Found problems: 85335
A company of $n$ soldiers is such that
(i) $n$ is a palindrome number (read equally in both directions);
(ii) if the soldiers arrange in rows of $3, 4$ or $5$ soldiers, then the last row contains $2, 3$ and $5$ soldiers, respectively.
Find the smallest $n$ satisfying these conditions and prove that there are infinitely many such numbers $n$.
A square is divided into $25$ small squares. We draw diagonals of some of the small squares so that no two diagonals share a common point (not even a common endpoint). What is the largest possible number of diagonals that we can draw?
(I Rubanov)
For positive integers $m$ and $n$ we define $T(m,n) = gcd \left(m, \frac{n}{gcd(m,n)} \right)$
(a) Prove that there are infinitely many pairs $(m,n)$ of positive integers for which $T(m,n) > 1$ and $T(n,m) > 1$.
(b) Do there exist positive integers $m,n$ such that $T(m,n) = T(n,m) > 1$?
In the non-isosceles triangle $ABC$,$D$ is the midpoint of side $BC$,$E$ is the midpoint of side $CA$,$F$ is the midpoint of side $AB$.The line(different from line $BC$) that is tangent to the inscribed circle of triangle $ABC$ and passing through point $D$ intersect line $EF$ at $X$.Define $Y,Z$ similarly.Prove that $X,Y,Z$ are collinear.
Find the smallest constant $ k$ such that
$ \frac {x}{\sqrt {x \plus{} y}} \plus{} \frac {y}{\sqrt {y \plus{} z}} \plus{} \frac {z}{\sqrt {z \plus{} x}}\leq k\sqrt {x \plus{} y \plus{} z}$
for all positive $ x$, $ y$, $ z$.
The three points $A, B, C$ form a triangle. $AB=4, BC=5, AC=6$. Let the angle bisector of $\angle A$ intersect side $BC$ at $D$. Let the foot of the perpendicular from $B$ to the angle bisector of $\angle A$ be $E$. Let the line through $E$ parallel to $AC$ meet $BC$ at $F$. Compute $DF$.
[b]a.)[/b] Let $a,b$ be real numbers. Define sequence $x_k$ and $y_k$ such that
\[x_0 = 1, y_0 = 0, x_{k+1} = a \cdot x_k - b \cdot y_l, \quad y_{k+1} = x_k - a \cdot y_k \text{ for } k = 0,1,2, \ldots \]
Prove that
\[x_k = \sum^{[k/2]}_{l=0} (-1)^l \cdot a^{k - 2 \cdot l} \cdot \left(a^2 + b \right)^l \cdot \lambda_{k,l}\]
where $\lambda_{k,l} = \sum^{[k/2]}_{m=l} \binom{k}{2 \cdot m} \cdot \binom{m}{l}$
[b]b.)[/b] Let $u_k = \sum^{[k/2]}_{l=0} \lambda_{k,l} $. For positive integer $m,$ denote the remainder of $u_k$ divided by $2^m$ as $z_{m,k}$. Prove that $z_{m,k},$ $k = 0,1,2, \ldots$ is a periodic function, and find the smallest period.
Prove that every rational number $x$ in the interval $(0, 1)$ can be written as a finite sum of different fractions of the type $\frac{1}{k(k + 1)}$ , that is, different elements in the sequence $\frac12$ , $\frac{1}{6}$ , $\frac{1}{12}$,$...$.
For the positive integers $x , y$ and $z$ apply $\frac{1}{x}+\frac{1}{y}=\frac{1}{z}$ .
Prove that if the three numbers $x , y,$ and $z$ have no common divisor greater than $1$, $x + y$ is the square of an integer.
A circle $\omega$ with center $O$ and radius $r$ is constructed. A point $P$ is chosen on the circumference $\omega$ and a point A is taken inside it, such that is outside the line that passes through $P$ and $O$. Point $B$ is constructed, the reflection of $A$ wrt $O$. and $P'$ is another point on the circumference such that the chord $PP'$ is perpendicular to $PA$. Let $Q$ be the point on the line $PP'$ that minimizes the sum of distances from $A$ to $Q$ and from $Q$ to $B$. Show that the value of the sum of the lengths $AQ+QB$ does not depend on the choice of points $P$ or $A$
Show that the number of positive integer solutions to $x_1 + 2^3x_2 + 3^3x_3 + \ldots + 10^3x_{10} = 3025$ (*) equals the number of non-negative integer solutions to the equation $y_1 + 2^3y_2 + 3^3y_3 + \ldots + 10^3y_{10} = 0$. Hence show that (*) has a unique solution in positive integers and find it.
It is known that each two of the 12 competitors, that participated in the finals of the competition “Mathematical duels”, have a common friend among the other 10. Prove that there is one of them that has at least 5 friends among the group.
