Found problems: 85335
Vitya and Masha are playing a game. At first, Vitya thinks of three different integers. In one move Masha can ask one of the following three numbers: the sum of the numbers, the product of the numbers or the sum of pairwise products of the numbers. Masha asks questions and Vitya immediately answers before Masha asks the next question.
a) Prove that Masha can always guess Vitya's numbers.
b) What is the least amount of questions Masha needs to ask to guaranteely guess them?
A $10$ digit number is called interesting if its digits are distinct and is divisible by $11111$. Then find the number of interesting numbers.
Let $ n$ be a natural number such that $ n \geq 2$. Show that
\[ \frac {1}{n \plus{} 1} \left( 1 \plus{} \frac {1}{3} \plus{} \cdot \cdot \cdot \plus{} \frac {1}{2n \minus{} 1} \right) > \frac {1}{n} \left( \frac {1}{2} \plus{} \frac {1}{4} \plus{} \cdot \cdot \cdot \plus{} \frac {1}{2n} \right).
\]
In the acute-angled triangle $ABC$ is drawn the altitude $CH$. A ray beginning at point $C$ that lies inside the $\angle BCA$ and intersects for second time the circles circumscribed circles of $\vartriangle BCH$ and $\vartriangle ABC$ at points $X$ and $Y$ respectively. It turned out that $2CX = CY$. Prove that the line $HX$ bisects the segment $AC$.
(Hilko Danilo)
In a sign pyramid a cell gets a "+" if the two cells below it have the same sign, and it gets a "-" if the two cells below it have different signs. The diagram below illustrates a sign pyramid with four levels. How many possible ways are there to fill the four cells in the bottom row to produce a "+" at the top of the pyramid?
[asy]
unitsize(2cm);
path box = (-0.5,-0.2)--(-0.5,0.2)--(0.5,0.2)--(0.5,-0.2)--cycle;
draw(box); label("$+$",(0,0));
draw(shift(1,0)*box); label("$-$",(1,0));
draw(shift(2,0)*box); label("$+$",(2,0));
draw(shift(3,0)*box); label("$-$",(3,0));
draw(shift(0.5,0.4)*box); label("$-$",(0.5,0.4));
draw(shift(1.5,0.4)*box); label("$-$",(1.5,0.4));
draw(shift(2.5,0.4)*box); label("$-$",(2.5,0.4));
draw(shift(1,0.8)*box); label("$+$",(1,0.8));
draw(shift(2,0.8)*box); label("$+$",(2,0.8));
draw(shift(1.5,1.2)*box); label("$+$",(1.5,1.2));
[/asy]
$\textbf{(A) } 2 \qquad \textbf{(B) } 4 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 12 \qquad \textbf{(E) } 16$
Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function such that $$f(x)=f \left( \frac{x}{2} \right) + \frac{x}{2} f'(x), ~\forall x \in \mathbb{R}.$$
Prove that $f$ is a polynomial function of degree at most one.
[hide=Note]The problem was posted quite a few times before:
[url]https://artofproblemsolving.com/community/c7h100225p566080[/url]
[url]https://artofproblemsolving.com/community/q11h564540p3300032[/url]
[url]https://artofproblemsolving.com/community/c7h2605212p22490699[/url]
[url]https://artofproblemsolving.com/community/c7h198927p1093788[/url]
I'm reposting it just to have a more suitable statement for the [url=https://artofproblemsolving.com/community/c13_contests]Contest Collections[/url].
[/hide]
Square $ABCD$ has side length $13$, and points $E$ and $F$ are exterior to the square such that $BE=DF=5$ and $AE=CF=12$. Find $EF^{2}$.
[asy]
size(200);
defaultpen(fontsize(10));
real x=22.61986495;
pair A=(0,26), B=(26,26), C=(26,0), D=origin, E=A+24*dir(x), F=C+24*dir(180+x);
draw(B--C--F--D--C^^D--A--E--B--A, linewidth(0.7));
dot(A^^B^^C^^D^^E^^F);
pair point=(13,13);
label("$A$", A, dir(point--A));
label("$B$", B, dir(point--B));
label("$C$", C, dir(point--C));
label("$D$", D, dir(point--D));
label("$E$", E, dir(point--E));
label("$F$", F, dir(point--F));[/asy]
A convex hexagon $A_1B_1A_2B_2A_3B_3$ it is inscribed in a circumference $\Omega$ with radius $R$. The diagonals $A_1B_2$, $A_2B_3$, $A_3B_1$ are concurrent in $X$. For each $i=1,2,3$ let $\omega_i$ tangent to the segments $XA_i$ and $XB_i$ and tangent to the arc $A_iB_i$ of $\Omega$ that does not contain the other vertices of the hexagon; let $r_i$ the radius of $\omega_i$.
$(a)$ Prove that $R\geq r_1+r_2+r_3$
$(b)$ If $R= r_1+r_2+r_3$, prove that the six points of tangency of the circumferences $\omega_i$ with the diagonals $A_1B_2$, $A_2B_3$, $A_3B_1$ are concyclic
On a $(4n + 2)\times (4n + 2)$ square grid, a turtle can move between squares sharing a side.The turtle begins in a corner square of the grid and enters each square exactly once, ending in the square where she started. In terms of $n$, what is the largest positive integer $k$ such that there must be a row or column that the turtle has entered at least $k$ distinct times?
