Found problems: 85335
Given an acute triangle $ABC$, let $P$ be an arbitrary point on segment $BC$. A line passing through $P$ and perpendicular to $AC$ intersects $AB$ at $P_b$. A line passing through $P$ and perpendicular to $AB$ intersects $AC$ at $P_c$. Prove that the circumcircle of triangle $AP_bP_c$ passes through a fixed point other than $A$ when $P$ varies on segment $BC$.
[i]Proposed by ltf0501[/i]
Let $x_1,x_2,\ldots ,x_{1997}$ be real numbers satisfying the following conditions:
i) $-\dfrac{1}{\sqrt{3}}\le x_i\le \sqrt{3}$ for $i=1,2,\ldots ,1997$;
ii) $x_1+x_2+\cdots +x_{1997}=-318 \sqrt{3}$ .
Determine (with proof) the maximum value of $x^{12}_1+x^{12}_2+\ldots +x^{12}_{1997}$ .
Show, for all positive integers $n = 1,2 , \dots ,$ that $14$ divides $ 3 ^ { 4 n + 2 } + 5 ^ { 2 n + 1 }$.
Show that
$$ \left\{ X\in\mathcal{M}_2\left( \mathbb{Z}_3 \right)\left| \begin{pmatrix} 1&1\\2&2 \end{pmatrix} X\begin{pmatrix} 1&2\\2&1 \end{pmatrix} =0 \right. \right\} $$
is a multiplicative ring.
[i]Cătălin Țigăeru[/i]
Let $\Delta ABC$ be a triangle with an inscribed circle centered at $I$. The line perpendicular to $AI$ at $I$ intersects $\odot (ABC)$ at $P,Q$ such that, $P$ lies closer to $B$ than $C$. Let $\odot (BIP) \cap \odot (CIQ) =S$. Prove that, $SI$ is the angle bisector of $\angle PSQ$
$(POL 3)$ Given a polynomial $f(x)$ with integer coefficients whose value is divisible by $3$ for three integers $k, k + 1,$ and $k + 2$. Prove that $f(m)$ is divisible by $3$ for all integers $m.$
[b]Problem 1.[/b] Points $D$ and $E$ are chosen on the sides $AB$ and $AC$, respectively, of a triangle $\triangle ABC$ such that $DE\parallel BC$. The circumcircle $k$ of triangle $\triangle ADE$ intersects the lines $BE$ and $CD$ at the points $M$ and $N$ (different from $E$ and $D$). The lines $AM$ and $AN$ intersect the side $BC$ at points $P$ and $Q$ such that $BC=2\cdot PQ$ and the point $P$ lies between $B$ and $Q$. Prove that the circle $k$ passes through the point of intersection of the side $BC$ and the angle bisector of $\angle BAC$.
[i]Nikolai Nikolov[/i]
Let $p>3$ is a prime number and $k=\lfloor\frac{2p}{3}\rfloor$. Prove that \[{p \choose 1}+{p \choose 2}+\cdots+{p \choose k}\] is divisible by $p^{2}$.
The sum of the first 2011 terms of a geometric series is 200. The sum of the first 4022 terms of the same series is 380. Find the sum of the first 6033 terms of the series.
Al, Betty, and Clare split $ \$1000$ among them to be invested in different ways. Each begins with a different amount. At the end of one year they have a total of $ \$1500$. Betty and Clare have both doubled their money, whereas Al has managed to lose $ \$100$. What was Al’s original portion?
$ \textbf{(A)}\ \$ 250 \qquad
\textbf{(B)}\ \$ 350 \qquad
\textbf{(C)}\ \$ 400 \qquad
\textbf{(D)}\ \$ 450 \qquad
\textbf{(E)}\ \$ 500$
Let $ a$, $ b$, $ c$ denote the sides of a triangle. Show that the quantity
\[ \frac{a}{b\plus{}c}\plus{}\frac{b}{c\plus{}a}\plus{}\frac{c}{a\plus{}b}\]
must lie between the limits $ 3/2$ and 2. Can equality hold at either limits?
Does the Cauchy problem $u|_{y=x^2}=1$, $(\nabla u)^2=1$ have a smooth solution in the domain $y\ge x^2$? In the domain $y\le x^2$?
Find all real polynomials $P(x)$ such that for every four distinct natural numbers $a, b, c, d$ such that $a^2 + b^2 + c^2 = 2d^2$ with $gcd(a, b, c, d) = 1$ the following equality holds:
$$2(P(d))^2 + 2P(ab + bc + ca) = (P(a + b + c))^2$$
.
