Found problems: 85335
Given is a triangle $ABC$ and a point $K$ within the triangle. The point $K$ is mirrored in the sides of the triangle: $P , Q$ and $R$ are the mirrorings of $K$ in $AB , BC$ and $CA$, respectively . $M$ is the center of the circle passing through the vertices of triangle $PQR$. $M$ is mirrored again in the sides of triangle $ABC$: $P', Q'$ and $R'$ are the mirror of $M$ in $AB$ respectively, $BC$ and $CA$.
a. Prove that $K$ is the center of the circle passing through the vertices of triangle $P'Q'R'$ .
b. Where should you choose $K$ within triangle $ABC$ so that $M$ and $K$ coincide? Prove your answer.
Let $ p$ be a prime number. Let $ h(x)$ be a polynomial with integer coefficients such that $ h(0),h(1),\dots, h(p^2\minus{}1)$ are distinct modulo $ p^2.$ Show that $ h(0),h(1),\dots, h(p^3\minus{}1)$ are distinct modulo $ p^3.$
Let $p$ be a fixed odd prime. A $p$-tuple $(a_1,a_2,a_3,\ldots,a_p)$ of integers is said to be [i]good[/i] if
[list]
[*] [b](i)[/b] $0\le a_i\le p-1$ for all $i$, and
[*] [b](ii)[/b] $a_1+a_2+a_3+\cdots+a_p$ is not divisible by $p$, and
[*] [b](iii)[/b] $a_1a_2+a_2a_3+a_3a_4+\cdots+a_pa_1$ is divisible by $p$.[/list]
Determine the number of good $p$-tuples.
At a party with $n$ people, it is known that among any $4$ people, there are either $3$ people who all know one another or $3$ people none of which knows another. Show that the $n$ people can be separated into two rooms, so that everyone in one room knows one another and no two people in the other room know each other.
The triangle $ABC$ has circumcentre $O$ and circumcircle $\Gamma$. Let $AI$ be a diameter of $\Gamma$. The ray $AI$ extends to intersect the circumcircle $\omega$ of $\triangle BOC$ for the second time at a point $P$.
Let $AD$ and $IQ$ be perpendicular to $BC$, with $D$ and $Q$ on $BC$. Let $M$ be the midpoint of $BC$.
(a) Prove that $|AD| \cdot |QI| = |CD| \cdot |CQ| = |BD| \cdot |BQ|$.
(b) Prove that $IM$ is parallel to $PD$.
Find all rational solutions to: \begin{eqnarray*} t^2 - w^2 + z^2 &=& 2xy \\ t^2 - y^2 + w^2 &=& 2xz \\ t^2 - w^2 + x^2 &=& 2yz . \end{eqnarray*}
Consider a rectangular table $2 \times n.$ Let every cell be dyed either by black or white color in a way that no $2\times 2$ square is completely black. Denote $P_n$ the number of such colorings. Prove that the number $P_{1989}$ is divisible by three and find the greatest power of three that divides them.
Evaluate the following integrals.
(1) Meiji University
$\int_{\frac{1}{e}}^{e}\frac{(\log x)^{2}}{x}dx.$
(2) Tokyo University of Science
$\int_{0}^{1}\frac{7x^{3}+23x^{2}+21x+15}{(x^{2}+1)(x+1)^{2}}dx.$
How many $5-$digit positive numbers which contain only odd numbers are there such that there is at least one pair of consecutive digits whose sum is $10$?
$
\textbf{(A)}\ 3125
\qquad\textbf{(B)}\ 2500
\qquad\textbf{(C)}\ 1845
\qquad\textbf{(D)}\ 1190
\qquad\textbf{(E)}\ \text{None of the preceding}
$
Given a triangle $ABC$ inscribed in $(O)$ with $2$ symmedians $AD, CF(D,F$ are on the sides $BC, AB,$ respectively$).$ The ray $DF$ intersects $(O)$ at $P.$ The line passing through $P$ and perpendicular to $OA$ intersects $AB,AC$ at $Q,R,$ respectively$.$
Compute the ratio $\dfrac{PR}{PQ}.$
Given a positive integer $k$, a pigeon and a seagull play a game on an $n\times n$ board. The pigeon goes first, and they take turns doing the operations. The pigeon will choose $m$ grids and lay an egg in each grid he chooses. The seagull will choose a $k\times k$ grids and eat all the eggs inside them. If at any point every grid in the $n\times n $ board has an egg in it, then the pigeon wins. Else, the seagull wins. For every integer $n\geq k$, find all $m$ such that the pigeon wins.
[i]Proposed by amano_hina[/i]
Determine all pairs $(f,g)$ of functions from the set of real numbers to itself that satisfy \[g(f(x+y)) = f(x) + (2x + y)g(y)\] for all real numbers $x$ and $y$.
