Prove that for coprime each positive integers $a, c$ there is a positive integer $b$ such that $c$ divides $\underbrace{b^{b^{b^{\ldots^b}}}}_\text{b times}-a$

Let $S$ be the set of points $A$ in the Cartesian plane such that the four points $A$, $(2, 3)$, $(-1, 0)$, and $(0, 6)$ form the vertices of a parallelogram. Let $P$ be the convex polygon whose vertices are the points in $S$. What is the area of $P$?

Denote by $[n]!$ the product $ 1 \cdot 11 \cdot 111\cdot ... \cdot \underbrace{111...1}_{\text{n ones}}$.($n$ factors in total). Prove that $[n + m]!$ is divisible by $ [n]! \times [m]!$ [i](8 points)[/i]

The intersection point $I$ of the angles bisectors of the triangle $ABC$ has reflections the points $P,Q,T$ wrt the triangle's sides . It turned out that the circle $s$ circumscribed around of the triangle $PQT$ , passes through the vertex $A$. Find the radius of the circumscribed circle of triangle $ABC$ if $BC = a$. (Gryhoriy Filippovskyi)

Suppose that $X$ and $Y$ are two metric spaces and $f:X \longrightarrow Y$ is a continious function. Also for every compact set $K \subseteq Y$, it's pre-image $f^{pre}(K)$ is a compact set in $X$. Prove that $f$ is a closed function, i.e for every close set $C\subseteq X$, it's image $f(C)$ is a closed subset of $Y$.

Given an acute triangle $ABC$ with altitudes $AD$ and $BE$ intersecting at $H$, $M$ is the midpoint of $AB$. A nine-point circle of $ABC$ intersects with a circumcircle of $ABH$ on $P$ and $Q$ where $P$ lays on the same side of $A$ (with respect to $CH$). Prove that $ED, PH, MQ$ are concurrent on circumcircle $ABC$

On a natural number $n$ you are allowed to operations : $(1)$ multiply $n$ by $2$ or $(2)$ subtract $3$ from $n$. For example starting with $8$ you can reach $13$ as follows : $8 \longrightarrow 16 \longrightarrow 13$. You need two steps and you cannot do in less than two steps. Starting from $11$, what is the least number of steps required to reach $121$?

[b]6.1 [/b] The bindery had 92 sheets of white paper and $135$ sheets of colored paper. It took a sheet of white paper to bind each book. and a sheet of colored paper. After binding several books of white Paper turned out to be half as much as the colored one. How many books were bound? [b]6.2[/b] Prove that if you multiply all the integers from $1$ to $1965$, you get the number, the last whose non-zero digit is even. [b]6.3[/b] The front tires of a car wear out after $25,000$ kilometers, and the rear tires after $15,000$ kilometers of travel. When should you swap tires so that they wear out at the same time? [b]6.4[/b] A rectangle $19$ cm $\times 65$ cm is divided by straight lines parallel to its sides into squares with side 1 cm. How many parts will this rectangle be divided into if you also draw a diagonal in it? [b]6.5[/b] Find the dividend, divisor and quotient in the example: [center][img]https://cdn.artofproblemsolving.com/attachments/2/e/de053e7e11e712305a89d3b9e78ac0901dc775.png[/img] [/center] [b]6.6[/b] Odd numbers from $1$ to $49$ are written out in table form $$\,\,\,1\,\,\,\,\,\, 3\,\,\,\,\,\, 5\,\,\,\,\,\, 7\,\,\,\,\,\, 9$$ $$11\,\,\, 13\,\,\, 15\,\,\, 17\,\,\, 19$$ $$21\,\,\, 23\,\,\, 25\,\,\, 27\,\,\, 29$$ $$31\,\,\, 33\,\,\, 35\,\,\, 37\,\,\, 39$$ $$41\,\,\, 43\,\,\, 45\,\,\, 47\,\,\, 49$$ $5$ numbers are selected, any two of which are not on the same line or in one column. What is their sum? PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988081_1965_leningrad_math_olympiad]here[/url].

Solve the equation: $$x \cdot 2^{\dfrac{1}{x}}+\dfrac{1}{x} \cdot 2^x=4$$

Let $ABC$ be an acute triangle and $T$ be a point interior to this triangle. that $\angle ATB = \angle BTC = \angle CTA$. Let $M,N$ and $P$ be the feet of the perpendiculars from $T$ to $BC$, $CA$ and $AB$ respectively. Prove that if the circle circumscribed around $\vartriangle MNP$ cuts again the sides $ BC$, $CA$ and $AB$ in $M_1$, $N_1$, $P_1$ respectively, then the $\vartriangle M_1N_1P_1$ It is equilateral.

Ebeneezer is painting the edges of a cube. He wants to paint the edges so that the colored edges form a loop that does not intersect itself. For example, the loop should not look like a “figure eight” shape. If two colorings are considered equivalent if there is a rotation of the cubes so that the colored edges are the same, what is the number of possible edge colorings?

