Found problems: 85335
1993 Brazil National Olympiad, 3
Given a circle and its center $O$, a point $A$ inside the circle and a distance $h$, construct a triangle $BAC$ with $\angle BAC = 90^\circ$, $B$ and $C$ on the circle and the altitude from $A$ length $h$.
2003 Estonia National Olympiad, 1
Jiiri and Mari both wish to tile an $n \times n$ chessboard with cards shown in the picture (each card covers exactly one square). Jiiri wants that for each two cards that have a common edge, the neighbouring parts are of different color, and Mari wants that the neighbouring parts are always of the same color. How many possibilities does Jiiri have to tile the chessboard and how many possibilities does Mari have?
[img]https://cdn.artofproblemsolving.com/attachments/7/3/9c076eb17ba7ae7c000a2893c83288a94df384.png[/img]
2013 Bogdan Stan, 1
$ M,N,P,Q,R,S $ are the midpoints of the sides $ AB,BC,CD,DE,EF,FA $ of a convex hexagon $ ABCDEF. $
[b]a)[/b] Show that with the segments $ MQ,NR,PS, $ it can be formed a triangle.
[b]b)[/b] Show that a triangle formed with the segments $ MQ,NR,PS $ is right if and only if ether $ MQ\perp NR $ or $ MQ\perp PS $ or $ PS\perp RN. $
[i]Vasile Pop[/i]
1999 Brazil Team Selection Test, Problem 1
For a positive integer n, let $w(n)$ denote the number of distinct prime
divisors of n. Determine the least positive integer k such that
$2^{w(n)} \leq k \sqrt[4]{n}$
for all positive integers n.
2022 BMT, 17
Midori and Momoi are arguing over chores. Each of $5$ chores may either be done by Midori, done by Momoi, or put off for tomorrow. Today, each of them must complete at least one chore, and more than half of the chores must be completed. How many ways can they assign chores for today? (The order in which chores are completed does not matter.)
2017 Dutch Mathematical Olympiad, 4
If we divide the number $13$ by the three numbers $5, 7$, and $9$, then these divisions leave remainders: when dividing by $5$ the remainder is $3$, when dividing by $7$ the remainder is $6$, and when dividing by $9$ the remainder is 4. If we add these remainders, we obtain $3 + 6 + 4 = 13$, the original number.
(a) Let $n$ be a positive integer and let $a$ and $b$ be two positive integers smaller than $n$. Prove: if you divide $n$ by $a$ and $b$, then the sum of the two remainders never equals $n$.
(b) Determine all integers $n > 229$ having the property that if you divide $n$ by $99, 132$, and $229$, the sum of the three remainders is $n$.
2024 USEMO, 2
Let $k$ be a fixed positive integer. For each integer $1 \leq i \leq 4$, let $x_i$ and $y_i$ be positive integers such that their least common multiple is $k$. Suppose that the four points $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$, $(x_4, y_4)$ are the vertices of a non-degenerate rectangle in the Cartesian plane. Prove that $x_1x_2x_3x_4$ is a perfect square.
[i]Andrei Chirita[/i]
2022 IFYM, Sozopol, 6
Let $n$ be a natural number and $P_1, P_2, ... , P_n$ are polynomials with integer coefficients, each of degree at least $2$. Let $S$ be the set of all natural numbers $N$ for which there exists a natural number $a$ and an index $1 \le i \le n$ such that $P_i(a) = N$. Prove, that there are infinitely many primes that do not belong to $S$.
2014 AIME Problems, 1
Abe can paint the room in 15 hours, Bea can paint 50 percent faster than Abe, and Coe can paint twice as fast as Abe. Abe begins to paint the room and works alone for the first hour and a half. Then Bea joins Abe, and they work together until half the room is painted. Then Coe joins Abe and Bea, and they work together until the entire room is painted. Find the number of minutes after Abe begins for the three of them to finish painting the room.
2005 All-Russian Olympiad, 3
Given 2005 distinct numbers $a_1,\,a_2,\dots,a_{2005}$. By one question, we may take three different indices $1\le i<j<k\le 2005$ and find out the set of numbers $\{a_i,\,a_j,\,a_k\}$ (unordered, of course). Find the minimal number of questions, which are necessary to find out all numbers $a_i$.
1984 AMC 12/AHSME, 4
A rectangle intersects a circle as shown: $AB=4$, $BC=5$, and $DE=3$. Then $EF$ equals:
[asy]size(200);
defaultpen(linewidth(0.7)+fontsize(10));
pair D=origin, E=(3,0), F=(10,0), G=(12,0), H=(12,1), A=(0,1), B=(4,1), C=(9,1), O=circumcenter(B,C,F);
draw(D--G--H--A--cycle);
draw(Circle(O, abs(O-C)));
label("$A$", A, NW);
label("$B$", B, NW);
label("$C$", C, NE);
label("$D$", D, SW);
label("$E$", E, SE);
label("$F$", F, SW);
label("4", (2,0.85), N);
label("3", D--E, S);
label("5", (6.5,0.85), N);
[/asy]
$\mathbf{(A)}\; 6\qquad \mathbf{(B)}\; 7\qquad \mathbf{(C)}\; \frac{20}3\qquad \mathbf{(D)}\; 8\qquad \mathbf{(E)}\; 9$
2012 CHMMC Fall, 9
For a positive integer $n$, let $f(n)$ be equal to $n$ if there is an integer $x$ such that $x^2-n$ is divisible by $2^{12}$, and let $f(n)$ be $0$ otherwise. Determine the remainder when $$\sum^{2^{12}-1}_{n=0}f(n)$$ is divided by $2^{12}$.
