Found problems: 85335
Note that $12^2=144$ ends in two $4$s and $38^2=1444$ end in three $4$s. Determine the length of the longest string of equal nonzero digits in which the square of an integer can end.
Let $ABC$ be an acute triangle with orthocenter $H$. Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$. Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$. Prove that $IJ = AH$.
Consider the nonconvex quadrilateral $ABCD$ with $\angle C>180$ degrees. Let the side $DC$ extended to meet $AB$ at $F$ and the side $BC$ extended to meet $AD$ at $E$. A line intersects the interiors of the sides $AB,AD,BC,CD$ at points $K,L,J,I$ respectively. Prove that if $DI=CF$ and $BJ=CE$, then $KJ=IL$
Find the number of odd three digit positive integers $x$ satisfying $x^2 \equiv 1 \pmod{8}$ and the product of the digits of $x$ is odd.
[i]Lightning 1.2[/i]
Let $B = (-1, 0)$ and $C = (1, 0)$ be fixed points on the coordinate plane. A nonempty, bounded subset $S$ of the plane is said to be [i]nice[/i] if
$\text{(i)}$ there is a point $T$ in $S$ such that for every point $Q$ in $S$, the segment $TQ$ lies entirely in $S$; and
$\text{(ii)}$ for any triangle $P_1P_2P_3$, there exists a unique point $A$ in $S$ and a permutation $\sigma$ of the indices $\{1, 2, 3\}$ for which triangles $ABC$ and $P_{\sigma(1)}P_{\sigma(2)}P_{\sigma(3)}$ are similar.
Prove that there exist two distinct nice subsets $S$ and $S'$ of the set $\{(x, y) : x \geq 0, y \geq 0\}$ such that if $A \in S$ and $A' \in S'$ are the unique choices of points in $\text{(ii)}$, then the product $BA \cdot BA'$ is a constant independent of the triangle $P_1P_2P_3$.
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.)
When three different numbers from the set $ \{-3,-2,-1, 4, 5\} $ are multiplied, the largest possible product is
$ \text{(A)}\ 10\qquad\text{(B)}\ 20\qquad\text{(C)}\ 30\qquad\text{(D)}\ 40\qquad\text{(E)}\ 60 $
Let $ ABC$ be a triangle with $ AB \equal{} 5$, $ BC \equal{} 4$ and $ AC \equal{} 3$. Let $ \mathcal P$ and $ \mathcal Q$ be squares inside $ ABC$ with disjoint interiors such that they both have one side lying on $ AB$. Also, the two squares each have an edge lying on a common line perpendicular to $ AB$, and $ \mathcal P$ has one vertex on $ AC$ and $ \mathcal Q$ has one vertex on $ BC$. Determine the minimum value of the sum of the areas of the two squares.
[asy]import olympiad;
import math;
import graph;
unitsize(1.5cm);
pair A, B, C;
A = origin;
B = A + 5 * right;
C = (9/5, 12/5);
pair X = .7 * A + .3 * B;
pair Xa = X + dir(135);
pair Xb = X + dir(45);
pair Ya = extension(X, Xa, A, C);
pair Yb = extension(X, Xb, B, C);
pair Oa = (X + Ya)/2;
pair Ob = (X + Yb)/2;
pair Ya1 = (X.x, Ya.y);
pair Ya2 = (Ya.x, X.y);
pair Yb1 = (Yb.x, X.y);
pair Yb2 = (X.x, Yb.y);
draw(A--B--C--cycle);
draw(Ya--Ya1--X--Ya2--cycle);
draw(Yb--Yb1--X--Yb2--cycle);
label("$A$", A, W);
label("$B$", B, E);
label("$C$", C, N);
label("$\mathcal P$", Oa, origin);
label("$\mathcal Q$", Ob, origin);[/asy]
If $a_1,a_2,\ldots,a_n$ are positive real numbers, prove the inequality
$$\dfrac1{\dfrac1{1+a_1}+\dfrac1{1+a_2}+\ldots+\dfrac1{1+a_n}}-\dfrac1{\dfrac1{a_1}+\dfrac1{a_2}+\ldots+\dfrac1{a_n}}\ge\frac1n.$$
Let $\overline{BE}$ and $\overline{CF}$ be altitudes of triangle $ABC$, and let $D$ be the antipodal point of $A$ on the circumcircle of $ABC$. The lines $\overleftrightarrow{DE}$ and $\overleftrightarrow{DF}$ intersect $\odot(ABC)$ again at $Y$ and $Z$, respectively. Show that $\overleftrightarrow{YZ}$, $\overleftrightarrow{EF}$ and $\overleftrightarrow{BC}$ intersect at a point.
find all functions $f:\mathbb{Z} \to \mathbb{Z} $
such that $f(m+n)+f(m)f(n)=n^2(f(m)+1)+m^2(f(n)+1)+mn(2-mn)$ holds for all $m,n \in \mathbb{Z}$
On the blackboard were six figures: a circle, a triangle, a square, a trapezoid, a pentagon and a hexagon, painted in six colors: blue, white, red, yellow, green and brown. Each figure had only one color and all the figures were of different colors. The next day he wondered what color each figure was.
Paul replied: “The circle was red, the triangle was blue, the square was white, the trapezoid was green, the pentagon was brown, and the hexagon was yellow.
