Found problems: 85335
2001 Belarusian National Olympiad, 7
The convex quadrilateral $ABCD$ is inscribed in the circle $S_1$. Let $O$ be the intersection of $AC$ and $BD$. Circle $S_2$ passes through $D$ and $ O$, intersecting $AD$ and $CD$ at $ M$ and $ N$, respectively. Lines $OM$ and $AB$ intersect at $R$, lines $ON$ and $BC$ intersect at $T$, and $R$ and $T$ lie on the same side of line $BD$ as $ A$.
Prove that $O$, $R$,$T$, and $B$ are concyclic.
2017 Ecuador NMO (OMEC), 5
Let the sequences $(x_n)$ and $(y_n)$ be defined by $x_0 = 0$, $x_1 = 1$, $x_{n + 2} = 3x_{n + 1}-2x_n$ for $n = 0, 1, ...$ and $y_n = x^2_n+2^{n + 2}$ for $n = 0, 1, ...,$ respectively. Show that for all n> 0, and n is the square of a odd integer.
2010 IFYM, Sozopol, 6
Let $A=\{ x\in \mathbb{N},x=a^2+2b^2,a,b\in \mathbb{Z},ab\neq 0 \}$ and $p$ is a prime number.
Prove that if $p^2\in A$, then $p\in A$.
Mathley 2014-15, 6
A quadrilateral is called bicentric if it has both an incircle and a circumcircle. $ABCD$ is a bicentric quadrilateral with $(O)$ being its circumcircle. Let $E, F$ be the intersections of $AB$ and $CD, AD$ and $BC$ respectively. Prove that there is a circle with center $O$ tangent to all of the circumcircles of the four triangles $EAD, EBC, FAB, FCD$.
Nguyen Van Linh, a student of the Vietnamese College, Ha Noi
2008 Harvard-MIT Mathematics Tournament, 2
Let $ f(n)$ be the number of times you have to hit the $ \sqrt {\ }$ key on a calculator to get a number less than $ 2$ starting from $ n$. For instance, $ f(2) \equal{} 1$, $ f(5) \equal{} 2$. For how many $ 1 < m < 2008$ is $ f(m)$ odd?
2008 AIME Problems, 15
A square piece of paper has sides of length $ 100$. From each corner a wedge is cut in the following manner: at each corner, the two cuts for the wedge each start at distance $ \sqrt {17}$ from the corner, and they meet on the diagonal at an angle of $ 60^\circ$ (see the figure below). The paper is then folded up along the lines joining the vertices of adjacent cuts. When the two edges of a cut meet, they are taped together. The result is a paper tray whose sides are not at right angles to the base. The height of the tray, that is, the perpendicular distance between the plane of the base and the plane formed by the upper edges, can be written in the form $ \sqrt [n]{m}$, where $ m$ and $ n$ are positive integers, $ m < 1000$, and $ m$ is not divisible by the $ n$th power of any prime. Find $ m \plus{} n$.
[asy]import math;
unitsize(5mm);
defaultpen(fontsize(9pt)+Helvetica()+linewidth(0.7));
pair O=(0,0);
pair A=(0,sqrt(17));
pair B=(sqrt(17),0);
pair C=shift(sqrt(17),0)*(sqrt(34)*dir(75));
pair D=(xpart(C),8);
pair E=(8,ypart(C));
draw(O--(0,8));
draw(O--(8,0));
draw(O--C);
draw(A--C--B);
draw(D--C--E);
label("$\sqrt{17}$",(0,2),W);
label("$\sqrt{17}$",(2,0),S);
label("cut",midpoint(A--C),NNW);
label("cut",midpoint(B--C),ESE);
label("fold",midpoint(C--D),W);
label("fold",midpoint(C--E),S);
label("$30^\circ$",shift(-0.6,-0.6)*C,WSW);
label("$30^\circ$",shift(-1.2,-1.2)*C,SSE);[/asy]
2015 HMNT, 5
Let $S$ be a subset of the set $\{1, 2, 3, \dots, 2015\}$ such that for any two elements $a, b \in S$, the difference $a - b$ does not divide the sum $a + b$. Find the maximum possible size of $S$.