Consider the operation $*$ defined by the following table:
\[\begin{tabular}{c|cccc}
* & 1 & 2 & 3 & 4 \\ \hline
1 & 1 & 2 & 3 & 4 \\
2 & 2 & 4 & 1 & 3 \\
3 & 3 & 1 & 4 & 2 \\
4 & 4 & 3 & 2 & 1
\end{tabular}\]
For example, $3*2=1$. Then $(2*4)*(1*3)=$
$\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad \text{(E)}\ 5$
Suppose $a,b,c$ are real numbers such that $a+b \ge 0, b+c \ge 0$, and $c+a \ge 0$.
Prove that $a+b+c \ge \frac{|a|+|b|+|c|}{3}$ .
(Note: $|x|$ is called the absolute value of $x$ and is defined as follows.
If $x \ge 0$ then $|x|= x$, and if $x < 0$ then $|x| = -x$. For example, $|6|= 6, |0| = 0$ and $|-6| = 6$.)
Let $ABC$ be a triangle with orthocenter $H,$ and let $M$ be the midpoint of $\overline{BC}.$ Suppose that $P$ and $Q$ are distinct points on the circle with diameter $\overline{AH},$ different from $A,$ such that $M$ lies on line $PQ.$ Prove that the orthocenter of $\triangle APQ$ lies on the circumcircle of $\triangle ABC.$
[i]Proposed by Michael Ren[/i]
A line with slope $ 3$ intersects a line with slope $ 5$ at the point $ (10, 15)$. What is the distance between the $ x$-intercepts of these two lines?
$ \textbf{(A)}\ 2 \qquad
\textbf{(B)}\ 5 \qquad
\textbf{(C)}\ 7 \qquad
\textbf{(D)}\ 12 \qquad
\textbf{(E)}\ 20$
Let $ABC$ be acute triangle. Let $E$ and $F$ be points on $BC$, such that angles $BAE$ and $FAC$ are equal. Lines $AE$ and $AF$ intersect cirumcircle of $ABC$ at points $M$ and $N$. On rays $AB$ and $AC$ we have points $P$ and $R$, such that angle $PEA$ is equal to angle $B$ and angle $AER$ is equal to angle $C$. Let $L$ be intersection of $AE$ and $PR$ and $D$ be intersection of $BC$ and $LN$. Prove that
$$\frac{1}{|MN|}+\frac{1}{|EF|}=\frac{1}{|ED|}.$$
Consider lattice points of a $6*7$ grid.We start with two points $A,B$.We say two points $X,Y$ connected if one can reflect several times WRT points $A,B$ and reach from $X$ to $Y$.Over all choices of $A,B$ what is the minimum number of connected components?
How many ordered pairs $ (m,n)$ of positive integers, with $ m > n$, have the property that their squares differ by $ 96$?
$ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 12$
Let $a, b, c$ be nonzero integers such that $M = \frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ and $N =\frac{a}{c}+\frac{b}{a}+\frac{c}{b}$ are both integers. Find $M$ and $N$.
Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.
Let $a_1,a_2,a_3,\ldots$ be an infinite sequence of positive integers such that $a_{n+2m}$ divides $a_{n}+a_{n+m}$ for all positive integers $n$ and $m.$ Prove that this sequence is eventually periodic, i.e. there exist positive integers $N$ and $d$ such that $a_n=a_{n+d}$ for all $n>N.$
Show that the equation \[ (9 - x^2) \left (\frac{\mathrm dy}{\mathrm dx} \right)^2 = (9 - y^2)\] characterizes a family of conics touching the four sides of a fixed square.
Given an acute triangle $ABC$. Points $B'$ and $C'$ are symmetrical, respectively, to vertices $B$ and $ C$ wrt straight lines $AC$ and $AB$. Let $P$ be the intersection point of the circumcircles of triangles $ABB'$ and $ACC'$, different from $A$. Prove that the center of the circumcircle of triangle $ABC$ lies on line $PA$.
Let $r$ be a positive real number. Prove that the number of right triangles with prime positive integer sides that have an inradius equal to $r$ are zero or a power of $2$.
[hide=original wording]Seja r um numero real positivo. Prove que o numero de triangulos retangulos com lados inteiros positivos primos entre si que possuem inraio igual a r e zero ou uma potencia de 2.[/hide]
It is known about the triangle $ABC$ that $3 BC = CA + AB$. Let the $A$-symmedian of triangle $ABC$ intersect the circumcircle of triangle $ABC$ at point $D$. Prove that $\frac{1}{BD}+ \frac{1}{CD}= \frac{6}{AD}$.
(Ercole Suppa, Italy)
In triangle $ABC$ the angular bisector from $A$ intersects the side $BC$ at the point $D$, and the angular bisector from $B$ intersects the side $AC$ at the point $E$. Furthermore $|AE| + |BD| = |AB|$. Prove that $\angle C = 60^o$
[img]https://1.bp.blogspot.com/-8ARqn8mLn24/XzP3P5319TI/AAAAAAAAMUQ/t71-imNuS18CSxTTLzYXpd806BlG5hXxACLcBGAsYHQ/s0/2018%2BMohr%2Bp5.png[/img]
We wish to tile a strip of $n$ $1$-inch by $1$-inch squares. We can use dominos which are made up of two tiles that cover two adjacent squares, or $1$-inch square tiles which cover one square. We may cover each square with one or two tiles and a tile can be above or below a domino on a square, but no part of a domino can be placed on any part of a different domino. We do not distinguish whether a domino is above or below a tile on a given square. Let $t(n)$ denote the number of ways the strip can be tiled according to the above rules. Thus for example, $t(1)=2$ and $t(2)=8$. Find a recurrence relation for $t(n)$, and use it to compute $t(6)$.