Found problems: 85335
Let $ABCD$ be a convex quadrilateral with $AC = 7$ and $BD = 17$. Let $M$, $P$, $N$, $Q$ be the midpoints of sides $AB$, $BC$, $CD$, $DA$ respectively. Compute $MN^2 + PQ^2$
[color = red]The official problem statement does not have the final period.[/color]
Let $n\ge 3$. Suppose $a_1, a_2, ... , a_n$ are $n$ distinct in pairs real numbers.
In terms of $n$, find the smallest possible number of different assumed values by the following $n$ numbers:
$$a_1 + a_2, a_2 + a_3,..., a_{n- 1} + a_n, a_n + a_1$$
Find a $10$-digit number, in which no digit is zero, that is divisible by the sum of their digits.
Allen flips a fair two sided coin and rolls a fair $6$ sided die with faces numbered $1$ through $6$. What is the probability that the coin lands on heads and he rolls a number that is a multiple of $5$?
$\textbf{(A) }\dfrac1{24}\qquad\textbf{(B) }\dfrac1{12}\qquad\textbf{(C) }\dfrac16\qquad\textbf{(D) }\dfrac14\qquad\textbf{(E) }\dfrac13$
A circle $\omega$ is inscribed in the acute-angled triangle $\vartriangle ABC$, which touches the side $BC$ at the point $K$. On the lines $AB$ and $AC$, the points $P$ and $Q$, respectively, are chosen so that $PK \perp AC$ and $QK \perp AB$. Denote by $M$ and $N$ the points of intersection of $KP$ and $KQ$ with the circle $\omega$. Prove that if $MN \parallel PQ$, then $\vartriangle ABC$ is isosceles.
(S. Slobodyanyuk)
A $1$ or $0$ is placed on each square of a $4 \times 4$ board. One is allowed to change each symbol in a row, or change each symbol in a column, or change each symbol in a diagonal (there are $14$ diagonals of lengths $1$ to $4$). For which arrangements can one make changes which end up with all $0$s?
Show that there exist infinitely many pairs of positive integers $(m,n)$ such that $\binom m{n-1}=\binom{m-1}n$.
Let $(f_n) _{n\ge 0}$ be the sequence defined by$ f_0 = 0, f_1 = 1, f_{n + 2 }= f_{n + 1} + f_n$ for $n> 0$ (Fibonacci string)
and let $t_n =$ ${n+1}\choose{2}$ for $n \ge 1$ . Prove that:
a) $f_1^2+f_2^2+...+f_n^2 = f_n \cdot f_{n+1}$ for $n \ge 1$
b) $\frac{1}{n^2} \cdot \Sigma_{k=1}^{n}\left( \frac{t_k}{f_k}\right)^2 \ge \frac{t_{n+1}^2}{9 f_n \cdot f_{n+1}}$
Consider three circles in the plane $\Gamma_1,\Gamma_2,\Gamma_3$ of radii $R$ passing through a point $O$, and denote by $\mathfrak D$ the set of points of the plane which belong to at least two of these circles. Find the position of $\Gamma_1,\Gamma_2,\Gamma_3$ for which the area of $\mathfrak D$ is the minimum possible. Justify your answer.
Vasya has a numeric expression
$$ \Box \cdot \Box +\Box \cdot \Box $$
and 4 cards with numbers that can be put on 4 free places in the expression. Vasya tried to put cards in all possible ways and all the time obtained the same value as a result. Prove that equal numbers are written on three of his cards.
Find the maximum possible number of edges of a simple graph with $8$ vertices and without any quadrilateral. (a simple graph is an undirected graph that has no loops (edges connected at both ends to the same vertex) and no more than one edge between any two different vertices.)
Let $M = F_1\times\cdots\times F_k$ be the product of $k$ smooth, closed surfaces (2-dimensional, $C^\infty$, compact, connected, manifold without boundary), $s$ of which are non-orientable. Prove that $M$ can be embedded in $\mathbb{R}^{2k+s+1}$.
Three $\text{A's}$, three $\text{B's}$, and three $\text{C's}$ are placed in the nine spaces so that each row and column contain one of each letter. If $\text{A}$ is placed in the upper left corner, how many arrangements are possible?
