From 0209df9e00068d0fa9116a7f7d4bca724af31770 Mon Sep 17 00:00:00 2001
From: Mark <mark@betalupi.com>
Date: Thu, 18 May 2023 09:56:29 -0700
Subject: [PATCH] Added problems

---
 Problems/problems/algebra.tex      | 24 ++++++++++
 Problems/problems/geometry.tex     |  8 ++++
 Problems/problems/numbertheory.tex | 76 ++++++++++++++++++++++++++++++
 3 files changed, 108 insertions(+)

diff --git a/Problems/problems/algebra.tex b/Problems/problems/algebra.tex
index 4134594..f9d26ad 100755
--- a/Problems/problems/algebra.tex
+++ b/Problems/problems/algebra.tex
@@ -50,4 +50,28 @@
 		Factor $x^8 + x^4 + 1$ into four quadratics.
 	}
 	\answer{$(x^2 - \sqrt{3} x + 1)~(x^2 + \sqrt{3} x + 1)~(x^2 - x + 1)~(x^2 + x + 1)$}
+}
+
+\problemdef{Algebra}{6}{
+	\statement{
+		Sophia bought a Greyhound ticket, but then her plans changed and she sold it back for $\$24$. The percent of the ticket's cost that she lost in the sale is equal to the dollar value of her initial ticket. How much did she buy it for? List all options.
+	}
+
+	\answer{$\$40$ or $\$60$}
+}
+
+\problemdef{Algebra}{7}{
+	\statement{
+		How do you cut a cake into 6 pieces so that it can be distributed equally to both three guests and four guests?
+	}
+
+	\answer{$\frac{3/12} + \frac{3}{4}$ or $\frac{2}{12} + \frac{2}{6} + \frac{2}{4}$}
+}
+
+\problemdef{Algebra}{8}{
+	\statement{
+		On the first day the grocery store sold $\frac{1}{2}$ of all the geese and half a goose, on the second --- $\frac{1}{3}$ of the remainder and another $\frac{1}{3}$ of the goose, on the third --- $\frac{1}{4}$ of the new remnant and another 3/4 of the goose, on the fourth --- $\frac{1}{5}$ of the remainder and another $\frac{1}{5}$ of the goose. On the fifth day, the store sold the remaining 19 geese. How many geese were there in the store?
+	}
+
+	\answer{101}
 }
\ No newline at end of file
diff --git a/Problems/problems/geometry.tex b/Problems/problems/geometry.tex
index d8d5475..6fa6e3b 100755
--- a/Problems/problems/geometry.tex
+++ b/Problems/problems/geometry.tex
@@ -28,4 +28,12 @@
 	}
 
 	\answer{12}
+}
+
+\problemdef{Geometry}{5}{
+	\statement{
+		What is the largest number of sides a polygon that is the intersection of a quadrilateral and a triangle can have?
+	}
+
+	\answer{8}
 }
\ No newline at end of file
diff --git a/Problems/problems/numbertheory.tex b/Problems/problems/numbertheory.tex
index d12d523..9b3a868 100755
--- a/Problems/problems/numbertheory.tex
+++ b/Problems/problems/numbertheory.tex
@@ -94,3 +94,79 @@
 	\answer{36899863}
 }
 
+\problemdef{NumberTheory}{10}{
+	\statement{
+		In the cells of a $5 \times 5$ square, the numbers are arranged so that the sums of the numbers in all rows and in all columns are the same. The sum of all the numbers in the upper left $2 \times 2$ square is $10$, and in the lower right $3 \times 3$ square is $15$. Find the sum of all the numbers in the table.
+	}
+
+	\answer{25}
+}
+
+
+
+\problemdef{NumberTheory}{11}{
+	\statement{
+		Several numbers consisting only of ones (like 1 or 1111), were added together, and the result was the number 2021. Find the smallest possible number of terms.
+	}
+
+	\answer{11}
+}
+
+
+
+\problemdef{NumberTheory}{12}{
+	\statement{
+		Find the largest ten-digit number whose first digit is divisible by 1, the second by 2, $...$ , and the tenth by 10.
+	}
+
+	\answer{9898567890}
+}
+
+
+
+\problemdef{NumberTheory}{13}{
+	\statement{
+		Find any positive integers $a$ and $b$ so that the fractions $\frac{a}{b}$, $\frac{a + 1}{b}$, $\frac{a+1}{b+1}$ are irreducible.
+	}
+
+	\answer{Most numbers work. Checking is easy.}
+}
+
+\problemdef{NumberTheory}{14}{
+	\statement{
+		The numbers $1, 2, ..., 25$ are written out in a $5 \times 5$ table so that in each line the numbers are arranged in ascending order. What is the largest value that the sum of the numbers in the third column can take?
+	}
+
+	\answer{85}
+}
+
+
+\problemdef{NumberTheory}{15}{
+	\statement{
+		Find the smallest natural number whose sum and product of digits are equal to 80.
+	}
+
+	\answer{$11...1258$, which contains 65 ones.}
+}
+
+
+\problemdef{NumberTheory}{16}{
+	\statement{
+		The math teacher uses a problem book, which contains one hundred problems with numbers from 1 to 100. At the beginning of each lesson, the teacher attaches the numbers of three problems to the magnetic board. To do this, he uses magnets in form of digits. What is the smallest number of magnets a teacher needs to buy so that he can compose the numbers of any three problems from this book? (digit 6 can be flipped)
+	}
+
+	\answer{38}
+}
+
+\problemdef{NumberTheory}{17}{
+	\statement{
+		Arrange the following numbers in ascending order: \par
+
+		$(-\frac{2}{3})^1$, $(-\frac{2}{3})^2$, $(-\frac{2}{3})^3$, $(-\frac{2}{3})^4$
+	}
+
+	\answer{$(-\frac{2}{3})^1 < (-\frac{2}{3})^3 < (-\frac{2}{3})^4 < (-\frac{2}{3})^2$}
+}
+
+
+