Added problems

Problems/all.tex

@@ -72,6 +72,7 @@
 \input{problems/combinatorics}
 \input{problems/pidgeonhole}
 \input{problems/misc}
+\input{problems/geometry}
 
 %\makeatletter
 %\newcommand{\shoppinglist}[1]{Shopping list: #1\checknextarg}
@@ -100,5 +101,6 @@
 	\q{Combinatorics}
 	\q{NumberTheory}
 	\q{Pidgeonhole}
+	\q{Geometry}
 	\q{Misc}
 \end{document}

Problems/problems/algebra.tex

@@ -16,7 +16,8 @@
 
 \problemdef{Algebra}{2}{
 	\statement{
-		$(a + b)(a + b - 1) = ab$ and $a^2 + b^2 = 3$. Find $a^3 + b^3$.
+		$(a + b)(a + b - 1) = ab$ and $a^2 + b^2 = 3$. \par
+		Find $a^3 + b^3$.
 	}
 	\answer{3}
 }

Problems/problems/geometry.tex

@@ -0,0 +1,31 @@
+\problemdef{Geometry}{1}{
+	\statement{
+		What is the largest number of obtuse angles that nine rays emanating from one point can form on a plane?
+	}
+
+	\answer{27}
+}
+
+\problemdef{Geometry}{2}{
+	\statement{
+		The three sides of the trapezoid are equal, and one of the angles is $40^\circ$ larger than the other. Find an acute angle between the diagonals of the trapezoid.
+	}
+
+	\answer{$70^\circ$}
+}
+
+\problemdef{Geometry}{3}{
+	\statement{
+		In a cyclic quadrilateral $ABCD$ angle $ABD$ is $30^\circ$ and angle $BCA$ is $42^\circ$. Find angle $DAB$.
+	}
+
+	\answer{$108^\circ$}
+}
+
+\problemdef{Geometry}{4}{
+	\statement{
+		In a triangle $ABC$, angle $B$ is $20^\circ$, and angle $C$ is $40^\circ$. The bisector $AD$ of angle $A$ is 12. Find the difference between the sides $BC$ and $AB$.
+	}
+
+	\answer{12}
+}

Problems/problems/misc.tex

@@ -59,4 +59,21 @@
 	}
 
 	\answer{310}
+}
+
+
+\problemdef{Misc}{6}{
+	\statement{
+		All faces of the cube are painted in different colors (each face is painted with the same color). If you look at this cube from one side, then you can see the blue, white and yellow faces, on the other side you can see the black, blue and red faces, and on the third side you can see the green, black and white faces. Which face is opposite to the white one?
+	}
+
+	\answer{Red}
+}
+
+\problemdef{Misc}{7}{
+	\statement{
+		A huge military band performed in a field. First, the musicians lined up in a square. Then they rearranged themselves into a rectangle, and the number of ranks increased by 5. How many musicians were in the orchestra?
+	}
+
+	\answer{400}
 }

Problems/problems/numbertheory.tex

@@ -59,7 +59,6 @@
 
 
 \problemdef{NumberTheory}{6}{
-
 	\statement{
 		How many natural numbers $n$ less than 10,000 satisfy $2^n - n^2 \equiv 0~~\text{(mod 7)}$?
 	}
@@ -68,3 +67,30 @@
 }
 
 
+
+\problemdef{NumberTheory}{7}{
+	\statement{
+		Kolya was supposed to multiply a single-digit number and a two-digit one, but instead, he wrote them down in a row and got a three-digit number, which turned out to be three times more than the product that he was supposed to compute. \par
+
+		What numbers could Kolya have? List all the possibilities.
+	}
+
+	\answer{$7 \times 35$ or $1 \times 50$ or $2 \times 40$}
+}
+
+\problemdef{NumberTheory}{8}{
+	\statement{
+		Represent the number 2021 as a sum of four positive integers so that all the digits in these numbers are different.
+	}
+
+	\answer{$2021 = 1987 + 23 + 6 + 5$ Others are possible.}
+}
+
+\problemdef{NumberTheory}{9}{
+	\statement{
+		Find the largest positive integer in which each internal digit is greater than half the sum of the two adjacent digits
+	}
+
+	\answer{36899863}
+}
+