Chapter 29 — Parting Shots

Problem 526. If $a + b = 1$, find $a^3 + b^3 + 3ab$. (MAΘ 1991)

Problem 527. What is the units digit of $7^{1991}$? (MATHCOUNTS 1991)

Problem 528. A triangle has sides 5, 12, 13. What is the length of the altitude to the hypotenuse? (MATHCOUNTS)

Problem 529. Find the value of $\dfrac{1}{1 \cdot 2} + \dfrac{1}{2 \cdot 3} + \dfrac{1}{3 \cdot 4} + \cdots + \dfrac{1}{99 \cdot 100}$. (Classic telescoping)

Problem 530. How many zeros are at the end of $100!$?

Problem 531. If $x + \dfrac{1}{x} = 3$, find $x^2 + \dfrac{1}{x^2}$. (AHSME)

Problem 532. Find the area of a triangle with vertices $(0, 0)$, $(4, 0)$, $(2, 5)$. (MATHCOUNTS)

Problem 533. How many 3-digit numbers are palindromes? (MATHCOUNTS)

Problem 534. If $\log_2 x + \log_2 x^2 + \log_2 x^3 = 36$, find $x$. (MAΘ)

Problem 535. A regular hexagon has perimeter 24. What is its area? (MATHCOUNTS)

Problem 536. Find all primes $p$ such that $p + 10$ and $p + 14$ are also prime. (AHSME)

Problem 537. Two fair dice are rolled. What is the probability that the product of the numbers is odd?

Problem 538. Solve for $x$: $\sqrt{x + 7} + \sqrt{x} = 7$. (MATHCOUNTS)

Problem 539. The sum of the angles of a polygon is $2340°$. How many sides does it have?

Problem 540. If $(a - b)^2 + (b - c)^2 + (c - a)^2 = 0$, what can you conclude about $a$, $b$, $c$?

Problem 541. Find the number of ordered pairs $(x, y)$ of positive integers such that $\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{6}$. (AHSME)

Problem 542. Ten points are on a circle. How many triangles can be formed using these points as vertices? (MATHCOUNTS)

Problem 543. Find the sum of all two-digit multiples of 7. (MAΘ)

Problem 544. If $a$, $b$, $c$ are the roots of $x^3 - 6x^2 + 11x - 6 = 0$, find $a^2 + b^2 + c^2$. (Vieta’s formulas)

Problem 545. Prove that $1^3 + 2^3 + \cdots + n^3 = (1 + 2 + \cdots + n)^2$ by induction.

Problem 546. A clock’s minute and hour hands coincide at 12:00. At what time do they next coincide? (MATHCOUNTS)

Problem 547. What is the maximum number of regions into which 10 lines can divide the plane? (AHSME)

Problem 548. Find the remainder when $3^{100}$ is divided by 7.

Problem 549. How many diagonals does a convex 15-gon have?

Problem 550. If $f(x) = 2x - 3$ and $g(x) = x^2 + 1$, find $f(g(2))$ and $g(f(2))$.

Problem 551. In a round-robin tournament with 8 teams, how many games are played?

Problem 552. Find all integers $n$ such that $n^2 + 5n + 6$ is a perfect square.

Problem 553. A 6 × 8 rectangle is inscribed in a circle. What is the area of the circle?

Problem 554. For how many integers $n$ between 1 and 100 inclusive is $n^2 - n$ divisible by 3?

Problem 555. If $i = \sqrt{-1}$, find $i + i^2 + i^3 + \cdots + i^{100}$. (AHSME)

Problem 556. Three circles of radius 1 are mutually tangent. Find the area of the region enclosed between them.

Problem 557. The average of $n$ numbers is 32. When 17 is added to the list, the average is 30. Find $n$. (MATHCOUNTS)

Problem 558. Find the smallest positive integer $n$ such that $n!$ is divisible by $10^{10}$.

Problem 559. A square and equilateral triangle have equal perimeters. If the triangle has area $16\sqrt{3}$, find the area of the square. (MATHCOUNTS)

Problem 560. Find $\displaystyle\sum_{k=1}^{n} k \cdot k!$

Problem 561. Two sides of a triangle are 7 and 11. The length of the third side is an integer. How many possible values are there?

Problem 562. If $a \star b = ab + a + b$, find $(2 \star 3) \star 4$. (MAΘ)

Problem 563. Prove that among any 5 consecutive integers, exactly one is divisible by 5.

Problem 564. A cube is painted red and then cut into 27 smaller cubes. How many small cubes have paint on exactly 2 faces? (MATHCOUNTS)

Problem 565. If $\log_{10} 2 \approx 0.301$ and $\log_{10} 3 \approx 0.477$, find the number of digits in $6^{20}$. (AHSME)

Problem 566. Find the coefficient of $x^4$ in $(1 + x)^{10}$.

Problem 567. The probability of rain each day is independently $1/3$. What is the probability it rains on at least one of three consecutive days?

Problem 568. Find all real $x$ such that $|x - 3| + |x + 1| = 6$. (MAΘ)

Problem 569. A trapezoid has parallel sides of length 8 and 14, and height 5. Find its area.

Problem 570. Prove that $\gcd(n, n+1) = 1$ for all positive integers $n$.

Problem 571. Find the number of positive divisors of $2^4 \cdot 3^3 \cdot 5^2 \cdot 7$.

Problem 572. Evaluate $\displaystyle\prod_{k=2}^{10} \frac{k^2 - 1}{k^2}$.

Problem 573. A bag contains 3 red, 4 blue, and 5 green marbles. What is the least number that must be drawn to guarantee at least 3 of one color?

Problem 574. If $x^2 + y^2 = 1$ and $x + y = t$, express $xy$ in terms of $t$. (AHSME)

Problem 575. In how many ways can you arrange the letters of ABRACADABRA?

Problem 576. Find the last two digits of $7^{2023}$.

Problem 577. A right triangle has legs $a$ and $b$ and hypotenuse $c$. If $a + b = 17$ and $c = 13$, find the area of the triangle.

Problem 578. Prove that $\sqrt{2} + \sqrt{3}$ is irrational.

Problem 579. How many lattice points $(x, y)$ with integer coordinates satisfy $x^2 + y^2 \leq 25$? (AHSME)

Problem 580. If $f(x) = x^2$ and $g(x) = 2x + 1$, find all $x$ such that $f(g(x)) = g(f(x))$.

Problem 581. A circle of radius $r$ is inscribed in a triangle with sides 3, 4, 5. Find $r$. (Classic)

Problem 582. Find the value of $2^{1/2} \cdot 4^{1/4} \cdot 8^{1/8} \cdot 16^{1/16} \cdots$ (Infinite product)

Problem 583. In a class of 30, each student takes at least one of French (18 students) and Spanish (15 students). How many take both? (MATHCOUNTS)

Problem 584. Find the maximum value of $xy$ if $x + y = 10$ and $x, y > 0$. (AM-GM)

Problem 585. Find the sum $\displaystyle\sum_{k=0}^{6} \binom{6}{k}$.

Problem 586. A circle has equation $x^2 + y^2 - 6x + 8y = 0$. Find its center and radius.

Problem 587. The harmonic series $1 + \frac{1}{2} + \frac{1}{3} + \cdots$ diverges. Prove that its partial sums grow without bound. (Hint: Group terms in powers of 2.)

Problem 588. Show that any convex polygon with more than 3 sides can be triangulated (divided into triangles by non-crossing diagonals), and that an $n$-gon is divided into $n - 2$ triangles.