In a competition, there were $2n+1$ teams. Every team plays exatly once against every other team. Every match finishes with the victory of one of the teams. We call cyclical a 3-subset of team ${ A,B,C }$ if $A$ won against $B$, $B$ won against $C$ , $C$ won against $A$.
(a) Find the minimum of cyclical 3-subset (depending on $n$);
(b) Find the maximum of cyclical 3-subset (depending on $n$).
Let $T$ be the answer to question $18$. Rectangle $ZOMR$ has $ZO = 2T$ and $ZR = T$. Point $B$ lies on segment $ZO$, $O'$ lies on segment $OM$, and $E$ lies on segment $RM$ such that $BR = BE = EO'$, and $\angle BEO' = 90^o$. Compute $2(ZO + O'M + ER)$.
PS. You had better calculate it in terms of $T$.
Show that \[(-1)^{\frac{p-1}{2}}{p-1 \choose{\frac{p-1}{2}}}\equiv 4^{p-1}\pmod{p^{3}}\] for all prime numbers $p$ with $p \ge 5$.
A [i]squidward[/i] is a piece that moves on a board in the following way: it advances three squares in one direction and then two squares in a perpendicular direction. For example, in the figure below, by making one move, the squidward can move to any of the $8$ squares indicated with arrows. Initially, there is one squidward on each of the $35$ squares of a $5 \times 7$ board. At the same time, each squidward makes exactly one move. What is the smallest possible number of empty squares after these moves?
[center][img]https://i.imgur.com/rqgG95C.png[/img][/center]
For two positive integers a and b, which are relatively prime, find all integer that can be the great common divisor of $a+b$ and $\frac{a^{2005}+b^{2005}}{a+b}$.
Given some cards with either "$-1$" or "$+1$" written on the opposite side. You are allowed to choose a triple of cards and ask about the product of the three numbers on the cards. What is the minimal number of questions allowing to determine all the numbers on the cards ...
a) for $30$ cards,
b) for $31$ cards,
c) for $32$ cards.
(You should prove, that you cannot manage with less questions.)
d) Fifty above mentioned cards are lying along the circumference. You are allowed to ask about the product of three consecutive numbers only. You need to determine the product af all the $50$ numbers. What is the minimal number of questions allowing to determine it?
A point $D$ is chosen on the side $AC$ of a triangle $ABC$ with $\angle C < \angle A < 90^\circ$ in such a way that $BD=BA$. The incircle of $ABC$ is tangent to $AB$ and $AC$ at points $K$ and $L$, respectively. Let $J$ be the incenter of triangle $BCD$. Prove that the line $KL$ intersects the line segment $AJ$ at its midpoint.
Let $P(x) = x^2 + ax + b$. The two zeros of $P$, $r_1$ and $r_2$, satisfy the equation $|r_1^2 -r_2^2| = 17$. Give that $a, b > 1$ and are both integers, find $P(1)$.
A set $S$ of rational numbers is ordered in a tree-diagram in such a way that each rational number $\frac{a}{b}$ (where $a$ and $b$ are coprime integers) has exactly two successors: $\frac{a}{a+b}$ and $\frac{b}{a+b}$. How should the initial element be selected such that this tree contains the set of all rationals $r$ with $0 < r < 1$? Give a procedure for determining the level of a rational number $\frac{p}{q}$ in this tree.
let $f(x) = \sum_{n=0}^{\infty} 2^{-n} ||2^n x||$ , where ||x|| is the distance between x and the closest integer to x. Are the level sets $\{ x \in [0,1] : f(x)=y \}$ Lebesgue measurable for almost all $y \in f(R)$?
Let $k$ be a positive integer. Prove that there exists a positive integer $\ell$ with the following property: if $m$ and $n$ are positive integers relatively prime to $\ell$ such that $m^m\equiv n^n \pmod{\ell}$, then $m\equiv n \pmod k$.
$100$ distinct natural numbers $a_1, a_2, a_3, \ldots, a_{100}$ are written on the board. Then, under each number $a_i$, someone wrote a number $b_i$, such that $b_i$ is the sum of $a_i$ and the greatest common factor of the other $99$ numbers. What is the least possible number of distinct natural numbers that can be among $b_1, b_2, b_3, \ldots, b_{100}$?