Found problems: 85335
Let $ ABC$ be a triangle with circumcentre $ O$. The points $ P$ and $ Q$ are interior points of the sides $ CA$ and $ AB$ respectively. Let $ K,L$ and $ M$ be the midpoints of the segments $ BP,CQ$ and $ PQ$. respectively, and let $ \Gamma$ be the circle passing through $ K,L$ and $ M$. Suppose that the line $ PQ$ is tangent to the circle $ \Gamma$. Prove that $ OP \equal{} OQ.$
[i]Proposed by Sergei Berlov, Russia [/i]
Let $A,B$ be two fixed points on a plane and $\Omega$ a fixed semicircle arc with diameter $AB$. Let $T$ be another fixed point on $\Omega$, and $\omega$ a fixed circle that passes through $A$ and $T$ and has its center in $\Delta ABT$. Let $P$ be a moving point on the arc $TB$ (endpoints excluded), and $C,D$ be two moving points on $\omega$ such that $C$ lies on segment $AP$, $C,D$ lies on different sides of line $AB$ and $CD\ \bot \ AB$. Denote the circumcenter of $\Delta CDP$ of $K$. Prove that
(i) $K$ lies on the circumcircle of $\Delta TDP$.
(ii) $K$ is a fixed point.
How many three-digit positive integers have digits which sum to a multiple of $10$?
For positive integers $a$ and $b$, if the expression $\frac{a^2+b^2}{(a-b)^2}$ is an integer, prove that the expression $\frac{a^3+b^3}{(a-b)^3}$ is an integer as well.
An American traveling in Italy wishes to exchange American money (dollars) for Italian money (lire). If 3000 lire = $ \$ 1.60$, how much lire will the traveler receive in exchange for $ \$ 1.00$?
$\text{(A)}\ 180 \qquad \text{(B)}\ 480 \qquad \text{(C)}\ 1800 \qquad \text{(D)}\ 1875 \qquad \text{(E)}\ 4875$
Let $E$ be the midpoint of the side $CD$ of a square $ABCD$. Consider the point $M$ inside the square such that $\angle MAB = \angle MBC = \angle BME = x$. Find the angle $x$.
Let $ABCD$ be a convex quadrilateral. Determine all points $M$, which lie inside $ABCD$, such that the areas of $ABCM$ and $AMCD$ are equal.
Let triangle $ABC$ have side lengths$ AB = 19$, $BC = 180$, and $AC = 181$, and angle measure $\angle ABC = 90^o$. Let the midpoints of $AB$ and $BC$ be denoted by $M$ and $N$ respectively. The circle centered at $ M$ and passing through point $C$ intersects with the circle centered at the $N$ and passing through point $A$ at points $D$ and $E$. If $DE$ intersects $AC$ at point $P$, find min $(DP,EP)$.
Positive integers $x>1$ and $y$ satisfy an equation $2x^2-1=y^{15}$. Prove that 5 divides $x$.
Let $n$ be a positive integer and $A$ a set of integers such that the set $\{x = a + b\ |\ a, b \in A\}$ contains $\{1^2, 2^2, \dots, n^2\}$. Prove that there is a positive integer $N$ such that if $n \ge N$, then $|A| > n^{0.666}$.
Let $A, B, C$ and $D$ denote the digits in a four-digit number $n = ABCD$. Determine the least $n$ greater than $2017$ satisfying that there exists an integer $x$ such that $$x =\sqrt{A +\sqrt{B +\sqrt{C +\sqrt{D + x}}}}.$$
Prove that if equations $$x^2 + mx + n = 0
\,\,\,\, and\,\, \,\,
x^2 + px + q = 0$$
have a common root, there is a relationship between the coefficients of these equations
$$ (n - q)^2 - (m - p) (np - mq) = 0.$$
The complex numbers $z$ and $w$ satisfy $z^{13} = w$, $w^{11} = z$, and the imaginary part of $z$ is $\sin\left(\frac{m\pi}n\right)$ for relatively prime positive integers $m$ and $n$ with $m < n$. Find $n$.
Let $a,b,c$ be real numbers. Show that $\sqrt[3]{a} + \sqrt[3]{b} +\sqrt[3]{c} = \sqrt[3]{a+b+c}$ if and only if $ a^3 + b^3 + c^3 = (a + b + c)^3 $
A polyhedron has $7n$ faces. Show that there exist $n + 1$ of the polyhedron's faces that all have the same number of edges.
