Found problems: 85335
Let $\triangle ABC$ be an acute triangle with $AB =AC$ and let $D$ be a point on the side $BC$. The circle with centre $D$ passing through $C$ intersects $\odot(ABD)$ at points $P$ and $Q$, where $Q$ is the point closer to $B$. The line $BQ$ intersects $AD$ and $AC$ at points $X$ and $Y$ respectively. Prove that quadrilateral $PDXY$ is cyclic.
Prove that from a set of ten distinct two-digit numbers, it is always possible to find two disjoint subsets whose members have the same sum.
Prove that the equation $x+x^2=y+y^2+y^3$ do not have any solutions in positive integers.
Consider a sequence of positive integers $a_1, a_2, a_3, . . .$ such that for $k \geq 2$ we have $a_{k+1} =\frac{a_k + a_{k-1}}{2015^i},$ where $2015^i$ is the maximal power of $2015$ that divides $a_k + a_{k-1}.$ Prove that if this sequence is periodic then its period is divisible by $3.$
Let $A$ and $C$ be two points on a circle $X$ so that $AC$ is not diameter and $P$ a segment point on $AC$ different from its middle. The circles $c_1$ and $c_2$, inner tangents in $A$, respectively $C$, to circle $X$, pass through the point $P$ ¸ and intersect a second time at point $Q$. The line $PQ$ intersects the circle $X$ in points $B$ and $D$. The circle $c_1$ intersects the segments $AB$ and $AD$ in $K$, respectively $N$, and circle $c_2$ intersects segments $CB$ and ¸ $CD $ in $L$, respectively $M$. Show that: a) the $KLMN$ quadrilateral is isosceles trapezoid; b) $Q$ is the middle of the segment $BD$.
Proposed by Thanos Kalogerakis
Right triangle ABC with a right angle at A has AB = 20 and AC = 15. Point D is on AB with BD = 2. Points E and F are placed on ray CA and ray CB, respectively, such that CD is a median of $\triangle$ CEF. Find the area of $\triangle$CEF.
Find the number of ten-digit natural numbers (which do not start with zero) containing no block $ 1991$.
Find the number of pairs of integer solutions $(x,y)$ that satisfy the equation \[(x-y+2)(x-y-2) = -(x-2)(y-2)\]
Judge the following statements. If true, prove it; If false, give a counter-example.
[b](a)[/b] $P,Q$ are two different points on the sime side of line $l$. There exists two different circles, passing $P,Q$, and tangent to line $l$.
[b](b)[/b] If $a>0,b>0,a\neq1,b\neq1$, then$\log_ab+\log_ba\geq2$.
[b](c)[/b] $A,B$ are two sets of points on coordinate plane. $C_r=\{(x,y)|x^2+y^2\leq r^2\}$. For any $r\geq0$, $C_r\cup A\subset C_r\cup B$, then $A\subset B$.
Prove that for all integers $ m$ and $ n$, the inequality
\[ \dfrac{\phi(\gcd(2^m \plus{} 1,2^n \plus{} 1))}{\gcd(\phi(2^m \plus{} 1),\phi(2^n \plus{} 1))} \ge \dfrac{2\gcd(m,n)}{2^{\gcd(m,n)}}\]
holds.
[i]Nanang Susyanto, Jogjakarta [/i]
Prove that infinitely many positive integers cannot be written in the form \[{x_{1}}^{3}+{x_{2}}^{5}+{x_{3}}^{7}+{x_{4}}^{9}+{x_{5}}^{11},\] where $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\in \mathbb{N}$.
Each different letter in the following addition represents a different decimal digit. The sum is a six-digit integer whose digits are all equal.
$$\begin{tabular}{ccccccc}
& P & U & R & P & L & E\\
+ & & C & O & M & E & T \\
\hline
\\
\end{tabular}$$
Find the greatest possible value that the five-digit number $COMET$ could represent.
