Found problems: 85335
Croatia MO (HMO) - geometry, 2012.7
Let the points $M$ and $N$ be the intersections of the inscribed circle of the right-angled triangle $ABC$, with sides $AB$ and $CA$ respectively , and points $P$ and $Q$ respectively be the intersections of the ex-scribed circles opposite to vertices $B$ and $C$ with direction $BC$. Prove that the quadrilateral $MNPQ$ is a cyclic if and only if the triangle $ABC$ is right-angled with a right angle at the vertex $A$.
2001 AIME Problems, 11
Club Truncator is in a soccer league with six other teams, each of which it plays once. In any of its 6 matches, the probabilities that Club Truncator will win, lose, or tie are each $\frac{1}{3}$. The probability that Club Truncator will finish the season with more wins than losses is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2009 Indonesia MO, 2
For any real $ x$, let $ \lfloor x\rfloor$ be the largest integer that is not more than $ x$. Given a sequence of positive integers $ a_1,a_2,a_3,\ldots$ such that $ a_1>1$ and
\[ \left\lfloor\frac{a_1\plus{}1}{a_2}\right\rfloor\equal{}\left\lfloor\frac{a_2\plus{}1}{a_3}\right\rfloor\equal{}\left\lfloor\frac{a_3\plus{}1}{a_4}\right\rfloor\equal{}\cdots\]
Prove that
\[ \left\lfloor\frac{a_n\plus{}1}{a_{n\plus{}1}}\right\rfloor\leq1\]
holds for every positive integer $ n$.
2025 Azerbaijan IZhO TST, 1
An arbitary point $D$ is selected on arc $BC$ not containing $A$ on $(ABC)$. $P$ and $Q$ are the reflections of point $B$ and $C$ with respect to $AD$, respectively. Circumcircles of $ABQ$ and $ACP$ intersect at $E\neq A$. Prove that $A;D;E$ is colinear
2000 Austrian-Polish Competition, 8
In the plane are given $27$ points, no three of which are collinear. Four of this points are vertices of a unit square, while the others lie inside the square. Prove that there are three points in this set forming a triangle with area not exceeding $1/48$.
2006 Estonia National Olympiad, 3
Let there be $ n \ge 2$ real numbers such that none of them is greater than the arithmetic mean of the other numbers. Prove that all the numbers are equal.
2023 AMC 12/AHSME, 16
In Coinland, there are three types of coins, each worth $6,$ $10,$ and $15.$ What is the sum of the digits of the maximum amount of money that is impossible to have?
$\textbf{(A) }11\qquad\textbf{(B) }6\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10$
(I forgot the order)
2004 All-Russian Olympiad Regional Round, 8.6
Let $ABCD$ be a quadrilateral with parallel sides $AD$ and $BC$, $M$ and $N$ are the midpoints of its sides $AB$ and $CD$, respectively. The straight line $MN$ bisects the segment connecting the centers of the circumcircles of triangles $ABC$ and $ADC$. Prove that $ABCD$ is a parallelogram.
2011 AMC 8, 2
Karl's rectangular vegetable garden is $20$ by $45$ feet, and Makenna's is $25$ by $40$ feet. Which garden is larger in area?
$\textbf{(A)}$ Karl's garden is larger by 100 square feet.
$\textbf{(B)}$ Karl's garden is larger by 25 square feet.
$\textbf{(C)}$ The gardens are the same size.
$\textbf{(D)}$ Makenna's garden is larger by 25 square feet.
$\textbf{(E)}$ Makenna's garden is larger by 100 square feet.
2016 CCA Math Bonanza, L5.2
In this problem, the symbol $0$ represents the number zero, the symbol $1$ represents the number seven, the symbol $2$ represents the number five, the symbol $3$ represents the number three, the symbol $4$ represents the number four, the symbol $5$ represents the number two, the symbol $6$ represents the number nine, the symbol $7$ represents the number one, the symbol $8$ represents an arbitrarily large positive integer, the symbol $9$ represents the number six, and the symbol $\infty$ represents the number eight. Compute the value of $\left|0-1+2-3^4-5+6-7^8\times9-\infty\right|$.
[i]2016 CCA Math Bonanza Lightning #5.2[/i]
2010 AMC 10, 15
On a 50-question multiple choice math contest, students receive 4 points for a correct answer, 0 points for an answer left blank, and -1 point for an incorrect answer. Jesse's total score on the contest was 99. What is the maximum number of questions that Jesse could have answered correctly?
