Merge pull request #223 from elkhrt/main

More minor fixes to informal problem solutions.

informal/putnam.json

@@ -1070,7 +1070,7 @@
     {
         "problem_name": "putnam_1974_b2",
         "informal_statement": "Let $y(x)$ be a continuously differentiable real-valued function of a real vairable $x$. Show that if $(y')^2 + y^3 \\to 0$ as $x \\to +\\infty$, then $y(x)$ and $y'(x) \\to 0$ as $x \\to +\\infty$.",
-        "informal_solution": "None",
+        "informal_solution": "None.",
         "tags": [
             "analysis"
         ]
@@ -1508,7 +1508,7 @@
     {
         "problem_name": "putnam_1979_b5",
         "informal_statement": "In the plane, let $C$ be a closed convex set that contains $(0,0) but no other point with integer coordinations. Suppose that $A(C)$, the area of $C$, is equally distributed among the four quadrants. Prove that $A(C) \\leq 4$.",
-        "informal_solution": "",
+        "informal_solution": "None.",
         "tags": [
             "geometry",
             "analysis"
@@ -2814,7 +2814,7 @@
     {
         "problem_name": "putnam_1995_b3",
         "informal_statement": "To each positive integer with $n^{2}$ decimal digits, we associate the determinant of the matrix obtained by writing the digits in order across the rows. For example, for $n=2$, to the integer 8617 we associate $\\det \\left( \\begin{array}{cc} 8 & 6 \\ 1 & 7 \\end{array} \\right) = 50$. Find, as a function of $n$, the sum of all the determinants associated with $n^{2}$-digit integers. (Leading digits are assumed to be nonzero; for example, for $n=2$, there are 9000 determinants.)",
-        "informal_solution": "Show that the solution is $45$ if $n = 1$, $45^2*10$ if $n = 2$, and $0$ if $n$ is greater than 3.",
+        "informal_solution": "Show that the solution is $45$ if $n = 1$, $45^2*10$ if $n = 2$, and $0$ if $n$ is greater than 2.",
         "tags": [
             "linear_algebra"
         ]
@@ -4538,7 +4538,7 @@
     {
         "problem_name": "putnam_2016_a1",
         "informal_statement": "Find the smallest positive integer $j$ such that for every polynomial $p(x)$ with integer coefficients and for every integer $k$, the integer \\[ p^{(j)}(k) = \\left. \\frac{d^j}{dx^j} p(x) \\right|_{x=k} \\] (the $j$-th derivative of $p(x)$ at $k$) is divisible by 2016.",
-        "informal_solution": "Show that the solution is $18$.",
+        "informal_solution": "Show that the solution is $8$.",
         "tags": [
             "algebra",
             "number_theory"
@@ -4880,7 +4880,7 @@
     {
         "problem_name": "putnam_2019_b6",
         "informal_statement": "Let \\( \\mathbb{Z}^n \\) be the integer lattice in \\( \\mathbb{R}^n \\). Two points in \\( \\mathbb{Z}^n \\) are called neighbors if they differ by exactly 1 in one coordinate and are equal in all other coordinates. For which integers \\( n \\geq 1 \\) does there exist a set of points \\( S \\subset \\mathbb{Z}^n \\) satisfying the following two conditions? \\begin{enumerate} \\item If \\( p \\) is in \\( S \\), then none of the neighbors of \\( p \\) is in \\( S \\). \\item If \\( p \\in \\mathbb{Z}^n \\) is not in \\( S \\), then exactly one of the neighbors of \\( p \\) is in \\( S \\). \\end{enumerate}",
-        "informal_solution": "Show that the above statement is True.",
+        "informal_solution": "Show that the statement is true for every \\(n \\geq 1\\)",
         "tags": [
             "algebra"
         ]
@@ -4954,7 +4954,7 @@
     {
         "problem_name": "putnam_2020_b6",
         "informal_statement": "Let $n$ be a positive integer. Prove that $\\sum_{k=1}^n(-1)^{\\lfloor k(\\sqrt{2}-1) \\rfloor} \\geq 0$.",
-        "informal_solution": "None",
+        "informal_solution": "None.",
         "tags": [
             "algebra"
         ]