Found problems: 85335
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 $\frac{[MONG]}{[U S A]}$
Diana and Apollo each roll a standard die obtaining a number at random from $1$ to $6$. What is the probability that Diana's number is larger than Apollo's number?
$\text{(A)}\ \dfrac{1}{3} \qquad \text{(B)}\ \dfrac{5}{12} \qquad \text{(C)}\ \dfrac{4}{9} \qquad \text{(D)}\ \dfrac{17}{36} \qquad \text{(E)}\ \dfrac{1}{2}$
A grid consists of all points of the form $(m, n)$ where $m$ and $n$ are integers with $|m|\le 2019,|n| \le 2019$ and $|m| +|n| < 4038$. We call the points $(m,n)$ of the grid with either $|m| = 2019$ or $|n| = 2019$ the [i]boundary points[/i]. The four lines $x = \pm 2019$ and $y= \pm 2019$ are called [i]boundary lines[/i]. Two points in the grid are called [i]neighbours [/i] if the distance between them is equal to $1$.
Anna and Bob play a game on this grid.
Anna starts with a token at the point $(0,0)$. They take turns, with Bob playing first.
1) On each of his turns. Bob [i]deletes [/i] at most two boundary points on each boundary line.
2) On each of her turns. Anna makes exactly three [i]steps[/i] , where a [i]step [/i] consists of moving her token from its current point to any neighbouring point, which has not been deleted.
As soon as Anna places her token on some boundary point which has not been deleted, the game is over and Anna wins.
Does Anna have a winning strategy?
[i]Proposed by Demetres Christofides, Cyprus[/i]
Prove that $2005^2$ can be written in at least $4$ ways as the sum of 2 perfect (non-zero) squares.
In triangle $ABC$, the incenter is $I$ and the circumcenter is $O$. Let $AI$ intersects $(ABC)$ second time at $P$ .
The line passes through $I$ and perpendicular to $AI$ intersects $BC$ at $X$. The feet of the perpendicular from $X$ to $IO$ is $Y$. Prove that $A,P,X,Y$ cyclic.
Let $ m$ be a positive odd integer, $ m > 2.$ Find the smallest positive integer $ n$ such that $ 2^{1989}$ divides $ m^n \minus{} 1.$
On the $xy$ plane, find the area of the figure bounded by the graphs of $y=x$ and $y=\left|\ \frac34 x^2-3\ \right |-2$.
[i]2011 Kyoto University entrance exam/Science, Problem 3[/i]
Let $ABCD$ be a convex quadrilateral. Show that there exists a square $A'B'C'D'$ (Vertices maybe ordered clockwise or counter-clockwise) such that $A \not = A', B \not = B', C \not = C', D \not = D'$ and $AA',BB',CC',DD'$ are all concurrent.
Positive integers m and n are both greater 50, have a least common multiple equal to 480, and have a
greatest common divisor equal to 12. Find $m + n$.
There are $ n$ students; each student knows exactly $d $ girl students and $d $ boy students ("knowing" is a symmetric relation). Find all pairs $ (n,d) $ of integers .
Prove that there are infinitely many triplets $ (a,b,c)$ that satisfy the following equalities:
$ \dfrac{2a\minus{}b\plus{}6}{4a\plus{}c\plus{}2}\equal{}\dfrac{b\minus{}2c}{a\minus{}c}\equal{}\dfrac{2a\plus{}b\plus{}2c\minus{}2}{6a\plus{}2c\minus{}2}$
Let $ABCD$ be a cyclic quadrilateral with $\angle BAD < \angle ADC$. Let $M$ be the midpoint of the arc $CD$ not containing $A$. Suppose there is a point $P$ inside $ABCD$ such that $\angle ADB = \angle CPD$ and $\angle ADP = \angle PCB$.
Prove that lines $AD, PM$, and $BC$ are concurrent.
In some city there are $2000000$ citizens. In every group of $2000$ citizens there are $3$ pairwise friends. Prove, that there are $4$ pairwise friends in city.
The interior angles of a convex polygon form an arithmetic progression with a common difference of $4^\circ$. Determine the number of sides of the polygon if its largest interior angle is $172^\circ.$
In the diagram below, the three circles and the three line segments are tangent as shown. Given that the radius of all of the three circles is $1$, compute the area of the triangle.
[img]https://cdn.artofproblemsolving.com/attachments/b/e/8af4ea38d9a4c675edd0957aaa5336caec0ae2.png[/img]
On the blackboard are written the $400$ integers $1, 2, 3, \cdots , 399, 400$. Luis erases $100$ of these numbers, then Martin erases another $100$. Martin wins if the sum of the $200$ erased numbers equals the sum of those not deleted; otherwise, he wins Luis. Which of the two has a winning strategy? What if Luis deletes $101$ numbers and Martín deletes $99$?
