Found problems: 85335
A circle is inscribed in a triangle with side lengths $8$, $13$, and $17$. Let the segments of the side of length $8$, made by a point of tangency, be $r$ and $s$, with $r<s$. What is the ratio $r:s$?
${{ \textbf{(A)}\ 1:3 \qquad\textbf{(B)}\ 2:5 \qquad\textbf{(C)}\ 1:2 \qquad\textbf{(D)}\ 2:3 }\qquad\textbf{(E)}\ 3:4 } $
A semicircle with diameter $AB$ is given. Two non-intersecting circles $k_1$ and $k_2$ with different radii touch the diameter $AB$ and touch the semicircle internally at $C$ and $D$, respectively. An interior common tangent $t$ of $k_1$ and $k_2$ touches $k_1$ at $E$ and $k_2$ at $F$. Prove that the lines $CE$ and $DF$ intersect on the semicircle.
In triangle $ABC,\angle B > \angle C, T$ is the midpoint of arc $BAC$ of the circumcicle of $ABC$, and $I$ is the incentre of $ABC$. Let $E$ be point such that $\angle AEI = 90^0$ and $AE$ is parallel to $BC$. If $TE$ intersects the circumcircle of $ABC$ at $P(\ne T)$ and $\angle B = \angle IPB$, determine $\angle A$.
Assume that $f : [0,1)\to R$ is a function such that $f(x)-x^3$ and $f(x)-3x$ are both increasing functions. Determine if $f(x)-x^2-x$ is also an increasing function.
What is the smallest positive integer that is divisible by $225$ and that has ony the numbers one and zero as digits?
$a$, $b$, $c$, $d$ are positive real numbers satisfying the following condition:
\[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}=4\]
Prove that:
\[\sqrt[3]{\frac{a^{3}+b^{3}}{2}}+\sqrt[3]{\frac{b^{3}+c^{3}}{2}}+\sqrt[3]{\frac{c^{3}+d^{3}}{2}}+\sqrt[3]{\frac{d^{3}+a^{3}}{2}}\leq 2(a+b+c+d)-4\]
Some of $n$ towns are connected by two-way airlines. There are $m$ airlines in total. For $i = 1, 2, \cdots, n$, let $d_i$ be the number of airlines going from town $i$. If $1\le d_i \le 2010$ for each $i = 1, 2,\cdots, 2010$, prove that
\[\displaystyle\sum_{i=1}^n d_i^2\le 4022m- 2010n\]
Find all $n$ for which equality can be attained.
[i]Proposed by Aleksandar Ilic[/i]
Given a positive integer $k$, find all polynomials $P$ of degree $k$ with integer coefficients such that for all positive integers $n$ where all of $P(n)$, $P(2024n)$, $P(2024^2n)$ are nonzero, we have
$$\frac{\gcd(P(2024n), P(2024^2n))}{\gcd(P(n), P(2024n))}=2024^k.$$
[i]Allen Wang[/i]
In triangle $ABC$, angles $A$ and $B$ measure 60 degrees and 45 degrees, respectively. The bisector of angle $A$ intersects $\overline{BC}$ at $T$, and $AT=24.$ The area of triangle $ABC$ can be written in the form $a+b\sqrt{c},$ where $a$, $b$, and $c$ are positive integers, and $c$ is not divisible by the square of any prime. Find $a+b+c.$
Suppose $n$ houses are to be assigned to $n$ people. Each person ranks the houses in the order of preference, with no ties. After the assignment is made, it is observed that every other assignment would assign to at least one person a less preferred house. Prove that there is at least one person who received the house he/she preferred most under this assignment.
Let $x, y, z$ be positive reals. Prove that
$$\frac{xz}{x^2 + xy + y^2 + 6z^2} + \frac{zx}{z^2 + zy + y^2 + 6x^2} + \frac{xy}{x^2 + xz + z^2 + 6y^2} \le \frac13$$
In a triangle $ABC$ with $AB<AC<BC$, the perpendicular bisectors of $AC$ and $BC$ intersect $BC$ and $AC$ at $K$ and $L$, respectively. Let $O$, $O_1$, and $O_2$ be the circumcentres of triangles $ABC$, $CKL$, and $OAB$, respectively. Prove that $OCO_1O_2$ is a parallelogram.
A natural number is called [i]good[/i] if it can be represented as sum of two coprime natural numbers, the first of which decomposes into odd number of primes (not necceserily distinct) and the second to even. Prove that there exist infinity many $n$ with $n^4$ being good.