PEN H Problems, 28
Let $a, b, c$ be positive integers such that $a$ and $b$ are relatively prime and $c$ is relatively prime either to $a$ or $b$. Prove that there exist infinitely many triples $(x, y, z)$ of distinct positive integers such that \[x^{a}+y^{b}= z^{c}.\]
2013 China Team Selection Test, 3
A point $(x,y)$ is a [i]lattice point[/i] if $x,y\in\Bbb Z$. Let $E=\{(x,y):x,y\in\Bbb Z\}$. In the coordinate plane, $P$ and $Q$ are both sets of points in and on the boundary of a convex polygon with vertices on lattice points. Let $T=P\cap Q$. Prove that if $T\ne\emptyset$ and $T\cap E=\emptyset$, then $T$ is a non-degenerate convex quadrilateral region.
1966 IMO Longlists, 24
There are $n\geq 2$ people at a meeting. Show that there exist two people at the meeting who have the same number of friends among the persons at the meeting. (It is assumed that if $A$ is a friend of $B,$ then $B$ is a friend of $A;$ moreover, nobody is his own friend.)
2006 Baltic Way, 10
$162$ pluses and $144$ minuses are placed in a $30\times 30$ table in such a way that each row and each column contains at most $17$ signs. (No cell contains more than one sign.) For every plus we count the number of minuses in its row and for every minus we count the number of pluses in its column. Find the maximum of the sum of these numbers.
2016 IMAR Test, 4
A positive integer $m$ is perfect if the sum of all its positive divisors, $1$ and $m$ inclusive, is equal to $2m$. Determine the positive integers $n$ such that $n^n + 1$ is a perfect number.
2009 Hong Kong TST, 5
Let $ a,b,c$ be the three sides of a triangle. Determine all possible values of $ \frac {a^2 \plus{} b^2 \plus{} c^2}{ab \plus{} bc \plus{} ca}$
2008 ITest, 85
Let $(a,b,c,d)$ be a solution to the system \begin{align*}a+b&=15,\\ab+c+d&=78,\\ad+bc&=160,\\cd&=96.\end{align*} Find the greatest possible value of $a^2+b^2+c^2+d^2$.
LMT Team Rounds 2010-20, B15
Let $\vartriangle AMO$ be an equilateral triangle. Let $U$ and $G$ lie on side $AM$, and let $S$ and $N$ lie on side $AO$ such that $AU =UG = GM$ and $AS = SN = NO$. Find the value of $\frac{[MONG]}{[U S A]}$
2014 National Olympiad First Round, 29
Let $ABC$ be a triangle such that $|AB|=13 , |BC|=12$ and $|CA|=5$. Let the angle bisectors of $A$ and $B$ intersect at $I$ and meet the opposing sides at $D$ and $E$, respectively. The line passing through $I$ and the midpoint of $[DE]$ meets $[AB]$ at $F$. What is $|AF|$?
$
\textbf{(A)}\ \dfrac{3}{2}
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ \dfrac{5}{2}
\qquad\textbf{(D)}\ 3
\qquad\textbf{(E)}\ \dfrac{7}{2}
$
2021 Math Prize for Girls Problems, 20
Let $G$ be the set of points $(x, y)$ such that $x$ and $y$ are positive integers less than or equal to 6. A [i]magic grid[/i] is an assignment of an integer to each point in $G$ such that, for every square with horizontal and vertical sides and all four vertices in $G$, the sum of the integers assigned to the four vertices is the same as the corresponding sum for any other such square. A magic grid is formed so that the product of all 36 integers is the smallest possible value greater than 1. What is this product?
2014 Online Math Open Problems, 21
Consider a sequence $x_1,x_2,\cdots x_{12}$ of real numbers such that $x_1=1$ and for $n=1,2,\dots,10$ let \[ x_{n+2}=\frac{(x_{n+1}+1)(x_{n+1}-1)}{x_n}. \] Suppose $x_n>0$ for $n=1,2,\dots,11$ and $x_{12}=0$. Then the value of $x_2$ can be written as $\frac{\sqrt{a}+\sqrt{b}}{c}$ for positive integers $a,b,c$ with $a>b$ and no square dividing $a$ or $b$. Find $100a+10b+c$.
[i]Proposed by Michael Kural[/i]
Geometry Mathley 2011-12, 9.3
Let $ABCD$ be a quadrilateral inscribed in a circle $(O)$. Let $(O_1), (O_2), (O_3), (O_4)$ be the circles going through $(A,B), (B,C),(C,D),(D,A)$. Let $X, Y,Z, T$ be the second intersection of the pairs of the circles: $(O_1)$ and $(O_2), (O_2)$ and $(O_3), (O_3)$ and $(O_4), (O_4)$ and $(O_1)$.
(a) Prove that $X, Y,Z, T$ are on the same circle of radius $I$.
(b) Prove that the midpoints of the line segments $O_1O_3,O_2O_4,OI$ are collinear.
Nguyễn Văn Linh
2017 India IMO Training Camp, 1
In an acute triangle $ABC$, points $D$ and $E$ lie on side $BC$ with $BD<BE$. Let $O_1, O_2, O_3, O_4, O_5, O_6$ be the circumcenters of triangles $ABD, ADE, AEC, ABE, ADC, ABC$, respectively. Prove that $O_1, O_3, O_4, O_5$ are con-cyclic if and only if $A, O_2, O_6$ are collinear.