In each case, explain how the player with the winning strategy can ensure victory.
What is the radius of the largest sphere that fits inside an octahedron of side length $1$?
Define real number $y$ as the fractional part of real number $x$ such that $0\leq y<1$ and $x-y$ is integer. Denote this by $<x>$.
For real number $a$, define an infinite sequence $\{a_n\}\ (n=1,\ 2,\ 3,\ \cdots)$ inductively as follows.
(i) $a_1=<a>$
(ii) If $a\n\neq 0$, then $a_{n+1}=\left<\frac{1}{a_n}\right>$,
if $a_n=0$, then $a_{n+1}=0$.
(1) For $a=\sqrt{2}$, find $a_n$.
(2) For any natural number $n$, find real number $a\geq \frac 13$ such that $a_n=a$.
(3) Let $a$ be a rational number. When we express $a=\frac{p}{q}$ with integer $p$, natural number $q$, prove that $a_n=0$ for any natural number $n\geq q$.
[i]2011 Tokyo University entrance exam/Science, Problem 2[/i]
Let $ p$ be the product of two consecutive integers greater than 2. Show that there are no integers $ x_1, x_2, \ldots, x_p$ satisfying the equation
\[ \sum^p_{i \equal{} 1} x^2_i \minus{} \frac {4}{4 \cdot p \plus{} 1} \left( \sum^p_{i \equal{} 1} x_i \right)^2 \equal{} 1
\]
[b]OR[/b]
Show that there are only two values of $ p$ for which there are integers $ x_1, x_2, \ldots, x_p$ satisfying
\[ \sum^p_{i \equal{} 1} x^2_i \minus{} \frac {4}{4 \cdot p \plus{} 1} \left( \sum^p_{i \equal{} 1} x_i \right)^2 \equal{} 1
\]
In the plane, there are $n$ points ($n\ge 4$) where no 3 of them are collinear. Let $A(n)$ be the number of parallelograms whose vertices are those points with area $1$. Prove the following inequality:
$A(n)\leq \frac{n^2-3n}{4}$ for all $n\ge 4$
When the natural numbers are written one after another in an increasing way, you get an infinite succession of digits $123456789101112 ....$ Denote $A_k$ the number formed by the first $k$ digits of this sequence . Prove that for all positive integer $n$ there is a positive integer $m$ which simultaneously verifies the following three conditions:
(i) $n$ divides $A_m$,
(ii) $n$ divides $m$,
(iii) $n$ divides the sum of the digits of $A_m$.
Let $ABCD$ be a convex quadrilateral, where $M$ is the midpoint of $DC$, $N$ is the midpoint of $BC$, and $O$ is the intersection of diagonals $AC$ and $BD$. Prove that $O$ is the centroid of the triangle $AMN$ if and only if $ABCD$ is a parallelogram.
Prove that if $A,B,$ and $C$ are angles of a triangle measured in radians then $A \cos B +\sin A \cos C >0.$
Let $ABC$ be a triangle and $D$ be a point on $BC$ such that $AB+BD=AC+CD$. The line $AD$ intersects the incircle of triangle $ABC$ at $X$ and $Y$ where $X$ is closer to $A$ than $Y$ i. Suppose $BC$ is tangent to the incircle at $E$, prove that:
1) $EY$ is perpendicular to $AD$;
2) $XD=2IM$ where $I$ is the incentre and $M$ is the midpoint of $BC$.
Given is a parallelogram $ABCD$ with $\angle A < 90^o$ and $|AB| < |BC|$. The angular bisector of angle $A$ intersects side $BC$ in $M$ and intersects the extension of $DC$ in $N$. Point $O$ is the centre of the circle through $M, C$, and $N$. Prove that $\angle OBC = \angle ODC$.
[asy]
unitsize (1.2 cm);
pair A, B, C, D, M, N, O;
A = (0,0);
B = (2,0);
D = (1,3);
C = B + D - A;
M = extension(A, incenter(A,B,D), B, C);
N = extension(A, incenter(A,B,D), D, C);
O = circumcenter(C,M,N);
draw(D--A--B--C);
draw(interp(D,N,-0.1)--interp(D,N,1.1));
draw(A--interp(A,N,1.1));
draw(circumcircle(M,C,N));
label("$\circ$", A + (0.45,0.15));
label("$\circ$", A + (0.25,0.35));
dot("$A$", A, SW);
dot("$B$", B, SE);
dot("$C$", C, dir(90));
dot("$D$", D, dir(90));
dot("$M$", M, SE);
dot("$N$", N, dir(90));
dot("$O$", O, SE);
[/asy]