MathNet 2008-part-03 — 165 problems (EN–VI)

The sequence $49$, $4489$, $444889$, ... is given so that the number $48$ is put in the middle of the preceding term. Prove that every term in the sequence is perfect square.

Find the value of $y$ so that $\sqrt{y^2 + 2y + 1}$, $\frac{y^2 + 3y - 1}{3}$, $y-1$ are consecutive terms in the arithmetic progression.

The sides of the triangle are consecutive terms in the arithmetic progression. Prove that the line connecting the centroid and the center of the incircle is parallel to the side of the triangle with middle length.

Solve the equation $\frac{x-1}{x} + \frac{x-2}{x} + \dots + \frac{1}{x} = \frac{3x-20}{4}$ in $\mathbb{N}$.

Prove that if $\left|\frac{a+b}{2}\right| + \left|\frac{a-b}{2}\right| \lt c$, for $a, b, c \in \mathbb{R}$, then $|a| \lt c$ and $|b| \lt c$.

Prove that if $2a_m = a_n$, then $a_{2m-n}$ is a perfect square, where $a_n = 1+2+...+n$, for every $n \in \mathbb{N}$.

In a school there are $32$ classes. Six of them have $29$ students each, eight of them have $30$ students each, two of them have $31$ students each and the rest have $32$ students each. How many students are there in the school?

If one of the sides of a square is increased two times and the other one is increased by $22\text{ mm}$, then the new rectangle has a perimeter that is $2000\text{ mm}$ greater than the perimeter of the square. What is the side length of the square?

There were 25 birds on two branches. After a while 5 birds from the first branch flew over to the second and after a while longer 7 birds from the second flew away. In that moment on the first branch there were twice as many birds than on the second. How many birds were there on each of the two branches in the beginning?

From the digits $3$, $7$, $1$, $9$, $0$ and $4$ Mila formed the biggest and the smallest six-digit number using each digit exactly once in each of the two numbers. Then she reduced their difference $9$ times. Which number did she get?

Four boys Andrej, Bojan, Vasko and Goce are collecting post stamps. Andrej has as many post stamps as Bojan and Vasko have together. Goce has five times less post stamps than Andrej, and Bojan has four times more post stamps than Vasko. If they together have 5016 post stamps, how many post stamps each of the four boys has?

There are two empty pots on disposal. The first pot can contain exactly 3 liters of liquid and the other exactly 5 liters. Is it possible with these pots to measure exactly 4 liters of liquid?

One student was multiplying two numbers. During the multiplication he switched the last digit of the first number, which was $4$, with $1$. So he obtained $525$ as a result instead of $600$. Which numbers did the student multiply?

On training in a football club there were 225 children and 105 balls. The children were split in few equal groups. The coaches gave to each group equal number of balls. How many groups were formed and how many balls did every group get? How many solutions does the problem have?

Two lines intersect in one point and form four angles, two acute and two obtuse. The sum of the two acute angles is half of the one obtuse angle. Calculate these angles?

One rectangle has side lengths $a$ cm and $b$ cm. If the side with length $a$ cm is enlarged by $b$ cm and the side with length $b$ cm is enlarged by $a$ cm then the resulting square has area of $100\,\mathrm{cm}^2$. Determine the rectangle that satisfies this condition with smallest area if its side lengths are positive integers.

Determine the sum of all odd 3-digit numbers whose product of digits is equal to $140$.

Maja went to the bookstore to buy two books. The price of the first book was $65\%$ and the price of the other $57.5\%$ from the money that Maja had with her. She needed additional $45$ denars to buy the two books. How much money did Maja have with her?

Two lines $p$ and $q$ intersect at $C$. The point $C$ divides the line $p$ into two half-lines that, together with one of the half-lines of $q$ with starting point at $C$, form two angles $pCq$ and $qCp$. On the bisector of the angle $pCq$ a point $M$ is chosen such that $MN \parallel p$. The segment $MN$ intersects the line $q$ at a point $D$. Prove that $D$ is the midpoint of the segment $MN$.

Ilina ate $\frac{1}{5}$ plus three of the candies from the bag. From the remaining candies she ate $\frac{1}{5}$ plus five the next day. The third day she ate the remaining 15 candies. How many candies were there in the bag in the beginning?