The [i]Cantor set[/i] is defined as the set of real numbers $x$ such that $0 \le x < 1$ and the digit $1$ does not appear in the base-$3$ expansion of $x.$ Two numbers are uniformly and independently selected at random from the Cantor set. Compute the expected value of their difference.
(Formally, one can pick a number $x$ uniformly at random from the Cantor set by first picking a real number $y$ uniformly at random from the interval $[0, 1)$, writing it out in binary, reading its digits as if they were in base-$3,$ and setting $x$ to $2$ times the result.)
All faces of the tetrahedron $ABCD$ are acute-angled. Take a point $X$ in the interior of the segment $AB$, and similarly $Y$ in $BC, Z$ in $CD$ and $T$ in $AD$.
[b]a.)[/b] If $\angle DAB+\angle BCD\ne\angle CDA+\angle ABC$, then prove none of the closed paths $XYZTX$ has minimal length;
[b]b.)[/b] If $\angle DAB+\angle BCD=\angle CDA+\angle ABC$, then there are infinitely many shortest paths $XYZTX$, each with length $2AC\sin k$, where $2k=\angle BAC+\angle CAD+\angle DAB$.
For every non-negative integer $i$, define the number $M(i)$ as follows:
write $i$ down as a binary number, so that we have a string of zeroes and ones, if the number of ones in this string is even, then set $M(i) = 0$, otherwise set $M(i) = 1$. (The first terms of the sequence $M(i)$, $i = 0, 1, 2, ...$ are $0, 1, 1, 0, 1, 0, 0, 1,...$ )
(a) Consider the finite sequence $M(O), M(1), . . . , M(1000) $.
Prove that there are at least $320$ terms in this sequence which are equal to their neighbour on the right : $M(i) = M(i + 1 )$ .
(b) Consider the finite sequence $M(O), M(1), . . . , M(1000000)$ .
Prove that the number of terms $M(i)$ such that $M(i) = M(i +7)$ is at least $450000$.
(A Kanel)
There is a board with three rows and $2023$ columns. In the first row the numbers are written from $1$ to $2023$, ordered from least to greatest. The devil writes those same numbers in the boxes in the second row, but ordered to his choice. Then, in each box in the third row he writes the difference between the two numbers already written in his own column (the largest minus the smallest). For example, if the first two boxes of a column are the numbers $21$ and $198$, in the third box it is written $198-21 = 177$. Explain why, no matter how the devil completed the second row of the board, it will never happen that multiplying them $2023$ numbers in the third row the result is odd.
Let a convex quadrilateral $ ABCD$ fixed such that $ AB \equal{} BC$, $ \angle ABC \equal{} 80, \angle CDA \equal{} 50$. Define $ E$ the midpoint of $ AC$; show that $ \angle CDE \equal{} \angle BDA$
[i](Paolo Leonetti)[/i]
$N$ cossacks split into $3$ groups to discuss various issues with their friends. Cossack Taras moved from the first group to the second, cossack Andriy moved from the second to the third, and cossack Ostap - from the third group to the first. It turned out that the average height of the cossacks in the first group decreased by $8$ cm, while in the second and third groups it increased by $5$ cm and $8$ cm, respectively.
What is $N$, if it is known that there were $9$ cossacks in the first group?
Can the cells of a $2002 \times 2002$ table be filled with the numbers from $1$ to $2002^2$ (one per cell) so that for any cell we can find three numbers $a, b, c$ in the same row or column (or the cell itself) with $a = bc$?
A convex $1993$-gon is divided into convex $7$-gons. Prove that there are $3$ neighbouring sides of the $1993$-gon belonging to one such $7$-gon. (A vertex of a $7$-gon may not be positioned on the interior of a side of the $1993$-gon, and two $7$-gons either have no common points, exactly one common vertex or a complete common side.)
(A Kanel-Belov)
Consider the function $f (x) = (x - F_1)(x - F_2) ...(x -F_{3030})$ with $(F_n)$ is the Fibonacci sequence, which defined as $F_1 = 1, F_2 = 2$, $F_{n+2 }=F_{n+1} + F_n$, $n \ge 1$. Suppose that on the range $(F_1, F_{3030})$, the function $|f (x)|$ takes on the maximum value at $x = x_0$. Prove that $x_0 > 2^{2018}$.
Find the largest prime number $p<1000$ for which there exists a complex number $z$ satisfying [list]
[*] the real and imaginary part of $z$ are both integers;
[*] $|z|=\sqrt{p}$, and
[*] there exists a triangle whose three side lengths are $p$, the real part of $z^{3}$, and the imaginary part of $z^{3}$.
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