Found problems: 55
MMPC Part II 1958 - 95, 1987
[b]p1.[/b] Let $D(n)$ denote the number of positive factors of the integer $n$. For example, $D(6) = 4$ , since the factors of $6$ are $1, 2, 3$ , and $6$ . Note that $D(n) = 2$ if and only if $n$ is a prime number.
(a) Describe the set of all solutions to the equation $D(n) = 5$ .
(b) Describe the set of all solutions to the equation $D(n) = 6$ .
(c) Find the smallest $n$ such that $D(n) = 21$ .
[b]p2.[/b] At a party with $n$ married couples present (and no one else), various people shook hands with various other people. Assume that no one shook hands with his or her spouse, and no one shook hands with the same person more than once. At the end of the evening Mr. Jones asked everyone else, including his wife, how many hands he or she had shaken. To his surprise, he got a different answer from each person. Determine the number of hands that Mr. Jones shook that evening,
(a) if $n = 2$ .
(b) if $n = 3$ .
(c) if $n$ is an arbitrary positive integer (the answer may depend on $n$).
[b]p3.[/b] Let $n$ be a positive integer. A square is divided into triangles in the following way. A line is drawn from one corner of the square to each of $n$ points along each of the opposite two sides, forming $2n + 2$ nonoverlapping triangles, one of which has a vertex at the opposite corner and the other $2n + 1$ of which have a vertex at the original corner. The figure shows the situation for $n = 2$ . Assume that each of the $2n + 1$ triangles with a vertex in the original corner has area $1$. Determine the area of the square,
(a) if $n = 1$ .
(b) if $n$ is an arbitrary positive integer (the answer may depend on $n$).
[img]https://cdn.artofproblemsolving.com/attachments/1/1/62a54011163cc76cc8d74c73ac9f74420e1b37.png[/img]
[b]p4.[/b] Arthur and Betty play a game with the following rules. Initially there are one or more piles of stones, each pile containing one or more stones. A legal move consists either of removing one or more stones from one of the piles, or, if there are at least two piles, combining two piles into one (but not removing any stones). Arthur goes first, and play alternates until a player cannot make a legal move; the player who cannot move loses.
(a) Determine who will win the game if initially there are two piles, each with one stone, assuming that both players play optimally.
(b) Determine who will win the game if initially there are two piles, each with $n$ stones, assuming that both players play optimally; $n$ is a positive integer, and the answer may depend on $n$ .
(c) Determine who will win the game if initially there are $n$ piles, each with one stone, assuming that both players play optimally; $n$ is a positive integer, and the answer may depend on $n$ .
[b]p5.[/b] Suppose $x$ and $y$ are real numbers such that $0 < x < y$. Define a sequence$ A_0 , A_1 , A_2, A_3, ...$ by-setting $A_0 = x$ , $A_1 = y$ , and then $A_n= |A_{n-1}| - A_{n-2}$ for each $n \ge 2$ (recall that $|A_{n-1}|$ means the absolute value of $A_{n-1}$ ).
(a) Find all possible values for $A_6$ in terms of $x$ and $y$ .
(b) Find values of $x$ and $y$ so that $A_{1987} = 1987$ and $A_{1988} = -1988$ (simultaneously).
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MMPC Part II 1958 - 95, 1963
[b]p1.[/b] Suppose $x \ne 1$ or $10$ and logarithms are computed to the base $10$. Define $y= 10^{\frac{1}{1-\log x}}$ and $z = ^{\frac{1}{1-\log y}}$ . Prove that $x= 10^{\frac{1}{1-\log z}}$
[b]p2.[/b] If $n$ is an odd number and $x_1, x_2, x_3,..., x_n$ is an arbitrary arrangement of the integers $1, 2,3,..., n$, prove that the product $$(x_1 -1)(x_2-2)(x_3- 3)... (x_n-n)$$ is an even number (possibly negative or zero).
[b]p3.[/b] Prove that $\frac{1 \cdot 3 \cdot 5 \cdot \cdot \cdot (2n-1)}{2 \cdot 4 \cdot 6 \cdot \cdot \cdot(2n} < \sqrt{\frac{1}{2n + 1}}$ for all integers $n = 1,2,3,...$
[b]p4.[/b] Prove that if three angles of a convex polygon are each $60^o$, then the polygon must be an equilateral triangle.
[b]p5.[/b] Find all solutions, real and complex, of $$4 \left(x^2+\frac{1}{x^2} \right)-4 \left( x+\frac{1}{x} \right)-7=0$$
[b]p6.[/b] A man is $\frac38$ of the way across a narrow railroad bridge when he hears a train approaching at $60$ miles per hour. No matter which way he runs he can [u]just [/u] escape being hit by the train. How fast can he run? Prove your assertion.
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MMPC Part II 1958 - 95, 1967
[b]p1.[/b] Consider the system of simultaneous equations
$$(x+y)(x+z)=a^2$$
$$(x+y)(y+z)=b^2$$
$$(x+z)(y+z)=c^2$$
, where $abc \ne 0$. Find all solutions $(x,y,z)$ in terms of $a$,$b$, and $c$.