Let $a, b, c, n$ be four integers, where n$\ge 2$, and let $p$ be a prime dividing both $a^2+ab+b^2$ and $a^n+b^n+c^n$, but not $a+b+c$. for instance, $a \equiv b \equiv -1 (mod \,\, 3), c \equiv 1 (mod \,\, 3), n$ a positive even integer, and $p = 3$ or $a = 4, b = 7, c = -13, n = 5$, and $p = 31$ satisfy these conditions. Show that $n$ and $p - 1$ are not coprime.
Let $(x_j,y_j)$, $1\le j\le 2n$, be $2n$ points on the half-circle in the upper half-plane. Suppose $\sum_{j=1}^{2n}x_j$ is an odd integer. Prove that $\displaystyle{\sum_{j=1}^{2n}y_j \ge 1}$.
Let $ n$ and $ k$ be positive integers such that $ \frac{1}{2} n < k \leq \frac{2}{3} n.$ Find the least number $ m$ for which it is possible to place $ m$ pawns on $ m$ squares of an $ n \times n$ chessboard so that no column or row contains a block of $ k$ adjacent unoccupied squares.
Suppose $a$, $b$, $c$, and $d$ are distinct positive integers such that $a^b$ divides $b^c$, $b^c$ divides $c^d$, and $c^d$ divides $d^a$.
[list](a) Is it possible to determine which of the numbers $a$, $b$, $c$, $d$ is the smallest?
(b) Is it possible to determine which of the numbers $a$, $b$, $c$, $d$ is the largest?[/list]
What is the maximum distance between the particle and the origin?
(A) $\text{2.00 m}$
(B) $\text{2.50 m}$
(C) $\text{3.50 m}$
(D) $\text{5.00 m}$
(E) $\text{7.00 m}$
Find all integers that can be written in the form $\frac{(x+y+z)^2}{xyz}$, where $x,y,z$ are positive integers.
Two circles intersect at points $P$ and $Q$. Point $A$ lies on the first circle, but outside the second. Lines $AP$ and $AQ$ intersect the second circle at points $B$ and $C$, respectively. Indicate the position of point $A$ at which triangle $ABC$ has the largest area.
(D. Prokopenko)
Let $a,b,c$ be positive real numbers such that $a+b+c=3$. Show that
$$\frac{a}{1+b^2}+\frac{b}{1+c^2}+\frac{c}{1+a^2}\geq \frac{3}{2}.$$
China Mathematical Olympiad 2018 Q6
Given the positive integer $n ,k$ $(n>k)$ and $ a_1,a_2,\cdots ,a_n\in (k-1,k)$ ,if positive number $x_1,x_2,\cdots ,x_n$ satisfying:For any set $\mathbb{I} \subseteq \{1,2,\cdots,n\}$ ,$|\mathbb{I} |=k$,have $\sum_{i\in \mathbb{I} }x_i\le \sum_{i\in \mathbb{I} }a_i$ , find the maximum value of $x_1x_2\cdots x_n.$
A regular hexagon of side length $1{ }$ is inscribed in a circle. Each minor arc of the circle determined by a side of the hexagon is reflected over that side. What is the area of the region bounded by these $6$ reflected arcs?
$(\textbf{A})\: \frac{5\sqrt{3}}{2} - \pi\qquad(\textbf{B}) \: 3\sqrt{3}-\pi\qquad(\textbf{C}) \: 4\sqrt{3}-\frac{3\pi}{2}\qquad(\textbf{D}) \: \pi - \frac{\sqrt{3}}{2}\qquad(\textbf{E}) \: \frac{\pi + \sqrt{3}}{2}$
It is known that the sequence $\{a_n\}$ satisfies $a_1=2$, $a_n=2^{2n}a_{n-1}+n\cdot 2^{n^2}$, $(n \ge 2)$, find the general term of $a_n$.
A ball of mass $M$ and radius $R$ has a moment of inertia of $I=\frac{2}{5}MR^2$. The ball is released from rest and rolls down the ramp with no frictional loss of energy. The ball is projected vertically upward off a ramp as shown in the diagram, reaching a maximum height $y_{max}$ above the point where it leaves the ramp. Determine the maximum height of the projectile $y_{max}$ in terms of $h$.
[asy]
size(250);
import roundedpath;
path A=(0,0)--(5,-12)--(20,-12)--(20,-10);
draw(roundedpath(A,1),linewidth(1.5));
draw((25,-10)--(12,-10),dashed+linewidth(0.5));
filldraw(circle((1.7,-1),1),lightgray);
draw((25,-1)--(-1.5,-1),dashed+linewidth(0.5));
draw((23,-9.5)--(23,-1.5),Arrows(size=5));
label(scale(1.1)*"$h$",(23,-6.5),2*E);
[/asy]
(A) $h$
(B) $\frac{25}{49}h$
(C) $\frac{2}{5}h$
(D) $\frac{5}{7}h$
(E) $\frac{7}{5}h$