The diagram below shows four congruent squares and some of their diagonals. Let $T$ be the number of triangles and $R$ be the number of rectangles that appear in the diagram. Find $T + R$. [img]https://cdn.artofproblemsolving.com/attachments/1/5/f756bbe67c09c19e811011cb6b18d0ff44be8b.png[/img]

Prove that if $x$ is such that $$x +\frac{1}{x}= 2\cos \alpha $$ then, for all $n = 0, 1, 2, . . . ,$ $$x^n ++\frac{1}{x^n}= 2\cos n \alpha .$$

The professor tells Peter the product of two positive integers and Sam their sum. At first, nobody of them knows the number of the other. One of them says: [i]You can't possibly guess my number[/i]. Then the other says: [i]You are wrong, the number is 136[/i]. Which number did the professor tell them respectively? Give reasons for your claim.

Two perpendicular lines are drawn through the orthocenter $H$ of triangle $ABC$, one of which intersects $BC$ at point $X$, and the other intersects $AC$ at point $Y$. Lines $AZ, BZ$ are parallel, respectively with $HX$ and $HY$. Prove that the points $X, Y, Z$ lie on the same line.

Solve equation $27\cdot3^{3\sin x}=9^{\cos^2x}$ where $x\in [0,2\pi )$

Let be a group and two coprime natural numbers $ m,n. $ Show that if the applications $ G\ni x\mapsto x^{m+1},x^{n+1} $ are surjective endomorphisms, then the group is commutative.

Let point $I$ be the center of the inscribed circle of an acute-angled triangle $ABC$. Rays $AI$ and $BI$ intersect the circumcircle $k$ of triangle $ABC$ at points $D$ and $E$ respectively. The segments $DE$ and $CA$ intersect at point $F$, the line through point $E$ parallel to the line $FI$ intersects the circle $k$ at point $G$, and the lines $FI$ and $DG$ intersect at point $H$. Prove that the lines $CA$ and $BH$ touch the circumcircle of the triangle $DFH$ at the points $F$ and $H$ respectively.

Alf was attending an eight-year elementary school on Melmac. At the end of each school year, he showed the certificate to his father. If he was promoted, his father gave him the number of cats equal to Alf’s age times the number of the grade he passed. During elementary education, Alf failed one grade and had to repeat it. When he finished elementary education he found out that the total number of cats he had received was divisible by $1998$. Which grade did Alf fail?

Two lines, $l_1$ and $l_2$, are tangent to the parabola $x^2-4(x+y)+y^2=2xy+8$ such that they intersect at a point whose coordinates sum to $-32$. The minimum possible sum of the slopes of $l_1$ and $l_2$ can be written as $\frac{m}{n}$ for relatively prime integers $m$ and $n$. Find $m+n$. [I] Proposed by [b]AOPS12142015[/b][/I]

Let $d_1,d_2,\ldots,d_k$ be all factors of a positive integer $n$ including $1$ and $n$. If $d_1+d_2+\ldots+d_k=72$ then $\frac1{d_1}+\frac1{d_2}+\ldots+\frac1{d_k}$ is $\textbf{(A)}~\frac{k^2}{72}$ $\textbf{(B)}~\frac{72}k$ $\textbf{(C)}~\frac{72}n$ $\textbf{(D)}~\text{None of the above}$

Let us consider one variable polynomials with the senior coefficient equal to one. We shall say that two polynomials $P(x)$ and $Q(x)$ commute, if $P(Q(x))=Q(P(x))$ (i.e. we obtain the same polynomial, having collected the similar terms). a) For every a find all $Q$ such that the $Q$ degree is not greater than three, and $Q$ commutes with $(x^2 - a)$. b) Let $P$ be a square polynomial, and $k$ is a natural number. Prove that there is not more than one commuting with $P$ $k$-degree polynomial. c) Find the $4$-degree and $8$-degree polynomials commuting with the given square polynomial $P$. d) $R$ and $Q$ commute with the same square polynomial $P$. Prove that $Q$ and $R$ commute. e) Prove that there exists a sequence $P_2, P_3, ... , P_n, ...$ ($P_k$ is $k$-degree polynomial), such that $P_2(x) = x^2 - 2$, and all the polynomials in this infinite sequence pairwise commute.

Let $x$ and $y$ be positive real numbers with $x + y =1 $. Prove that $$\frac{(3x-1)^2}{x}+ \frac{(3y-1)^2}{y} \ge1.$$ For which $x$ and $y$ equality holds? (K. Czakler, GRG 21, Vienna)

Let $a \geq b \geq 1 \geq c \geq 0$ be real numbers such that $a+b+c=3$. Show that $$3 \left( \frac{a}{b}+\frac{b}{a} \right ) \geq 4c^2+\frac{a^2}{b}+\frac{b^2}{a}$$

In the triangle $ABC$, the respective mid points of the sides $BC$, $AB$ and $AC$ are $D$, $E$ and $F$. Let $M$ be the point where the internal bisector of the angle $\angle ADB$ intersects the side $AB$, and $N$ the point where the internal bisector of the angle $\angle ADC$ intersects the side $AC$. Also, let $O$ be the intersection point of $AD$ and $MN$, $P$ the intersection point of $AB$ and $FO$, and $R$ the intersection point of $AC$ and $EO$. Prove that $PR=AD$.

