Found problems: 85335
Let $ABC$ be an acute triangle. The points $M$ and $N$ are midpoints of $AB$ and $BC$ respectively, and $BH$ is an altitude of $ABC$. The circumcircles of $AHN$ and $CHM$ meet in $P$ where $P\ne H$. Prove that $PH$ passes through the midpoint of $MN$.
[i]V. Filimonov[/i]
Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that $(f(x + y))^2 = f(x^2) + f(y^2)$ for all $x, y \in \mathbb{Z}$.
Let $n > 3$ be a positive integer. Suppose that $n$ children are arranged in a circle, and $n$ coins are distributed between them (some children may have no coins). At every step, a child with at least 2 coins may give 1 coin to each of their immediate neighbors on the right and left. Determine all initial distributions of the coins from which it is possible that, after a finite number of steps, each child has exactly one coin.
Let $f$ be a polynomial with degree at most $n-1$. Show that
$$
\sum_{k=0}^n\left(\begin{array}{l}
n \\
k
\end{array}\right)(-1)^k f(k)=0
$$
Let $S$ be the statement
"If the sum of the digits of the whole number $n$ is divisible by 6, then $n$ is divisible by 6."
A value of $n$ which shows $S$ to be false is
$ \textbf{(A)}\ 30 \qquad
\textbf{(B)}\ 33 \qquad
\textbf{(C)}\ 40 \qquad
\textbf{(D)}\ 42 \qquad
\textbf{(E)}\ \text{None of these} $
On complex plane, path equation of moving point $Z_1$ is $|Z_1-Z_0|=|Z_1|$, where $Z_0(Z_0\neq0)$ is a fixed point. Another moving point $Z$ satisfies that $ZZ_1=-1$. Find the path of $Z$ and describe its location and shape.
For a set $A$, we define $A + A = \{a + b: a, b \in A \}$. Determine whether there exists a set $A$ of positive integers such that $$\sum_{a \in A} \frac{1}{a} = +\infty \quad \text{and} \quad \lim_{n \rightarrow +\infty} \frac{|(A+A) \cap \{1,2,\cdots,n \}|}{n}=0.$$
[hide=Note]Google translated from [url=http://ciim.uan.edu.co/ciim-2020-pruebas-virtuales/pruebas-virtuales]http://ciim.uan.edu.co/ciim-2020-pruebas-virtuales/pruebas-virtuales[/url][/hide]
Determine the locus of vertices of triangles which have prescribed orthocenter and center of circumcircle.
Find all functions $f: \mathbb{Z}\to \mathbb{Z}$ such that for all $m,n\in \mathbb{Z}$: \[f(m+f(n)) = f(m)+n.\]
[b]a)[/b] Show that the sum of the squares of the minimum distances from a point that is situated on a sphere to the faces of the cube that circumscribe the sphere doesn't depend on the point.
[b]b)[/b] Show that the sum of the cubes of the minimum distances from a point that is situated on a sphere to the faces of the cube that circumscribe the sphere doesn't depend on the point.
[i]Alexandru Sergiu Alamă[/i]
Find all functions $f$ from integers to integers such that
\[ f(m+n) + f(m-n) - 2f(m) = 6mn^2\]
for all integers $m$ and $n$.
Let $ABCD$ be a square and let $\ell$ be a line. Let $M$ be the centre of the square. The diagonals of the square have length 2 and the distance from $M$ to $\ell$ exceeds 1. Let $A',B',C',D'$ be the orthogonal projections of $A,B,C,D$ onto $\ell$. Suppose that one rotates the square, such that $M$ is invariant. The positions of $A,B,C,D,A',B',C',D'$ change. Prove that the value of $AA'^2 + BB'^2 + CC'^2 + DD'^2$ does not change.
If you take a subset of $4002$ numbers from the whole numbers $1$ to $6003$, then there is always a subset of $2001$ numbers within that subset with the following property:
If you order the $2001$ numbers from small to large, the numbers are alternately even and odd (or odd and even).
Prove this.
Let $ABC$ be any triangle and $D, E, F$ the midpoints of $BC, CA, AB$. The medians $AD, BE$ and $CF$ intersect at point $S$. At least two of the quadrilaterals $AF SE, BDSF, CESD$ are cyclic. Show that the triangle $ABC$ is equilateral.
There is an $n\times n$ chessboard where $n\geq 4$ is a positive even number. The cells of the chessboard are coloured black and white such that adjacent cells sharing a common side have different colours. Let $A$ and $B$ be two interior cells (which means cells not lying on an edge of the chessboard) of distinct colours. Prove that a chess piece can move from $A$ to $B$ by moving across adjacent cells such that every cell of the chessboard is passed through exactly once.