2014 AMC 8, 23
Three members of the Euclid Middle School girls' softball team had the following conversation.
Ashley: I just realized that our uniform numbers are all $2$-digit primes.
Bethany: And the sum of your two uniform numbers is the date of my birthday earlier this month.
Caitlin: That's funny. The sum of your two uniform numbers is the date of my birthday later this month.
Ashley: And the sum of you two uniform numbers is today's date.
What number does Caitlin wear?
$\textbf{(A) }11\qquad\textbf{(B) }13\qquad\textbf{(C) }17\qquad\textbf{(D) }19\qquad \textbf{(E) }23$
2008 Harvard-MIT Mathematics Tournament, 1
How many different values can $ \angle ABC$ take, where $ A,B,C$ are distinct vertices of a cube?
1935 Moscow Mathematical Olympiad, 001
Find the ratio of two numbers if the ratio of their arithmetic mean to their geometric mean is $25 : 24$
2002 Iran MO (3rd Round), 4
$a_{n}$ ($n$ is integer) is a sequence from positive reals that \[a_{n}\geq \frac{a_{n+2}+a_{n+1}+a_{n-1}+a_{n-2}}4\] Prove $a_{n}$ is constant.
2022 Germany Team Selection Test, 2
The kingdom of Anisotropy consists of $n$ cities. For every two cities there exists exactly one direct one-way road between them. We say that a [i]path from $X$ to $Y$[/i] is a sequence of roads such that one can move from $X$ to $Y$ along this sequence without returning to an already visited city. A collection of paths is called [i]diverse[/i] if no road belongs to two or more paths in the collection.
Let $A$ and $B$ be two distinct cities in Anisotropy. Let $N_{AB}$ denote the maximal number of paths in a diverse collection of paths from $A$ to $B$. Similarly, let $N_{BA}$ denote the maximal number of paths in a diverse collection of paths from $B$ to $A$. Prove that the equality $N_{AB} = N_{BA}$ holds if and only if the number of roads going out from $A$ is the same as the number of roads going out from $B$.
[i]Proposed by Warut Suksompong, Thailand[/i]
2016 Turkmenistan Regional Math Olympiad, Problem 4
Let $ABC$ is isosceles triangle $AB=AC$. The point $P$ inside $ABC$ triangle such that angle $\widehat{BCP}=30^o$ , $\widehat{APB}=150^o$ and $\widehat{CAP}=39^o$ . Find $\widehat{BAP}$
2015 Vietnam National Olympiad, 3
Given $m\in\mathbb{Z}^+$. Find all natural numbers $n$ that does not exceed $10^m$ satisfying the following conditions:
i) $3|n.$
ii) The digits of $n$ in decimal representation are in the set $\{2,0,1,5\}$.
2005 iTest, 30
How many of the following statements are false?
a. $2005$ distinct positive integers exist such that the sum of their squares is a cube and the sum of their cubes is a square.
b. There are $2$ integral solutions to $x^2 + y^2 + z^2 = x^2y^2$.
c. If the vertices of a triangle are lattice points in a plane, the diameter of the triangle’s circumcircle will never exceed the product of the triangle’s side lengths.
2002 Tournament Of Towns, 2
[list]
[*] A test was conducted in class. It is known that at least $\frac{2}{3}$ of the problems were hard. Each such problems were not solved by at least $\frac{2}{3}$ of the students. It is also known that at least $\frac{2}{3}$ of the students passed the test. Each such student solved at least $\frac{2}{3}$ of the suggested problems. Is this possible?
[*] Previous problem with $\frac{2}{3}$ replaced by $\frac{3}{4}$.
[*] Previous problem with $\frac{2}{3}$ replaced by $\frac{7}{10}$.[/list]
2022 Brazil National Olympiad, 6
Determine the largest positive integer $k$ for which the following statement is true: given
$k$ distinct subsets of the set $\{1, 2, 3, \dots , 2023\}$, each with $1011$ elements, it is possible
partition the subsets into two collections so that any two subsets in one same collection have some element in common.