Point $O$ is the foot of the altitude of a quadrilateral pyramid. A sphere with center $O$ is tangent to all lateral faces of the pyramid. Points $A,B,C,D$ are taken on successive lateral edges so that segments $AB$, $BC$, and $CD$ pass through the three corresponding tangency points of the sphere with the faces. Prove that the segment $AD$ passes through the fourth tangency point
Let $ABC$ be an acute triangle for which $AB \neq AC$, and let $O$ be its circumcenter. Line $AO$ meets the circumcircle of $ABC$ again in $D$, and the line $BC$ in $E$. The circumcircle of $CDE$ meets the line $CA$ again in $P$. The lines $PE$ and $AB$ intersect in $Q$. Line passing through $O$ parallel to the line $PE$ intersects the $A$-altitude of $ABC$ in $F$.
Prove that $FP = FQ$.
Consider a sequence of positive integers with total sum $20$ such that no number and no sum of a set of consecutive numbers is equal to $3$. Is it possible for such a sequence to contain more than $10$ numbers?
(Alexandr Shapovalov)
Let $ABCD$ be a square with side $4$. Find, with proof, the biggest $k$ such that no matter how we place $k$ points into $ABCD$, such that they are on the interior but not on the sides, we always have a square with sidr length $1$, which is inside the square $ABCD$, such that it contains no points in its interior(they can be on the sides).
Determine the minimal and maximal values the expression $\frac{|a+b|+|b+c|+|c+a|}{|a|+|b|+|c|}$ can take, where $a,b,c$ are real numbers.
Let $a, b, c$ be positive real numbers such that $a^2 + b^2 + c^2 = 3$. Prove that
$$\frac{1}{1+2ab}+\frac{1}{1+2bc}+\frac{1}{1+2ca}\ge 1$$
$n$ is a natural number. We shall call a permutation $a_1,\dots,a_n$ of $1,\dots,n$ a quadratic(cubic) permutation if $\forall 1\leq i \leq n-1$ we have $a_ia_{i+1}+1$ is a perfect square(cube).
$(a)$ Prove that for infinitely many natural numbers $n$ there exists a quadratic permutation.
$(b)$ Prove that for no natural number $n$ exists a cubic permutation.
An infinite sequence of integers $a_1,a_2,a_3, ...$ is given with $a_1 = 0$ and further holds for every natural number $n$ that $a_{n+1} = a_n - n$ if $a_n \ge n$ and $a_{n+1} = a_n + n$ if $a_n < n$ .
(a) Prove that there are infinitely many numbers in the sequence equal to $0$.
(b) Express in terms of $k$ the ordinal number of the $k^e$ number from the sequence, which is equal to $0$.
Find all functions $f:R\to R$ such that for any real $x, y$ , $$f(x+2^y)=f(2^x)+f(y)$$
Suppose that $a_1,a_2,\cdots,a_{100}\in\mathbb{R}^+$ such that $a_i\geq a_{101-i}\,(i=1,2,\cdots,50).$
Let $x_k=\frac{ka_{k+1}}{a_1+a_2+\cdots+a_k}\,(k=1,2,\cdots,99).$ Prove that
$$x_1x_2^2\cdots x_{99}^{99}\leq 1.$$
For each positive integer $k$, let $g(k)$ be the maximum possible number of points in the plane such that pairwise distances between these points have only $k$ different values. Prove that there exists $k$ such that $g(k) > 2k + 2020$.
There are $2000$ components in a circuit, every two of which were initially joined by a wire. The hooligans Vasya and Petya cut the wires one after another. Vasya, who starts, cuts one wire on his turn, while Petya cuts two or three. The hooligan who cuts the last wire from some component loses. Who has the winning strategy?
We call a pentagon [i]distinguished [/i] if either all side lengths or all angles are equal. We call it [i]very distinguished[/i] if in addition two of the other parts are equal. i.e. $5$ sides and $2$ angles or $2$ sides and $5$ angles.Show that every very distinguished pentagon has an axis of symmetry.