Found problems: 85335
Prove that for some positive integer \(N\), \(N\) points can be chosen on a circle such that there are at least \(1000N^2\) unordered quadruples \((A,B,C,D)\) of distinct selected points for which \(\displaystyle \frac{AC}{BC} = \frac{AD}{BD}\).
The side lengths of a triangle are the roots of a cubic polynomial with rational coefficients. Prove that the altitudes of this triangle are roots of a polynomial of sixth degree with rational coefficients.
Consider two circles $k_{1},k_{2}$ touching externally at point $T$. a line touches $k_{2}$ at point $X$ and intersects $k_{1}$ at points $A$ and $B$. Let $S$ be the second intersection point of $k_{1}$ with the line $XT$ . On the arc $\widehat{TS}$ not containing $A$ and $B$ is chosen a point $C$ . Let $\ CY$ be the tangent line to $k_{2}$ with $Y\in k_{2}$ , such that the segment $CY$ does not intersect the segment $ST$ . If $I=XY\cap SC$ . Prove that :
(a) the points $C,T,Y,I$ are concyclic.
(b) $I$ is the excenter of triangle $ABC$ with respect to the side $BC$.
Given a $m \times n$ rectangle where $m,n\geq 2023$. The square in the $i$-th row and $j$-th column is filled with the number $i+j$ for $1\leq i \leq m, 1\leq j \leq n$. In each move, Alice can pick a $2023 \times 2023$ subrectangle and add $1$ to each number in it. Alice wins if all the numbers are multiples of $2023$ after a finite number of moves. For which pairs $(m,n)$ can Alice win?
[i]Proposed by Boon Qing Hong[/i]
Let $S_5$ be the set of permutations of $\{1,2,3,4,5\}$, and let $C$ be the convex hull of the set
$$\{(\sigma(1),\sigma(2),\ldots,\sigma(5))\,|\,\sigma\in S_5\}.$$
Then $C$ is a polyhedron. What is the total number of $2$-dimensional faces of $C$?
The numbers $a_1,a_2$ and $a_3$ are distinct positive integers, such that
(i) $a_1$ is a divisor of $a_2+a_3+a_2a_3$;
(ii) $a_2$ is a divisor of $a_3+a_1+a_3a_1$;
(iii) $a_3$ is a divisor of $a_1+a_2+a_1a_2$.
Prove that $a_1,a_2$ and $a_3$ cannot all be prime.
Let $a, b, c$ be real numbers such that $a+b+c > 0$, $ab+bc+ca > 0$, $abc > 0$. Show that $a, b, c$ are all positive.
Rectangle $ ABCD$ has $ AB\equal{}8$ and $ BC\equal{}6$. Point $ M$ is the midpoint of diagonal $ \overline{AC}$, and E is on $ \overline{AB}$ with $ \overline{ME}\perp\overline{AC}$. What is the area of $ \triangle AME$?
$ \textbf{(A)}\ \frac{65}{8} \qquad
\textbf{(B)}\ \frac{25}{3} \qquad
\textbf{(C)}\ 9 \qquad
\textbf{(D)}\ \frac{75}{8} \qquad
\textbf{(E)}\ \frac{85}{8}$
Find all nonnegative integer numbers such that $7^x- 2 \cdot 5^y = -1$
On her blackboard, Alice has written $n$ integers strictly greater than $1$. Then, she can, as often as she likes, erase two numbers $a$ and $b$ such that $a \neq b$, and replace them with $q$ and $q^2$, where $q$ is the product of the prime factors of $ab$ (each prime factor is counted only once). For instance, if Alice erases the numbers $4$ and $6$, the prime factors of $ab = 2^3 \times 3$ and $2$ and $3$, and Alice writes $q = 6$ and $q^2 =36$.
Prove that, after some time, and whatever Alice's strategy is, the list of numbers written on the blackboard will never change anymore.
[i]Note: The order of the numbers of the list is not important.[/i]
Points $A,B,C$are chosen inside the triangle $ A_{1}B_{1}C_{1},$ so that the quadrilaterals $B_{1}CBC_{1}, C_{1}ACA_{1}$ and $A_{1}BAB_{1}$ are inscribed in the circles $\Omega _{A}, \Omega _{B}$ and $\Omega _{C},$ respectively. The circle $Y_{A}$ internally touches the circles $\Omega _{B}, \Omega _{C}$ and externally touches the circle $\Omega _{A}.$ The common interior tangent to the circles $Y_{A}$ and $\Omega _{A}$ intersects the line $BC$ at point $A'.$ Points $B'$ and $C'$ are analogously defined. Prove that points $A',B'$ and $C'$ are lying on the same line.