2016 Online Math Open Problems, 30
In triangle $ABC$, $AB=3\sqrt{30}-\sqrt{10}$, $BC=12$, and $CA=3\sqrt{30}+\sqrt{10}$. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $AC$. Denote $l$ as the line passing through the circumcenter $O$ and orthocenter $H$ of $ABC$, and let $E$ and $F$ be the feet of the perpendiculars from $B$ and $C$ to $l$, respectively. Let $l'$ be the reflection of $l$ in $BC$ such that $l'$ intersects lines $AE$ and $AF$ at $P$ and $Q$, respectively. Let lines $BP$ and $CQ$ intersect at $K$. $X$, $Y$, and $Z$ are the reflections of $K$ over the perpendicular bisectors of sides $BC$, $CA$, and $AB$, respectively, and $R$ and $S$ are the midpoints of $XY$ and $XZ$, respectively. If lines $MR$ and $NS$ intersect at $T$, then the length of $OT$ can be expressed in the form $\frac{p}{q}$ for relatively prime positive integers $p$ and $q$. Find $100p+q$.
[i]Proposed by Vincent Huang and James Lin[/i]
1999 Swedish Mathematical Competition, 1
Solve $|||||x^2-x-1| - 2| - 3| - 4| - 5| = x^2 + x - 30$.
2016 India Regional Mathematical Olympiad, 5
a.) A 7-tuple $(a_1,a_2,a_3,a_4,b_1,b_2,b_3)$ of pairwise distinct positive integers with no common factor is called a shy tuple if $$ a_1^2+a_2^2+a_3^2+a_4^2=b_1^2+b_2^2+b_3^2$$and for all $1 \le i<j \le 4$ and $1 \le k \le 3$, $a_i^2+a_j^2 \not= b_k^2$. Prove that there exists infinitely many shy tuples.
b.) Show that $2016$ can be written as a sum of squares of four distinct natural numbers.
1988 All Soviet Union Mathematical Olympiad, 464
$ABCD$ is a convex quadrilateral. The midpoints of the diagonals and the midpoints of $AB$ and $CD$ form another convex quadrilateral $Q$. The midpoints of the diagonals and the midpoints of $BC$ and $CA$ form a third convex quadrilateral $Q'$. The areas of $Q$ and $Q'$ are equal. Show that either $AC$ or $BD$ divides $ABCD$ into two parts of equal area.
2009 AMC 10, 9
Positive integers $ a$, $ b$, and $ 2009$, with $ a<b<2009$, form a geometric sequence with an integer ratio. What is $ a$?
$ \textbf{(A)}\ 7 \qquad
\textbf{(B)}\ 41 \qquad
\textbf{(C)}\ 49 \qquad
\textbf{(D)}\ 289 \qquad
\textbf{(E)}\ 2009$
2016 Singapore Junior Math Olympiad, 5
Determine the minimum number of lines that can be drawn on the plane so that they intersect in exactly $200$ distinct points.
(Note that for $3$ distinct points, the minimum number of lines is $3$ and for $4$ distinct points, the minimum is $4$)
2017 Indonesia MO, 8
A field is made of $2017 \times 2017$ unit squares. Luffy has $k$ gold detectors, which he places on some of the unit squares, then he leaves the area. Sanji then chooses a $1500 \times 1500$ area, then buries a gold coin on each unit square in this area and none other. When Luffy returns, a gold detector beeps if and only if there is a gold coin buried underneath the unit square it's on. It turns out that by an appropriate placement, Luffy will always be able to determine the $1500 \times 1500$ area containing the gold coins by observing the detectors, no matter how Sanji places the gold coins. Determine the minimum value of $k$ in which this is possible.
1991 Cono Sur Olympiad, 2
Two people, $A$ and $B$, play the following game: $A$ start choosing a positive integrer number and then, each player in it's turn, say a number due to the following rule:
If the last number said was odd, the player add $7$ to this number;
If the last number said was even, the player divide it by $2$.
The winner is the player that repeats the first number said. Find all numbers that $A$ can choose in order to win. Justify your answer.