Practice (EndingDigits,TheDivideByNineMethod,MODBasic)

3520

Find all solutions to

$(m^2+n)(m + n^2)= (m - n)^3$ , where m and n are non-zero integers.

3521

The feet of the angle bisectors of

$\Delta ABC$ form a right-angled triangle. If the right-angle is at

$X$ , where

$AX$ is the bisector of

$\angle A$ , find all possible values for

$\angle A$ .

3522

$X$ is the smallest set of polynomials

$p(x)$ such that: 1.

$p(x) = x$ belongs to

$X$ . 2. If

$r(x)$ belongs to

$X$ , then

$x\cdot r(x)$ and

$(x + (1 - x) \cdot r(x) )$ both belong to

$X$ . Show that if

$r(x)$ and

$s(x)$ are distinct elements of

$X$ , then

$r(x) \neq s(x)$ for any

$0 < x < 1$ .

3523

M is the midpoint of XY. The points P and Q lie on a line through Y on opposite sides of Y, such that

$|XQ| = 2|MP|$ and

$\frac{|XY|}2 < |MP| < \frac{3|XY|}2$ . For what value of

$\frac{|PY|}{|QY|}$ is

$|PQ|$ a minimum?

3524

$a_1, a_2, \cdots, a_n$ is a sequence of 0's and 1's. T is the number of triples

$(a_i, a_j, a_k)$ with $ii

$with$ a_j\neq a_i

$. Show that$ T=\sum_{i=1}^n f(i)\cdot\left(\frac{f(i)-1}2\right)$. If n is odd, what is the smallest value of T?

3525

The repeating decimal

$0.ab \cdots k \overline {pq \cdots u} = \dfrac {m}{n}$ , where

$m$ and

$n$ are relatively prime integers, and there is at least one decimal before the repeating part. Show that

$n$ is divisible by

$2$ or

$5$ (or both). (For example,

$0.011 \overline {36} = 0.01136363636 \cdots = \dfrac {1}{88}$ , and

$88$ is divisible by

$2$ .)

3526

The cubic polynomial

$x^3 + ax^2 + bx + c$ has real coefficients and three real roots

$r \ge s \ge t$ . Show that

$k = a^2 - 3b \ge 0$ and that

$\sqrt {k} \le r - t$ .

3527

Let

$X$ be the set

$\{ 1, 2, \cdots, 20 \}$ and let

$P$ be the set of all 9-element subsets of

$X$ . Show that for any map

$f: P \mapsto X$ we can find a 10-element subset

$Y$ of

$X$ , such that

$f(Y - \{ k \}) \ne k$ for any

$k$ in

$Y$ .

3528

$\triangle ABC$ is a triangle with incenter

$I$ . Show that the circumcenters of

$\triangle IAB, \triangle IBC, \triangle ICA$ lie on a circle whose center is the circumcenter of

$\triangle ABC$ .

3529

A polynomial product of the form

$(1-z)^{b_1}(1-z^2)^{b_2}(1-z^3)^{b_3}(1-z^4)^{b_4}(1-z^5)^{b_5}\cdots(1-z^{32})^{b_{32}},$ where the

$b_k$ are positive integers, has the surprising property that if we multiply it out and discard all terms involving

$z$ to a power larger than

$32$ , what is left is just

$1-2z$ . Determine, with proof,

$b_{32}$ .

3530

For each positive integer

$n$ , let

$\begin{eqnarray*} S_n &=& 1 + \frac 12 + \frac 13 + \cdots + \frac 1n, \\ T_n &=& S_1 + S_2 + S_3 + \cdots + S_n, \\ U_n &=& \frac{T_1}{2} + \frac{T_2}{3} + \frac{T_3}{4} + \cdots + \frac{T_n}{n+1}. \end{eqnarray*}$ Find, with proof, integers

$0 < a, b,c, d < 1000000$ such that

$T_{1988} = a S_{1989} - b$ and

$U_{1988} = c S_{1989} - d$ .

3531

The 20 members of a local tennis club have scheduled exactly 14 two-person games among themselves, with each member playing in at least one game. Prove that within this schedule there must be a set of 6 games with 12 distinct players.

3532

Let

$P(z)= z^n + c_1 z^{n-1} + c_2 z^{n-2} + \cdots + c_n$ be a polynomial in the complex variable

$z$ , with real coefficients

$c_k$ . Suppose that

$|P(i)| < 1$ . Prove that there exist real numbers

$a$ and

$b$ such that

$P(a + bi) = 0$ and

$(a^2 + b^2 + 1)^2 < 4 b^2 + 1$ .

