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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-116661

Problems Fun Problems Word Problems Solving problems by inequalities. Going through the cases. Algebra Sequences Progressions Arithmetic progression Methods Pigeonhole principle Pigeonhole principle (other) Proof by contradiction

10–12 3.0

Can 100 weights of masses 1, 2, 3, ..., 99, 100 be arranged into 10 piles of different masses so that the following condition is fulfilled: the heavier the pile, the fewer weights in it?

Problem #PRU-30744

Problems Number Theory Numeral systems Decimal number system Algebra Sequences Progressions Geometric progression

11–13 2.0

How many integers are there from 0 to 999999, in the decimal notation of which there are no two identical numbers next to each other?

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-88105

Problems Algebra Sequences Progressions Arithmetic progression

12–15 2.0

Is the sum of the numbers $1 + 2 + 3 + \dots + 1999$ divisible by 1999?

Problem #PRU-98427

Problems Algebra Sequences Recurrent relations Linear recurrent relations Calculus Functions of one variable. Continuity Periodicity and aperiodicity

12–14 3.0

In a row there are 2023 numbers. The first number is 1. It is known that each number, except the first and the last, is equal to the sum of two neighboring ones. Find the last number.

Problem #WSP-5286

Problems Algebra Sequences Fun Problems Word Problems

10–13 2.0

In good conditions, bacteria in a Petri cup spread quite fast, doubling every second. If there was initially one bacterium, then in $32$ seconds the bacteria will cover the whole surface of the cup.

Now suppose that there are initially $4$ bacteria. At what time will the bacteria cover the surface of the cup?

Problem #WSP-000112

Problems Algebra Sequences Fibonacci

10–14 1.0

Among the first $20$ Fibonacci numbers: $F_{0} = 0, F_{1} = 1, F_{2} = 1, F_{3} = 2, F_{4} = 3, . . ., F_{20} = 6765$ find all numbers whose digit-sum is equal to their index. For example, $F_{1} = 1$ fits the description, but $F_{20} = 6765$ does not, since $6 + 7 + 6 + 5 \neq 20$ .