Find all real numbers $(x,y)$ satisfying the following: $$x+\frac{3x-y}{x^2+y^2}=3$$ $$y-\frac{x+3y}{x^2+y^2}=0$$
How many triangles are in this figure?
[asy]
import olympiad;
pair A = (0,0);
pair B = (0,1);
pair C = (0,2);
pair D = (0,3);
pair E = (0,4);
pair F = (1,0);
pair G = (2,0);
pair H = (3,0);
pair I = (4,0);
pair J = (1,4);
pair K = (2,4);
pair L = (3,4);
pair M = (4,4);
pair N = (4,3);
pair O = (4,2);
pair P = (4,1);
draw(A--E--I--A);
draw(M--E--I--M);
draw(B--F);
draw(C--G);
draw(D--H);
draw(L--N);
draw(O--K);
draw(P--J);
draw(B--F);
draw(B--F);
draw(H--P);
draw(G--O);
draw(F--N);
draw(B--L);
draw(C--K);
draw(D--J);
draw(A--M);
[/asy]
$(A) 56$
$(B) 60$
$(C) 64$
$(D) 68$
Consider $9$ points in space, no four of which are coplanar. Each pair of points is joined by an edge (that is, a line segment) and each edge is either colored blue or red or left uncolored. Find the smallest value of $\,n\,$ such that whenever exactly $\,n\,$ edges are colored, the set of colored edges necessarily contains a triangle all of whose edges have the same color.
Let $ M,N,P,Q $ be points on the sides $ AB,BC,CD,DA $ (respectively) of a convex quadrilateral $ ABCD $ so that:
$$ \frac{MA}{MB} =\frac{NB}{NC} =\frac{PD}{PC} =\frac{QA}{QD}\neq 1 $$
Show that the area of $ MNPQ $ is half the area of $ ABCD $ if and only if $ ABD $ and $ BCD $ have equal areas.
[i]Petre Asaftei[/i]
Each side of a unit square (side length 1) is also one side of an equilateral triangle that lies in the square. Compute the area of the intersection of (the interiors of) all four triangles.
In a country, the time for presidential elections has approached. There are exactly 20 million voters in the country, of which only one percent supports the current president, Miraflores. Naturally, he wants to be elected again, but on the other hand, he wants the elections to seem democratic.
Miraflores established the following voting process: all the voters are divided into several equal groups, then each of these groups is again divided into a number of equal groups, and so on. In the smallest groups, a representative is chosen. Then, the chosen electors choose representatives in the second-smallest groups, to vote in an even larger group, and so on. Finally, the representatives of the largest groups choose the president.
Miraflores divides voters into groups as he wants and instructs his supporters how to vote. Will he be able to organize the elections in such a way that he will be elected president? (If the votes are equal, the opposition wins.)
[i]From the 32nd Moscow Mathematical Olympiad[/i]
Show that any integer can be expressed as the form $a^{2}+b^{2}-c^{2}$, where $a, b, c \in \mathbb{Z}$.
In a given plane, points $A$ and $B$ are $10$ units apart. How many points $C$ are there in the plane such that the perimeter of $\triangle ABC$ is $50$ units and the area of $\triangle ABC$ is $100$ square units?
$\textbf{(A) }0\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }8\qquad\textbf{(E) }\text{infinitely many}$
(A.Myakishev, 9--10) Given a triangle $ ABC$. Point $ A_1$ is chosen on the ray $ BA$ so that segments $ BA_1$ and $ BC$ are equal. Point $ A_2$ is chosen on the ray $ CA$ so that segments $ CA_2$ and $ BC$ are equal. Points $ B_1$, $ B_2$ and $ C_1$, $ C_2$ are chosen similarly. Prove that lines $ A_1A_2$, $ B_1B_2$, $ C_1C_2$ are parallel.
Given a set $M$ of $1985$ distinct positive integers, none of which has a prime divisor greater than $23$, prove that $M$ contains a subset of $4$ elements whose product is the $4$th power of an integer.
One country has $n$ cities and every two of them are linked by a railroad. A railway worker should travel by train exactly once through the entire railroad system (reaching each city exactly once). If it is impossible for worker to travel by train between two cities, he can travel by plane. What is the minimal number of flights that the worker will have to use?
Let $a$, $b$, and $c$ be positive numbers satisfying $abc=1$. Show that
\[\frac{a^3}{a^3+2}+\frac{b^3}{b^3+2}+\frac{c^3}{c^3+2}\ge1\text{ and }\frac1{a^3+2}+\frac1{b^3+2}+\frac1{c^3+2}\le1\]
Find all functions $f:R \to R$ such that $$f(x^2+f(y)-y) =(f(x))^2-f(y)$$ for all $x,y \in R$