Let $\omega$ be a circle with diameter $\overline{AB},$ center $O,$ and cyclic quadrilateral $ABCD$ inscribed in it, with $C$ and $D$ on the same side of $\overline{AB}.$ Let $AB=20, BC=13, AD=7.$ Let $\overleftrightarrow{BC}$ and $\overleftrightarrow{AD}$ intersect at $E.$ Let the $E$-excircle of $ECD$ have its center at $L.$ Find $OL.$
A positive integer is called [i]monotonic[/i] if when written in base $10$, the digits are weakly increasing. Thus $12226778$ is monotonic. Note that a positive integer cannot have first digit $0$. Prove that for every positive integer $n$, there is an $n$-digit monotonic number which is a perfect square.
If $0<\theta<\pi$, then the maximum value of $\sin\frac{\theta}{2}(1+\cos\theta)$ is________.
Compute the value of
$$\sin^2\left(\frac{\pi}{7}\right) + \sin^2\left(\frac{3\pi}{7}\right) + \sin^2\left(\frac{5\pi}{7}\right).$$
Your answer should not involve any trigonometric functions.
[i]Proposed by Howard Halim[/i]
Find the minimum value of $\int_{-1}^1 \sqrt{|t-x|}\ dt$
Right isosceles triangle $T$ is placed in the first quadrant of the coordinate plane. Suppose that the projection of $T$ onto the $x$-axis has length $6$, while the projection of $T$ onto the $y$-axis has length $8$. What is the sum of all possible areas of the triangle $T$?
[asy]
import olympiad;
size(120);
defaultpen(linewidth(0.8));
pair A = (0.9,0.6), B = (1.7, 0.8), C = rotate(270, B)*A;
pair PAx = (A.x,0), PBx = (B.x,0), PAy = (0, A.y), PCy = (0, C.y);
draw(PAx--A--PAy^^PCy--C^^PBx--B, linetype("4 4"));
draw(rightanglemark(A,B,C,3));
draw(A--B--C--cycle);
draw((0,2)--(0,0)--(2,0),Arrows(size=8));
label("$6$",(PAx+PBx)/2,S);
label("$8$",(PAy+PCy)/2,W);
[/asy]
Let $a_1,a_2,\dots,a_n$ be positive real numbers whose product is $1$. Show that the sum \[\textstyle\frac{a_1}{1+a_1}+\frac{a_2}{(1+a_1)(1+a_2)}+\frac{a_3}{(1+a_1)(1+a_2)(1+a_3)}+\cdots+\frac{a_n}{(1+a_1)(1+a_2)\cdots(1+a_n)}\] is greater than or equal to $\frac{2^n-1}{2^n}$.
On a $9\times 9$ square lake composed of unit squares, there is a $2\times 4$ rectangular iceberg also composed of unit squares (it could be in either orientation; that is, it could be $4\times 2$ as well). The sides of the iceberg are parallel to the sides of the lake. Also, the iceberg is invisible. Lily is trying to sink the iceberg by firing missiles through the lake. Each missile fires through a row or column, destroying anything that lies in its row or column. In particular, if Lily hits the iceberg with any missile, she succeeds. Lily has bought $n$ missiles and will fire all $n$ of them at once. Let $N$ be the smallest possible value of $n$ such that Lily can guarantee that she hits the iceberg. Let $M$ be the number of ways for Lily to fire $N$ missiles and guarantee that she hits the iceberg. Compute $100M+N$.
[i]Proposed by Brandon Wang[/i]
Let $n\in\mathbb{N}$ be given. Prove that the following two conditions are equivalent:
$\quad(\text{i})\: n|a^n-a$ for any positive integer $a$;
$\quad(\text{ii})\:$ For any prime divisor $p$ of $n$, $p^2 \nmid n$ and $p-1|n-1$.