[b]7.1[/b] Given a convex $n$-gon all of whose angles are obtuse. Prove that the sum of the lengths of the diagonals in it is greater than the sum of the lengths of the sides. [b]7.2[/b] Find all integer values for $x$ and $y$ such that $x^4 + 4y^4$ is a prime number[b]. (typo corrected)[/b] [b]7.3.[/b] Given a triangle $ABC$. Parallelograms $ABKL$, $BCMN$ and $ACFG$ are constructed on the sides, Prove that the segments $KN$, $MF$ and $GL$ can form a triangle. [img]https://cdn.artofproblemsolving.com/attachments/a/f/7a0264b62754fafe4d559dea85c67c842011fc.png[/img] [b]7.4 / 6.2[/b] Prove that a $10 \times 10$ chessboard cannot be covered with $ 25$ figures like [img]https://cdn.artofproblemsolving.com/attachments/0/4/89aafe1194628332ec13ad1c713bb35cbefff7.png[/img]. [b]7.5[/b] Find the greatest number of different natural numbers, each of which is less than $50$, and every two of which are coprime. [b]7.6.[/b] Given a triangle $ABC$.$ D$ and $E$ are the midpoints of the sides $AB$ and $BC$. Point$ M$ lies on $AC$ , $ME > EC$. Prove that $MD < AD$. [img]https://cdn.artofproblemsolving.com/attachments/e/c/1dd901e0121e5c75a4039d21b954beb43dc547.png[/img] PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983461_1964_leningrad_math_olympiad]here[/url].

Find the sum of all $4$-digit numbers using the digits $2,3,4,5,6$ without a repetition of any of those digits.

Every bacterium has a horizontal body with natural length and some nonnegative number of vertical feet, each with nonnegative (!) natural length, that lie below its body. In how many ways can these bacteria fill an $m\times n$ table such that no two of them overlap? [i]proposed by Mahyar Sefidgaran[/i]

Let $a$ be a positive number, and we show decimal part of the $a$ with $\left\{a\right\}$.For a positive number $x$ with $\sqrt 2< x <\sqrt 3$ such that, $\left\{\frac{1}{x}\right\}$=$\left\{x^2\right\}$.Find value of the $$x(x^7-21)$$

Given that $\tan A=\frac{12}{5}$, $\cos B=-\frac{3}{5}$ and that $A$ and $B$ are in the same quadrant, find the value of $\cos (A-B)$. $ \textbf{(A) }-\frac{63}{65}\qquad\textbf{(B) }-\frac{64}{65}\qquad\textbf{(C) }\frac{63}{65}\qquad\textbf{(D) }\frac{64}{65}\qquad\textbf{(E) }\frac{65}{63} $

In a right circular cone of wood, the radius of the circumference $T$ of the base circle measures $10$ cm, while every point on said circumference is $20$ cm away. from the apex of the cone. A red ant and a termite are located at antipodal points of $T$. A black ant is located at the midpoint of the segment that joins the vertex with the position of the termite. If the red ant moves to the black ant's position by the shortest possible path, how far does it travel?

Find all pairs of positive integers ($x$,$y$) such that $y^3=x^3+7x^2+4x+15$.

For what value of $ x$ does \[ \log_{\sqrt{2}} \sqrt{x} \plus{} \log_2 x \plus{} \log_4 (x^2) \plus{} \log_8 (x^3) \plus{} \log_{16} (x^4) \equal{} 40?\] $ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 256 \qquad \textbf{(E)}\ 1024$

Let $ABC$ be a triangle with circumcircle $\omega$, circumcenter $O{}$, and orthocenter $H{}$. Let $K{}$ be the midpoint of $AH{}$. The perpendicular to $OK{}$ at $K{}$ intersects $AB{}$ and $AC{}$ at $P{}$ and $Q{}$, respectively. The lines $BK$ and $CK$ intersect $\omega$ again at $X{}$ and $Y{}$, respectively. Prove that the second intersection of the circumcircles of triangles $KPY$ and $KQX$ lies on $\omega$. [i]Stefan Lozanovski[/i]

There is a game. The magician let the participant think up a positive integer (at least two digits). For example, an integer $ \displaystyle\overline{a_1a_2 \cdots a_n}$ is rearranged as $ \overline{a_{i_1}a_{i_2} \cdots a_{i_n}}$, that is, $ i_1, i_2, \cdots, i_n$ is a permutation of $ 1,2, \cdots, n$. Then we get $ n!\minus{}1$ integers. The participant is asked to calculate the sum of the $ n!\minus{}1$ numbers, then tell the magician the sum $ S$. The magician claims to be able to know the original number when he is told the sum $ S$. Try to decide whether the magician can be successful or not.

Which of the following changes will result in an [i]increase[/i] in the period of a simple pendulum? (A) Decrease the length of the pendulum (B) Increase the mass of the pendulum (C) Increase the amplitude of the pendulum swing (D) Operate the pendulum in an elevator that is accelerating upward (E) Operate the pendulum in an elevator that is moving downward at constant speed.