}}}}}}}}}$, for natural $n$. Prove that $\left| {g(n) - k(n)} \right| \le \frac{1}{{(n - 1)!n!}}$.
1995 Kurschak Competition, 3
Points $A$, $B$, $C$, $D$ are such that no three of them are collinear. Let $E=AB\cap CD$ and $F=BC\cap DA$. Let $k_1$, $k_2$ and $k_3$ denote the circles with diameter $\overline{AC}$, $\overline{BD}$ and $\overline{EF}$, respectively. Prove that either $k_1,k_2,k_3$ pass through one point, or no two of them intersect.
2020 Ukrainian Geometry Olympiad - April, 4
Inside triangle $ABC$, the point $P$ is chosen such that $\angle PAB = \angle PCB =\frac14 (\angle A+ \angle C)$. Let $BL$ be the bisector of $\vartriangle ABC$. Line $PL$ intersects the circumcircle of $\vartriangle APC$ at point $Q$. Prove that the line $QB$ is the bisector of $\angle AQC$.
2019 IFYM, Sozopol, 2
In $\Delta ABC$ with $\angle ACB=135^\circ$, are chosen points $M$ and $N$ on side $AB$, so that
$\angle MCN=90^\circ$. Segments $MD$ and $NQ$ are angle bisectors of $\Delta AMC$ and $\Delta NBC$ respectively. Prove that the reflection of $C$ in line $PQ$ lies on the line $AB$.
1966 Kurschak Competition, 3
Do there exist two infinite sets of non-negative integers such that every non-negative integer can be uniquely represented in the form $a + b$ with $a$ in $A$ and $b$ in $B$?
2018 Romanian Master of Mathematics, 6
Fix a circle $\Gamma$, a line $\ell$ to tangent $\Gamma$, and another circle $\Omega$ disjoint from $\ell$ such that $\Gamma$ and $\Omega$ lie on opposite sides of $\ell$. The tangents to $\Gamma$ from a variable point $X$ on $\Omega$ meet $\ell$ at $Y$ and $Z$. Prove that, as $X$ varies over $\Omega$, the circumcircle of $XYZ$ is tangent to two fixed circles.
2020 LMT Fall, 4
At the Lexington High School, each student is given a unique five-character ID consisting of uppercase letters. Compute the number of possible IDs that contain the string "LMT".
[i]Proposed by Alex Li[/i]
2006 Irish Math Olympiad, 3
Prove that a square of side 2.1 units can be completely covered by seven squares of side 1 unit.
Extra: Try to prove that 7 is the minimal amount.
2018 Malaysia National Olympiad, B2
Let $a$ and $b$ be positive integers such that
(i) both $a$ and $b$ have at least two digits;
(ii) $a + b$ is divisible by $10$;
(iii) $a$ can be changed into $b$ by changing its last digit.
Prove that the hundreds digit of the product $ab$ is even.
2009 Indonesia TST, 2
For every positive integer $ n$, let $ \phi(n)$ denotes the number of positive integers less than $ n$ that is relatively prime to $ n$ and $ \tau(n)$ denote the sum of all positive divisors of $ n$. Let $ n$ be a positive integer such that $ \phi(n)|n\minus{}1$ and that $ n$ is not a prime number. Prove that $ \tau(n)>2009$.
2022 Princeton University Math Competition, A3 / B5
Daeun draws a unit circle centered at the origin and inscribes within it a regular hexagon $ABCDEF$. Then Dylan chooses a point $P$ within the circle of radius $2$ centered at the origin. Let $M$ be the maximum possible value of $|PA| \cdot |PB| \cdot |PC| \cdot |PD| \cdot |PE| \cdot |PF|$, and let $N$ be the number of possible points $P$ for which this maximal value is obtained. Find $M + N^2$.
2019 Estonia Team Selection Test, 7
An acute-angled triangle $ABC$ has two altitudes $BE$ and $CF$. The circle with diameter $AC$ intersects the segment $BE$ at point $P$. A circle with diameter $AB$ intersects the segment $CF$ at point $Q$ and the extension of this altitude at point $Q'$. Prove that $\angle PQ'Q = \angle PQB$.
2014 Thailand Mathematical Olympiad, 3
Let $M$ and $N$ be positive integers. Pisut walks from point $(0, N)$ to point $(M, 0)$ in steps so that
$\bullet$ each step has unit length and is parallel to either the horizontal or the vertical axis, and
$\bullet$ each point ($x, y)$ on the path has nonnegative coordinates, i.e. $x, y > 0$.