[b]p2.[/b] Shown in the figure is a triangle $PQR$ upon whose sides squares of areas $13$, $25$, and $36$ sq. units have been constructed. Find the area of the hexagon $ABCDEF$ .
[img]https://cdn.artofproblemsolving.com/attachments/b/6/ab80f528a2691b07430d407ff19b60082c51a1.png[/img]
[b]p3.[/b] Suppose $p,q$, and $r$ are positive integers no two of which have a common factor larger than $1$. Suppose $P,Q$, and $R$ are positive integers such that $\frac{P}{p}+\frac{Q}{q}+\frac{R}{r}$ is an integer. Prove that each of $P/p$, $Q/q$, and $R/r$ is an integer.
[b]p4.[/b] An isosceles tetrahedron is a tetrahedron in which opposite edges are congruent. Prove that all face angles of an isosceles tetrahedron are acute angles.
[img]https://cdn.artofproblemsolving.com/attachments/7/7/62c6544b7c3651bba8a9d210cd0535e82a65bd.png[/img]
[b]p5.[/b] Suppose that $p_1$, $p_2$, $p_3$ and $p_4$ are the centers of four non-overlapping circles of radius $1$ in a plane and that, $p$ is any point in that plane. Prove that $$\overline{p_1p}^2+\overline{p_2p}^2+\overline{p_3p}^2+\overline{p_4p}^2 \ge 6.$$
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MMPC Part II 1958 - 95, 1970
[b]p1.[/b] Show that the $n \times n$ determinant
$$\begin{vmatrix}
1+x & 1 & 1 & . & . & . & 1 \\
1 & 1+x & 1 & . & . & . & 1 \\
. & . & . & . & . & . & . \\
. & . & . & . & . & . & . \\
1 & 1 & . & . & . & . & 1+x \\
\end{vmatrix}$$
has the value zero when $x = -n$
[b]p2.[/b] Let $c > a \ge b$ be the lengths of the sides of an obtuse triangle. Prove that $c^n = a^n + b^n$ for no positive integer $n$.
[b]p3.[/b] Suppose that $p_1 = p_2^2+ p_3^2 + p_4^2$ , where $p_1$, $p_2$, $p_3$, and $p_4$ are primes. Prove that at least one of $p_2$, $p_3$, $p_4$ is equal to $3$.
[b]p4.[/b] Suppose $X$ and $Y$ are points on tJhe boundary of the triangular region $ABC$ such that the segment $XY$ divides the region into two parts of equal area. If $XY$ is the shortest such segment and $AB = 5$, $BC = 4$, $AC = 3$ calculate the length of $XY$.
Hint: Of all triangles having the same area and same vertex angle the one with the shortest base is isosceles.
Clearly justify all claims.
[b]p5.[/b] Find all solutions of the following system of simultaneous equations
$$x + y + z = 7\,\, , \,\, x^2 + y^2 + z^2 = 31\,\,, \,\,x^3 + y^3 + z^3 = 154$$
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MMPC Part II 1958 - 95, 1965
[b]p1.[/b] For what integers $x$ is it possible to find an integer $y$ such that $$x(x + 1) (x + 2) (x + 3) + 1 = y^2 ?$$
[b]p2.[/b] Two tangents to a circle are parallel and touch the circle at points $A$ and $B$, respectively. A tangent to the circle at any point $X$, other than $A$ or $B$, meets the first tangent at $Y$ and the second tangent at $Z$. Prove $AY \cdot BZ$ is independent of the position of $X$.
[b]p3.[/b] If $a, b, c$ are positive real numbers, prove that $$8abc \le (b + c) (c + a) (a + b)$$ by first verifying the relation in the special case when $c = b$.
[b]p4.[/b] Solve the equation $$\frac{x^2}{3}+\frac{48}{x^2}=10 \left( \frac{x}{3}-\frac{4}{x}\right)$$
[b]p5.[/b] Tom and Bill live on the same street. Each boy has a package to deliver to the other boy’s house. The two boys start simultaneously from their own homes and meet $600$ yards from Bill's house. The boys continue on their errand and they meet again $700$ yards from Tom's house. How far apart do the boy's live?
[b]p6.[/b] A standard set of dominoes consists of $28$ blocks of size $1$ by $2$. Each block contains two numbers from the set $0,1,2,...,6$. We can denote the block containing $2$ and $3$ by $[2, 3]$, which is the same block as $[3, 2]$. The blocks $[0, 0]$, $[1, 1]$,..., $[6, 6]$ are in the set but there are no duplicate blocks.
a) Show that it is possible to arrange the twenty-eight dominoes in a line, end-to-end, with adjacent ends matching, e. g., $... [3, 1]$ $[1, 1]$ $[1, 0]$ $[0, 6] ...$ .
b) Consider the set of dominoes which do not contain $0$. Show that it is impossible to arrange this set in such a line.
c) Generalize the problem and prove your generalization.
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