In triangles ABC and DEF, DE = 4AB, EF = 4BC, and F D = 4CA. The area of
DEF is 360 units more than the area of ABC. Compute the area of ABC.
Triangles $ABC$ and $DEF$ share circumcircle $\Omega$ and incircle $\omega$ so that points $A,F,B,D,C,$ and $E$ occur in this order along $\Omega$. Let $\Delta_A$ be the triangle formed by lines $AB,AC,$ and $EF,$ and define triangles $\Delta_B, \Delta_C, \ldots, \Delta_F$ similarly. Furthermore, let $\Omega_A$ and $\omega_A$ be the circumcircle and incircle of triangle $\Delta_A$, respectively, and define circles $\Omega_B, \omega_B, \ldots, \Omega_F, \omega_F$ similarly.
(a) Prove that the two common external tangents to circles $\Omega_A$ and $\Omega_D$ and the two common external tangents to $\omega_A$ and $\omega_D$ are either concurrent or pairwise parallel.
(b) Suppose that these four lines meet at point $T_A$, and define points $T_B$ and $T_C$ similarly. Prove that points $T_A,T_B$, and $T_C$ are collinear.
[i]Nikolai Beluhov[/i]
Let $k_1, k_2$ and $k_3$ be three circles with centers $O_1, O_2$ and $O_3$ respectively, such that no center is inside of the other two circles. Circles $k_1$ and $k_2$ intersect at $A$ and $P$, circles $k_1$ and $k_3$ intersect and $C$ and $P$, circles $k_2$ and $k_3$ intersect at $B$ and $P$. Let $X$ be a point on $k_1$ such that the line $XA$ intersects $k_2$ at $Y$ and the line $XC$ intersects $k_3$ at $Z$, such that $Y$ is nor inside $k_1$ nor inside $k_3$ and $Z$ is nor inside $k_1$ nor inside $k_2$.
a) Prove that $\triangle XYZ$ is simular to $\triangle O_1O_2O_3$
b) Prove that the $P_{\triangle XYZ} \le 4P_{\triangle O_1O_2O_3}$. Is it possible to reach equation?$
*Note: $P$ denotes the area of a triangle*
A rectangle is said to be inscribed in a parallelogram if its vertices lie one on each side of the parallelogram. On the larger side $AB$ of the parallelogram $ABCD$, find all those points $K$ that are the vertices of the rectangles inscribed in $ABCD$.
Each square in a $3\times 3$ grid of squares is colored red, white, blue, or green so that every $2\times 2$ square contains one square of each color. One such coloring is shown on the right below. How many different colorings are possible?\\
[asy]
size(8cm);
pen grey1, grey2, grey3;
grey1 = RGB(211, 211, 211);
grey2 = RGB(173, 173, 173);
grey3 = RGB(138, 138, 138);
for(int i = 0; i < 4; ++i) {
draw((i, 0)--(i, 3));
draw((0, i)--(3, i));
}
filldraw((5, 3)--(6, 3)--(6, 2)--(5, 2)--cycle, grey1);
label('B', (5.5, 2.5));
filldraw((6, 3)--(7, 3)--(7, 2)--(6, 2)--cycle, grey2);
label('R', (6.5, 2.5));
filldraw((7, 3)--(8, 3)--(8, 2)--(7, 2)--cycle, grey1);
label('B', (7.5, 2.5));
filldraw((5, 2)--(6, 2)--(6, 1)--(5, 1)--cycle, grey3);
label('G', (5.5, 1.5));
filldraw((6, 2)--(7, 2)--(7, 1)--(6, 1)--cycle, white);
filldraw((7, 2)--(8, 2)--(8, 1)--(7, 1)--cycle, grey3);
label('G', (7.5, 1.5));
filldraw((5, 1)--(6, 1)--(6, 0)--(5, 0)--cycle, grey2);
label('R', (5.5, 0.5));
filldraw((6, 1)--(7, 1)--(7, 0)--(6, 0)--cycle, grey1);
label('B', (6.5, 0.5));
filldraw((7, 1)--(8, 1)--(8, 0)--(7, 0)--cycle, grey2);
label('R', (7.5, 0.5));
[/asy]
$\textbf{(A) }24\qquad\textbf{(B) }48\qquad\textbf{(C) }60\qquad\textbf{(D) }72\qquad\textbf{(E) }96$