Sofía answered: “The circle was yellow, the triangle was green, the square was red, the trapezoid was blue, the pentagon was brown, and the hexagon was white.”
Pablo was wrong three times and Sofia twice, and it is known that the pentagon was brown.
Determine if it is possible to know with certainty what the color of each of the figures was.
Dorothy organized a party for the birthday of Duck Mom and she also prepared a cylindershaped cake. Since she was originally expecting to have $15$ guests, she divided the top of the cake into this many equal circular sectors, marking where the cuts need to be made. Just for fun Dorothy’s brother Donald split the top of the cake into $10$ equal circular sectors in such a way that some of the radii that he marked coincided with Dorothy’s original markings. Just
before the arrival of the guests Douglas cut the cake according to all markings, and then he placed the cake into the fridge. This way they forgot about the cake and only got to eating it when only $6$ of them remained. Is it possible for them to divide the cake into $6$ equal parts without making any further cuts?
In a convex quadrilateral $ABCD$ we are given that $\angle CAD=10^\circ$, $\angle DBC=20^\circ$, $\angle BAD=40^\circ$, $\angle ABC=50^\circ$. Compute angle $BDC$.
A week ago, Sandy’s seasonal Little League batting average was $360$. After five more at bats this week, Sandy’s batting average is up to $400$. What is the smallest number of hits that Sandy could have had this season?
The rows and columns of a $7\times 7$ grid are each numbered $1,2,\ldots, 7$. In how many ways can one choose 8 cells of this grid such that for every two chosen cells $X$ and $Y$, either the positive difference of their row numbers is at least $3$, or the positive difference of their column numbers is at least $3$?
[hide="Clarifications"]
[list]
[*] The ``or'' here is inclusive (as by convention, despite the ``either''), i.e. $X$ and $Y$ are permitted if and only if they satisfy the row condition, the column condition, or both.[/list][/hide]
[i]Ray Li[/i]
Let $P$ be a parabola with focus $F$ and directrix $\ell$. A line through $F$ intersects $P$ at two points $A$ and $B$. Let $D$ and $C$ be the feet of the altitudes from $A$ and $B$ onto $\ell$, respectively. Given that $AB = 20$ and $CD = 14$, compute the area of $ABCD$.
A positive integer $n$ is good if $n$ can be written as the sum of $2004$ positive integers $a_1$, $a_2$, ..., $a_{2004}$ such that $1 \le a_1 < a_2<...<a_{2004}$ and $a_i$ divides $a_{i+1}$ for $i=1$, $2$, ..., $2003$. Show that there are only finitely many positive integers that are not good.
Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$
A triangle $ABC$ is given. Find all the segments $XY$ that lie inside the triangle such that $XY$ and five of the segments $XA,XB, XC, YA,YB,YC$ divide the triangle $ABC$ into $5$ regions with equal areas. Furthermore, prove that all the segments $XY$ have a common point.
Let $ABC$ be an acute triangle with $AB = AC$. Let $D$ be the point on $BC$ such that $AD$ is perpendicular to $BC$. Let $O,H,G$ be the circumcenter, orthocenter and centroid of triangle $ABC$ respectively. Suppose that $2 \cdot OD = 23 \cdot HD$. Prove that $G$ lies on the incircle of triangle $ABC$.
Let $ a\geq 2$ be a real number; with the roots $ x_{1}$ and $ x_{2}$ of the equation $ x^2\minus{}ax\plus{}1\equal{}0$ we build the sequence with $ S_{n}\equal{}x_{1}^n \plus{} x_{2}^n$.
[b]a)[/b]Prove that the sequence $ \frac{S_{n}}{S_{n\plus{}1}}$, where $ n$ takes value from $ 1$ up to infinity, is strictly non increasing.
[b]b)[/b]Find all value of $ a$ for the which this inequality hold for all natural values of $ n$ $ \frac{S_{1}}{S_{2}}\plus{}\cdots \plus{}\frac{S_{n}}{S_{n\plus{}1}}>n\minus{}1$
We say that a finite set $\mathcal{S}$ of points in the plane is [i]balanced[/i] if, for any two different points $A$ and $B$ in $\mathcal{S}$, there is a point $C$ in $\mathcal{S}$ such that $AC=BC$. We say that $\mathcal{S}$ is [i]centre-free[/i] if for any three different points $A$, $B$ and $C$ in $\mathcal{S}$, there is no points $P$ in $\mathcal{S}$ such that $PA=PB=PC$.
(a) Show that for all integers $n\ge 3$, there exists a balanced set consisting of $n$ points.
(b) Determine all integers $n\ge 3$ for which there exists a balanced centre-free set consisting of $n$ points.
Proposed by Netherlands
Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that
\[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]
4. Suppose that four people A, B, C and D decide to play games of tennis
doubles. They might first play the team A and B against the team C
and D. Next A and C might play B and D. Finally A and D might play
B and C. The advantage of this arrangement is that two conditions are
satisfied.
(a) Each player is on the same team as each other player exactly once.
(b) Each player is on the opposing team to each other player exactly
twice.
Is it possible to arrange a collection of tennis matches satisfying both
condition (a) and condition (b) in the following circumstances?
(i) There are five players.
(ii) There are seven players.
(iii) There are nine players.