3533

Let

$ABC$ be an acute-angled triangle whose side lengths satisfy the inequalities

$AB < AC < BC$ . If point

$I$ is the center of the inscribed circle of triangle

$ABC$ and point

$O$ is the center of the circumscribed circle, prove that line

$IO$ intersects segments

$AB$ and

$BC$ .

3534

Let

$u$ and

$v$ be real numbers such that

$(u + u^2 + u^3 + \cdots + u^8) + 10u^9 = (v + v^2 + v^3 + \cdots + v^{10}) + 10v^{11} = 8.$ Determine, with proof, which of the two numbers,

$u$ or

$v$ , is larger.

3535

A certain state issues license plates consisting of six digits (from 0 to 9). The state requires that any two license plates differ in at least two places. (For instance, the numbers 027592 and 020592 cannot both be used.) Determine, with proof, the maximum number of distinct license plates that the state can use.

3536

A sequence of functions

$\, \{f_n(x) \} \,$ is defined recursively as follows:

$\begin{align*}f_1(x) &= \sqrt{x^2 + 48}, \quad \mbox{and} \\ f_{n+1}(x) &= \sqrt{x^2 + 6f_n(x)} \quad \mbox{for } n \geq 1.\end{align*}$ (Recall that

$\sqrt{\makebox[5mm]{}}$ is understood to represent the positive square root.) For each positive integer

$n$ , find all real solutions of the equation

$\, f_n(x) = 2x \,$ .

3537

Suppose that necklace

$\, A \,$ has 14 beads and necklace

$\, B \,$ has 19. Prove that for any odd integer

$n \geq 1$ , there is a way to number each of the 33 beads with an integer from the sequence

$\{ n, n+1, n+2, \dots, n+32 \}$ so that each integer is used once, and adjacent beads correspond to relatively prime integers. (Here a ``necklace'' is viewed as a circle in which each bead is adjacent to two other beads.)

3538

Find, with proof, the number of positive integers whose base-

$n$ representation consists of distinct digits with the property that, except for the leftmost digit, every digit differs by

$\pm 1$ from some digit further to the left. (Your answer should be an explicit function of

$n$ in simplest form.)

3539

An acute-angled triangle

$ABC$ is given in the plane. The circle with diameter

$\, AB \,$ intersects altitude

$\, CC' \,$ and its extension at points

$\, M \,$ and

$\, N \,$ , and the circle with diameter

$\, AC \,$ intersects altitude

$\, BB' \,$ and its extensions at

$\, P \,$ and

$\, Q \,$ . Prove that the points

$\, M, N, P, Q \,$ lie on a common circle.

3540

In triangle

$\, ABC, \,$ angle

$\,A\,$ is twice angle

$\,B,\,$ angle

$\,C\,$ is obtuse, and the three side lengths

$\,a,b,c\,$ are integers. Determine, with proof, the minimum possible perimeter.

3541

For any nonempty set

$\,S\,$ of numbers, let

$\,\sigma(S)\,$ and

$\,\pi(S)\,$ denote the sum and product, respectively, of the elements of

$\,S\,$ . Prove that

$\sum \frac{\sigma(S)}{\pi(S)} = (n^2 + 2n) - \left(1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \right) (n+1),$ where ``

$\Sigma$ '' denotes a sum involving all nonempty subsets

$S$ of

$\{1,2,3, \ldots,n\}$ .

3542

Show that, for any fixed integer

$\,n \geq 1,\,$ the sequence

$2, \; 2^2, \; 2^{2^2}, \; 2^{2^{2^2}}, \ldots (\mbox{mod} \; n)$ is eventually constant. [The tower of exponents is defined by

$a_1 = 2, \; a_{i+1} = 2^{a_i}$ . Also

$a_i \; (\mbox{mod} \; n)$ means the remainder which results from dividing

$a_i$ by

$n$ .]

3543

Let

$a = \frac{m^{m+1} + n^{n+1}}{m^m + n^n}$ , where

$m$ and

$n$ are positive integers. Prove that

$a^m + a^n \geq m^m + n^n$ .

3544

Let

$\, D \,$ be an arbitrary point on side

$\, AB \,$ of a given triangle

$\, ABC, \,$ and let

$\, E \,$ be the interior point where

$\, CD \,$ intersects the external common tangent to the incircles of triangles

$\, ACD \,$ and

$\, BCD$ . As

$\, D \,$ assumes all positions between

$\, A \,$ and

$\, B \,$ , prove that the point

$\, E \,$ traces the arc of a circle.

USAMO

With Solutions

Practice (EndingDigits,TheDivideByNineMethod,MODBasic)

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