Let $ n$ be a positive integer. We consider the set $ P$ of all partitions of $ n$ into a sum of positive integers (the order is irrelevant). For every partition $ \alpha$, let $ a_{k}(\alpha)$ be the number of summands in $ \alpha$ that are equal to $ k, k = 1,2,...,n.$ Prove that
$ \sum_{\alpha\in P}\frac{1}{1^{a_{1}(\alpha)}a_{1}(\alpha)!\cdot 2^{a_{2}(\alpha)}a_{2}(\alpha)!...n^{a_{n}(\alpha)}a_{n}(\alpha)!}=1.$
Let $a_1,a_2,...,a_{15}$ be positive integers for which the number $a_k^{k+1} - a_k$ is not divisible by $17$ for any $k = 1,...,15$. Show that there are integers $b_1,b_2,...,b_{15}$ such that:
(i) $b_m - b_n$ is not divisible by $17$ for $1 \le m < n \le 15$, and
(ii) each $b_i$ is a product of one or more terms of $(a_i)$.
Let $\triangle ABC$ be a right triangle with right angle at $C$. Let $D$ and $E$ be points on $\overline{AB}$ with $D$ between $A$ and $E$ such that $\overline{CD}$ and $\overline{CE}$ trisect $\angle C$. If $\frac{DE}{BE} = \frac{8}{15}$, then $\tan B$ can be written as $\frac{m\sqrt{p}}{n}$, where $m$ and $n$ are relatively prime positive integers, and $p$ is a positive integer not divisible by the square of any prime. Find $m+n+p$.
How many numbers $\overline{abcd}$ with different digits satisfy the following property:
if we replace the largest digit with the digit $1$ results in a multiple of $30$?
Find all functions $f: \mathbb{Q}[x] \to \mathbb{R}$ such that:
(a) for all $P, Q \in \mathbb{Q}[x]$, $f(P \circ Q) = f(Q \circ P);$
(b) for all $P, Q \in \mathbb{Q}[x]$ with $PQ \neq 0$, $f(P\cdot Q) = f(P) + f(Q).$
($P \circ Q$ indicates $P(Q(x))$.)
Let $\gamma,\gamma_0,\gamma_1,\gamma_2$ be four circles in plane,such that $\gamma_i$ is interiorly tangent to $\gamma$ in point $A_i$,and $\gamma_i$ and $\gamma_{i+1}$ are exteriorly tangent in point $B_{i+2}$,$i=0,1,2$(the indexes are reduced modulo $3$).The tangent in $B_i$,common for circles $\gamma_{i-1}$ and $\gamma_{i+1}$,intersects circle $\gamma$ in point $C_i$,situated in the opposite semiplane of $A_i$ with respect to line $A_{i-1}A_{i+1}$.Prove that the three lines $A_iC_i$ are concurrent.
Let $ABC$ be a triangle with circumcenter $O$. Point $X$ is the intersection of the parallel line from $O$ to $AB$ with the perpendicular line to $AC$ from $C$. Let $Y$ be the point where the external bisector of $\angle BXC$ intersects with $AC$. Let $K$ be the projection of $X$ onto $BY$. Prove that the lines $AK, XO, BC$ have a common point.
Which of the following is equivalent to $$(2+3)(2^2+3^2)(2^4+3^4)(2^8+3^8)(2^{16}+3^{16})(2^{32}+3^{32})(2^{64}+3^{64})?$$
$\textbf{(A) }3^{127}+2^{127} \qquad \textbf{(B) }3^{127}+2^{127}+2\cdot 3^{63}+3\cdot 2^{63} \qquad \textbf{(C) }3^{128}-2^{128} \qquad \textbf{(D) }3^{128}+2^{128} \qquad \textbf{(E) }5^{127}$
Given that $40!=\overline{abcdef283247897734345611269596115894272pqrstuvwx}$, find $a,b,c,d,e,f,p,q,r,s,t,u,v,w,x$.
Let $D$ be a non-empty subset of positive integers and let $d$ be the greatest common divisor of $D$, and let $d\mathbb{Z}=[dn: n \in \mathbb{Z} ]$. Prove that there exists a bijection $f: \mathbb{Z} \rightarrow d\mathbb{Z} $ such that $| f(n+1)-f(n)|$ is member of $D$ for every integer $n$.