Let $ABC$ be an acute angled triangle with circumcircle $O$. Line $AO$ intersects the circumcircle of triangle $ABC$ again at point $D$. Let $P$ be a point on the side $BC$. Line passing through $P$ perpendicular to $AP$ intersects lines $DB$ and $DC$ at $E$ and $F$ respectively . Line passing through $D$ perpendicular to $BC$ intersects $EF$ at point $Q$. Prove that $EQ = FQ$ if and only if $BP = CP$.

Define the sequence $(a_n)_{n=1}^\infty$ of positive integers by $a_1=1$ and the condition that $a_{n+1}$ is the least integer such that \[\mathrm{lcm}(a_1, a_2, \ldots, a_{n+1})>\mathrm{lcm}(a_1, a_2, \ldots, a_n)\mbox{.}\] Determine the set of elements of $(a_n)$.

There are given $111$ coins and a $n\times n$ table divided into unit cells. This coins are placed inside the unit cells (one unit cell may contain one coin, many coins, or may be empty), such that the difference between the number of coins from two neighbouring cells (that have a common edge) is $1$. Find the maximal $n$ for this to be possible.

A $4\times4\times4$ cube is composed of $64$ unit cubes. The faces of $16$ unit cubes are to be coloured red. A colouring is called interesting if there is exactly $1$ red unit cube in every $1\times1\times 4$ rectangular box composed of $4$ unit cubes. Determine the number of interesting colourings.

In a sports meeting a total of $m$ medals were awarded over $n$ days. On the first day one medal and $\frac{1}{7}$ of the remaining medals were awarded. On the second day two medals and $\frac{1}{7}$ of the remaining medals were awarded, and so on. On the last day, the remaining $n$ medals were awarded. How many medals did the meeting last, and what was the total number of medals ?

A square $S_a$ is inscribed in an acute-angled triangle $ABC$ with two vertices on side $BC$ and one on each of sides $AC$ and $AB$. Squares $S_b$ and $S_c$ are analogously inscribed in the triangle. For which triangles are the squares $S_a,S_b$, and $S_c$ congruent?

here is the most difficult and the most beautiful problem occurs in 21th iranian (2003) olympiad assume that P is n-gon ,lying on the plane ,we name its edge 1,2,..,n. if S=s1,s2,s3,.... be a finite or infinite sequence such that for each i, si is in {1,2,...,n}, we move P on the plane according to the S in this form: at first we reflect P through the s1 ( s1 means the edge which iys number is s1)then through s2 and so on like the figure below. a)show that there exist the infinite sequence S sucth that if we move P according to S we cover all the plane b)prove that the sequence in a) isn't periodic. c)assume that P is regular pentagon ,which the radius of its circumcircle is 1,and D is circle ,with radius 1.00001 ,arbitrarily in the plane .does exist a sequence S such that we move P according to S then P reside in D completely?

A large urn contains $100$ balls, of which $36\%$ are red and the rest are blue. How many of the blue balls must be removed so that the percentage of red balls in the urn will be $72\%?$ (No red balls are to be removed.) $ \textbf{(A) }28 \qquad \textbf{(B) }32 \qquad \textbf{(C) }36 \qquad \textbf{(D) }50 \qquad \textbf{(E) }64 \qquad $

$S=\{1,2, \ldots, 6\} .$ Then find out the number of unordered pairs of $(A, B)$ such that $A, B \subseteq S$ and $A \cap B=\phi$ [list=1] [*] 360 [*] 364 [*] 365 [*] 366 [/list]

The triangle $ABC$ is isosceles and has a right angle at the vertex $A$. Construct all points that simultaneously satisfy the following two conditions: (i) are equidistant from points $A$ and $B$ (ii) heve distance exactly three times from point $C$ as far as from point $B$.

In each square of an $11\times 11$ board, we are to write one of the numbers $-1$, $0$, or $1$ in such a way that the sum of the numbers in each column is nonnegative and the sum of the numbers in each row is nonpositive. What is the smallest number of zeros that can be written on the board? Justify your answer.

A jacket was originally priced $\textdollar 100$. The price was reduced by $10\%$ three times and increased by $10\%$ four times in some order. To the nearest cent, what was the final price?

For $n \ge 1$ call a finite sequence $(a_1, a_2 \ldots a_n)$ of positive integers [i]progressive[/i] if $a_i < a_{i+1}$ and $a_i$ divides $a_{i+1}$ for all $1 \le i \le n-1$. Find the number of progressive sequences such that the sum of the terms in the sequence is equal to $360$.

Find all the real $a,b,c$ such that the equality $$|ax+by+cz| + |bx+cy+az| + |cx+ay+bz| = |x|+|y|+|z|$$ is valid for all the real $x,y,z$.