[asy]
size((80));
draw((0,0)--(9,0)--(9,9)--(0,9)--(0,0));
draw((3,0)--(3,9));
draw((6,0)--(6,9));
draw((0,3)--(9,3));
draw((0,6)--(9,6));
label("A", (1.5,7.5));
[/asy]
$ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6 $
Find all nonnegative integers $m,n$ so that \[\sum_{k=1}^{2^{m}}\lfloor \frac{kn}{2^{m}}\rfloor\in \{28,29,30\}\]
Let $ ABCD$ be a unit square (that is, the labels $ A, B, C, D$ appear in that order around the square). Let $ X$ be a point outside of the square such that the distance from $ X$ to $ AC$ is equal to the distance from $ X$ to $ BD$, and also that $ AX \equal{} \frac {\sqrt {2}}{2}$. Determine the value of $ CX^2$.
Points $D,E,F$ are midpoints of the sides $AB,BC,CA$ of triangle $ABC$. Angle bisectors of the angles $BDC$ and $ADC$ intersect the lines $BC$ and $AC$ respectively at the points $M$ and $N$, and the line $MN$ intersects the line $CD$ at the point $O$. Let the lines $EO$ and $FO$ intersect respectively the lines $AC$ and $BC$ at the points $P$ and $Q$. Prove that $CD=PQ$. [i](Plamen Koshlukov)[/i]
Let $m$, $n$ be positive integers. Prove that, for some positive integer $a$, each of $\phi(a)$, $\phi(a+1)$, $\cdots$, $\phi(a+n)$ is a multiple of $m$.
Two men, $A$ and $B$, set out from town $M$ to town $N$, which is $15$ km away. Their walking speed is $6$ km/hr. They also have a bicycle which they can ride at $15$ km/hr. Both $A$ and $B$ start simultaneously, $A$ walking and $B$ riding a bicycle until $B$ meets a pedestrian girl, $C$, going from $N$ to $M$. Then $B$ lends his bicycle to $C$ and proceeds on foot; $C$ rides the bicycle until she meets $A$ and gives $A$ the bicycle which $A$ rides until he reaches $N$. The speed of $C$ is the same as that of $A$ and $B$. The time spent by $A$ and $B$ on their trip is measured from the moment they started from $M$ until the arrival of the last of them at $N$.
a) When should the girl $C$ leave $N$ for $A$ and $B$ to arrive simultaneously in $N$?
b) When should $C$ leave $N$ to minimize this time?
Given a convex quadrilateral $MNPQ$. Assume that there exists 2 points $U, V$ inside $MNPQ$ satifying:$$\angle MUN = \angle MUV = \angle NUV = \angle QVU = \angle PVU = \angle PVQ$$Consider another 2 points $X, Y$ in the plane. Prove that the sum$$XM + XN + XY + YP + YQ$$get its minimum value iff $X\equiv U, Y\equiv V$.
Suppose that $p$ is an odd prime such that $2p+1$ is also prime. Show that the equation $x^{p}+2y^{p}+5z^{p}=0$ has no solutions in integers other than $(0,0,0)$.
Compute
$$\sum_{i=0}^\infty(-1)^i\sum_{j=i}^\infty(-1)^j\frac{2}{j^2+4j+3}.$$
The teacher wrote on the blackboard quadratic polynomial $x^2 + 10x + 20$. Then in turn each student in the class either increased or decreased by $1$ either the coefficient of $x$ or the constant term. At the end the quadratic polynomial became $x^2+20x+10$. Is it true that at certain moment a quadratic polynomial with integer roots was on the board?
A finite family of finite sets $F$ is given, satisfying two conditions:
(i) if $A, B \in F$, then $A \cup B \in F$;
(ii) if $A \in F$, then the number of elements $| A |$ is not a multiple of $3$.
Prove that you can specify at most two elements so that every set of the family $F$ contains at least one of them.
Solve for real $x$ : $2^{2x} \cdot 2^{3\{x\}} = 11 \cdot 2^{5\{x\}} + 5 \cdot 2^{2[x]}$
(For a real number $x, [x]$ denotes the greatest integer less than or equal to x. For instance, $[2.5] = 2$, $[-3.1] = -4$, $[\pi ] = 3$. For a real number $x, \{x\}$ is defined as $x - [x]$.)
Let $A, B, C$, and $D$ be four different points in the plane. Three of the line segments $AB, AC, AD, BC, BD$,
and $CD$ have length $a$. The other three have length $b$, where $b > a$. Determine all possible values of the quotient $\frac{b}{a}$.
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