During each step, Pisut measures his distance from the axis parallel to the direction of his step, if after the step he ends up closer from the origin (compared to before the step) he records the distance as a positive number, else he records it as a negative number.
Prove that, after Pisut completes his walk, the sum of the signed distances Pisut measured is zero.
2009 Philippine MO, 3
Each point of a circle is colored either red or blue.
[b](a)[/b] Prove that there always exists an isosceles triangle inscribed in this circle such that all its vertices are colored the same.
[b](b)[/b] Does there always exist an equilateral triangle inscribed in this circle such that all its vertices are colored the same?
2017 Hanoi Open Mathematics Competitions, 7
Determine two last digits of number $Q = 2^{2017} + 2017^2$
2013 Online Math Open Problems, 49
In $\triangle ABC$, $CA=1960\sqrt{2}$, $CB=6720$, and $\angle C = 45^{\circ}$. Let $K$, $L$, $M$ lie on $BC$, $CA$, and $AB$ such that $AK \perp BC$, $BL \perp CA$, and $AM=BM$. Let $N$, $O$, $P$ lie on $KL$, $BA$, and $BL$ such that $AN=KN$, $BO=CO$, and $A$ lies on line $NP$. If $H$ is the orthocenter of $\triangle MOP$, compute $HK^2$.
[hide="Clarifications"]
[list]
[*] Without further qualification, ``$XY$'' denotes line $XY$.[/list][/hide]
[i]Evan Chen[/i]
2008 JBMO Shortlist, 5
Is it possible to cover a given square with a few congruent right-angled triangles with acute angle equal to ${{30}^{o}}$? (The triangles may not overlap and may not exceed the margins of the square.)
2023 VN Math Olympiad For High School Students, Problem 9
Given a quadrilateral $ABCD$ inscribed in $(O)$. Let $L, J$ be the [i]Lemoine[/i] point of $\triangle ABC$ and $\triangle ACD$.
Prove that: $AC, BD, LJ$ are concurrent.
2004 Baltic Way, 10
Is there an infinite sequence of prime numbers $p_1$, $p_2$, $\ldots$, $p_n$, $p_{n+1}$, $\ldots$ such that $|p_{n+1}-2p_n|=1$ for each $n \in \mathbb{N}$?
2011 China Second Round Olympiad, 4
Let $A$ be a $3 \times 9$ matrix. All elements of $A$ are positive integers. We call an $m\times n$ submatrix of $A$ "ox" if the sum of its elements is divisible by $10$, and we call an element of $A$ "carboxylic" if it is not an element of any "ox" submatrix. Find the largest possible number of "carboxylic" elements in $A$.
2014 Swedish Mathematical Competition, 1
Determine all polynomials $p(x)$ with non-negative integer coefficients such that $p (1) = 7$ and $p (10) = 2014$.
2008 Rioplatense Mathematical Olympiad, Level 3, 3
Consider a collection of stones whose total weight is $65$ pounds and each of whose stones is at most $w$ pounds. Find the largest number $w$ for which any such collection of stones can be divided into two groups whose total weights differ by at most one pound.
Note: The weights of the stones are not necessarily integers.
Kvant 2022, M2709
There are $n > 2022$ cities in the country. Some pairs of cities are connected with straight two-ways airlines. Call the set of the cities {\it unlucky}, if it is impossible to color the airlines between them in two colors without monochromatic triangle (i.e. three cities $A$, $B$, $C$ with the airlines $AB$, $AC$ and $BC$ of the same color).
The set containing all the cities is unlucky. Is there always an unlucky set containing exactly 2022 cities?
2010 Polish MO Finals, 3
Real number $C > 1$ is given. Sequence of positive real numbers $a_1, a_2, a_3, \ldots$, in which $a_1=1$ and $a_2=2$, satisfy the conditions
\[a_{mn}=a_ma_n, \] \[a_{m+n} \leq C(a_m + a_n),\]
for $m, n = 1, 2, 3, \ldots$. Prove that $a_n = n$ for $n=1, 2, 3, \ldots$.
1992 Poland - First Round, 6
The sequence $(x_n)$ is determined by the conditions:
$x_0=1992,x_n=-\frac{1992}{n} \cdot \sum_{k=0}^{n-1} x_k$ for $n \geq 1$.
Find $\sum_{n=0}^{1992} 2^nx_n$.