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Problems Algebra Sequences Progressions Arithmetic progression

There are one hundred natural numbers, they are all different, and sum up to 5050. Can you find those numbers? Are they unique, or is there another bunch of such numbers?

Problem #PRU-107992

Problems Calculus Real numbers Integer and fractional parts. Archimedean property Methods Pigeonhole principle Algebra Sequences Algebraic methods Symmetry and involutions

14–16 5.0

For each pair of real numbers $a$ and $b$ , consider the sequence of numbers $p_{n} = ⌊ 2 {a n + b} ⌋$ . Any $k$ consecutive terms of this sequence will be called a word. Is it true that any ordered set of zeros and ones of length $k$ is a word of the sequence given by some $a$ and $b$ for $k = 4$ ; when $k = 5$ ?

Note: $⌊ c ⌋$ is the integer part, ${c}$ is the fractional part of the number $c$ .

Problem #PRU-109599

Problems Calculus Number sequences Boundedness, monotonicity Algebra Polynomials Algebraic identities for polynomials Identical transformations Sequences

14–16 5.0

Prove that for any natural number $a_{1} > 1$ there exists an increasing sequence of natural numbers $a_{1}, a_{2}, a_{3}, \dots$ , for which $a_{1}^{2} + a_{2}^{2} + \dots + a_{k}^{2}$ is divisible by $a_{1} + a_{2} + \dots + a_{k}$ for all $k \geq 1$ .

Problem #PRU-109715

Problems Algebra Sequences Progressions Geometric progression Calculus Real numbers Integer and fractional parts. Archimedean property Sums of number sequences and difference series

13–15 4.0

Find the sum $1 / 3 + 2 / 3 + 2^{2} / 3 + 2^{3} / 3 + \dots + 2^{1000} / 3$ .

Problem #PRU-116433

Problems Calculus Functions of one variable. Continuity Functions. Continuity (other) Algebra Sequences Progressions Geometric progression

14–16 3.0

The function $f (x)$ is defined on the positive real $x$ and takes only positive values. It is known that $f (1) + f (2) = 10$ and $f (a + b) = f (a) + f (b) + 2 \sqrt{f (a) f (b)}$ for any $a$ and $b$ . Find $f (2^{2011})$ .

Problem #PRU-116589

Problems Number Theory Divisibility Divisibility of a number. General properties Algebra Sequences Recurrent relations Linear recurrent relations Calculus Number sequences Number sequences (other)

13–15 3.0

The sequence of numbers $a_{1}, a_{2}, \dots$ is given by the conditions $a_{1} = 1$ , $a_{2} = 143$ and

for all $n \geq 2$ .

Prove that all members of the sequence are integers.

Problem #PRU-34834

Problems Algebra Sequences Progressions Arithmetic progression Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

12–14 3.0

A road of length 1 km is lit with streetlights. Each streetlight illuminates a stretch of road of length 1 m. What is the maximum number of streetlights that there could be along the road, if it is known that when any single streetlight is extinguished the street will no longer be fully illuminated?

Problem #PRU-34854

Problems Methods Pigeonhole principle Pigeonhole principle (other) Algebra Sequences Recurrent relations Recurrent relations (other)

12–14 3.0

In the number $1234096 \dots$ each digit, starting with the 5th digit, is equal to the final digit of the sum of the previous 4 digits. Will the digits 8123 ever occur in a row in this number?

Problem #PRU-60364

Problems Algebra Sequences Progressions Arithmetic progression Methods Pigeonhole principle Pigeonhole principle (other) Fun Problems Word Problems Tables and tournaments Tournament tables

12–14 3.0

In a volleyball tournament teams play each other once. A win gives the team 1 point, a loss 0 points. It is known that at one point in the tournament all of the teams had different numbers of points. How many points did the team in second last place have at the end of the tournament, and what was the result of its match against the eventually winning team?

Problem #PRU-60468

Problems Algebra Sequences Progressions Arithmetic progression Calculus Functions of one variable. Continuity Certain properties of a function and recurrence relations.

13–15 3.0

Suppose that there are 15 prime numbers forming an arithmetic progression with a difference of $d$ . Prove that $d > 30, 000$ .