[i]Proposed by Japan[/i]
In triangle $ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:
${{ \textbf{(A)}\ 1:3 \qquad\textbf{(B)}\ 3:4 \qquad\textbf{(C)}\ 4:3 \qquad\textbf{(D)}\ 3:1 }\qquad\textbf{(E)}\ 7:1 } $
Find the greatest integer which doesn't exceed $\frac{3^{100}+2^{100}}{3^{96}+2^{96}}$
(A)$81$
(B)$80$
(C)$79$
(D)$82$
Let $ABCD$ be a rhombus where $\angle DAB = 60^\circ$, and $P$ be the intersection between $AC$ and $BD$. Let $Q,R,S$ be three points on the boundary of $ABCD$ such that $PQRS$ is a rhombus. Prove that exactly one of $Q,R,S$ lies on one of $A,B,C,D$.
Let $ x_1, x_2, \ldots, x_n$ be positive real numbers with sum $ 1$. Find the integer part of:
$ E\equal{}x_1\plus{}\dfrac{x_2}{\sqrt{1\minus{}x_1^2}}\plus{}\dfrac{x_3}{\sqrt{1\minus{}(x_1\plus{}x_2)^2}}\plus{}\cdots\plus{}\dfrac{x_n}{\sqrt{1\minus{}(x_1\plus{}x_2\plus{}\cdots\plus{}x_{n\minus{}1})^2}}$
Find all sets of real numbers $\{a_1,a_2,\ldots, _{1375}\}$ such that
\[2 \left( \sqrt{a_n - (n-1)}\right) \geq a_{n+1} - (n-1), \quad \forall n \in \{1,2,\ldots,1374\},\]
and
\[2 \left( \sqrt{a_{1375}-1374} \right) \geq a_1 +1.\]
Find all positive intgers $a,b$ and $c$ such that $\frac{a+b}{a+c}=\frac{b+c}{b+a}$ and $ab+bc+ca$ is a prime number
Define for positive integer $n$, a function $f_n(x)=\frac{\ln x}{x^n}\ (x>0).$ In the coordinate plane, denote by $S_n$ the area of the figure enclosed by $y=f_n(x)\ (x\leq t)$, the $x$-axis and the line $x=t$ and denote by $T_n$ the area of the rectagle with four vertices $(1,\ 0),\ (t,\ 0),\ (t,\ f_n(t))$ and $(1,\ f_n(t))$.
(1) Find the local maximum $f_n(x)$.
(2) When $t$ moves in the range of $t>1$, find the value of $t$ for which $T_n(t)-S_n(t)$ is maximized.
(3) Find $S_1(t)$ and $S_n(t)\ (n\geq 2)$.
(4) For each $n\geq 2$, prove that there exists the only $t>1$ such that $T_n(t)=S_n(t)$.
Note that you may use $\lim_{x\to\infty} \frac{\ln x}{x}=0.$
Let $ABC$ be an acute triangle with circumcenter $O$ and circumcircle $\Omega$. Choose points $D, E$ from sides $AB, AC$, respectively, and let $\ell$ be the line passing through $A$ and perpendicular to $DE$. Let $\ell$ intersect the circumcircle of triangle $ADE$ and $\Omega$ again at points $P, Q$, respectively. Let $N$ be the intersection of $OQ$ and $BC$, $S$ be the intersection of $OP$ and $DE$, and $W$ be the orthocenter of triangle $SAO$.
Prove that the points $S$, $N$, $O$, $W$ are concyclic.
[i]Proposed by Li4 and me.[/i]
Which reaction has the most positive entropy change under standard conditions?
$ \textbf{(A)}\hspace{.05in}\ce{H2O}_{(g)}+\ce{CO}_{(g)} \rightarrow \ce{H2}_{(g)}+ \ce{CO2}_{(g)}\qquad$
$\textbf{(B)}\hspace{.05in}\ce{CaCO3}_{(s)} \rightarrow \ce{CaO}_{(s)} + \ce{CO2}_{(g)} \qquad$
$\textbf{(C)}\hspace{.05in}\ce{NH3}_{(g)} \rightarrow \ce{NH3}_{(aq)}\qquad$
$\textbf{(D)}\hspace{.05in}\ce{C8H18}_{(l)} \rightarrow \ce{C8H18}_{(s)}\qquad$
Show that there exists a constant $a > 1$ such that, for any positive integers $m$ and $n$, $\frac{m}{n} < \sqrt7$ implies that $$7-\frac{m^2}{n^2} \ge \frac{a}{n^2} .$$
An ant starts at the point $(1, 0)$. Each minute, it walks from its current position to one of the four adjacent lattice points until it reaches a point $(x, y)$ with $|x| + |y| \le 2$. What is the probability that the ant ends at the point $(1, 1)$?
Given two circles ${\omega _1},{\omega _2}$, $S$ denotes all $\Delta ABC$ satisfies that ${\omega _1}$ is the circumcircle of $\Delta ABC$, ${\omega _2}$ is the $A$- excircle of $\Delta ABC$ , ${\omega _2}$ touches $BC,CA,AB$ at $D,E,F$.
$S$ is not empty, prove that the centroid of $\Delta DEF$ is a fixed point.
Let $\Delta ABC$ be a triangle. The internal bisectors of angles $\angle CAB$ and $\angle ABC$ intersect segments $BC$, respectively $AC$ in $D$, respectively $E$. Prove that \[DE\leq (3-2\sqrt{2})(AB+BC+CA).\]