$ \textbf{(A)}\ 25\qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 29\qquad\textbf{(D)}\ 31\qquad\textbf{(E)}\ 33$
2018 PUMaC Team Round, 10
For how many ordered quadruplets $(a,b,c,d)$ of positive integers such that $2\leq a\leq b \leq c$ and $1 \leq d \leq 418$ do we have that $bcd+abd+acd=abc+abcd?$
2014 Contests, 2
A segment $AB$ is given in (Euclidean) plane. Consider all triangles $XYZ$ such, that $X$ is an inner point of $AB$, triangles $XBY$ and $XZA$ are similar (in this order of vertices), and points $A, B, Y, Z$ lie on a circle in this order. Find the locus of midpoints of all such segments $YZ$.
(Day 1, 2nd problem
authors: Michal Rolínek, Jaroslav Švrček)
LMT Team Rounds 2010-20, B3
Find the number of ways to arrange the letters in $LE X I NGTON$ such that the string $LE X$ does not appear.
2019 Purple Comet Problems, 24
A $12$-sided polygon is inscribed in a circle with radius $r$. The polygon has six sides of length $6\sqrt3$ that alternate with six sides of length $2$. Find $r^2$.
LMT Team Rounds 2010-20, B13
Compute the number of ways there are to completely fill a $3\times 15$ rectangle with non-overlapping $1\times 3$ rectangles
2022 Princeton University Math Competition, 10
Let $\alpha, \beta, \gamma \in C$ be the roots of the polynomial $x^3 - 3x2 + 3x + 7$. For any complex number $z$, let $f(z)$ be defined as follows:
$$f(z) = |z -\alpha | + |z - \beta|+ |z-\gamma | - 2 \underbrace{\max}_{w \in \{\alpha, \beta, \gamma}\} |z - w|.$$
Let $A$ be the area of the region bounded by the locus of all $z \in C$ at which $f(z)$ attains its global minimum. Find $\lfloor A \rfloor$.
2021-IMOC, A10
For any positive reals $x$, $y$, $z$ with $xyz + xy + yz + zx = 4$, prove that
$$\sqrt{\frac{xy+x+y}{z}}+\sqrt{\frac{yz+y+z}{x}}+\sqrt{\frac{zx+z+x}{y}}\geq 3\sqrt{\frac{3(x+2)(y+2)(z+2)}{(2x + 1)(2y + 1)(2z + 1).
}}$$
2000 Moldova National Olympiad, Problem 5
Find all functions $f\colon \mathbb{R}\to\mathbb{R}$ that satisfy $f(x+y)-f(x-y)=2y(3x^2+y^2)$ for all $x,y{\in}R$
______________________________________
Azerbaijan Land of the Fire :lol:
2014 Purple Comet Problems, 18
Let $f$ be a real-valued function such that $4f(x)+xf\left(\tfrac1x\right)=x+1$ for every positive real number $x$. Find $f(2014)$.
1981 IMO, 1
[b]a.)[/b] For which $n>2$ is there a set of $n$ consecutive positive integers such that the largest number in the set is a divisor of the least common multiple of the remaining $n-1$ numbers?
[b]b.)[/b] For which $n>2$ is there exactly one set having this property?
2002 Flanders Junior Olympiad, 3
Is it possible to number the $8$ vertices of a cube from $1$ to $8$ in such a way that the value of the sum on every edge is different?
2012 Regional Olympiad of Mexico Center Zone, 2
Let $m, n$ integers such that:
$(n-1)^3+n^3+(n+1)^3=m^3$
Prove that 4 divides $n$
2014 JBMO TST - Turkey, 3
Let a line $\ell$ intersect the line $AB$ at $F$, the sides $AC$ and $BC$ of a triangle $ABC$ at $D$ and $E$, respectively and the internal bisector of the angle $BAC$ at $P$. Suppose that $F$ is at the opposite side of $A$ with respect to the line $BC$, $CD = CE$ and $P$ is in the interior the triangle $ABC$. Prove that
\[FB \cdot FA+CP^2 = CF^2 \iff AD \cdot BE = PD^2.\]
Estonia Open Senior - geometry, 2020.2.5
The bisector of the interior angle at the vertex $B$ of the triangle $ABC$ and the perpendicular line on side $BC$ passing through the vertex $C$ intersects at $D$. Let $M$ and $N$ be the midpoints of the segments $BC$ and $BD$, respectively, with $N$ on the side $AC$. Find all possibilities of the angles of the triangles $ABC$, if it is known that $\frac{| AM |}{| BC |}=\frac{|CD|}{|BD|}$.
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