Hints are better than ready-made answers!

A good doctor Powderpill is delivering four barrels of medicines to Africa. An evil pirate managed to add and dissolve sleeping pills into one of the barrels. The sleeping draft kicks in in an hour. The good doctor has only two assistants: a parrot and a hamster. He can pour medicine from any barrel only once to his assistants, but he must do it simultaneously — he cannot give medicine to only one of the assistants and wait for results.

1st hint
Number the barrels: 1, 2, 3, 4.
2nd hint

There are four possible outcomes: 1. The first assistant fell asleep; 2. The second assistant fell asleep; 3. Both assistants fell asleep; 4. Nobody fell asleep.

The factory produced a batch of marker pens. If this batch is laid out in boxes of 6 marker pens, then 3 marker pens will remain; if in boxes of 7, then 4 marker pens will remain; if in boxes of 8, then only 1 marker pen will remain. It is known that there are no more than 350 marker pens in this batch.

1st hint

The problem is equivalent to a system of comparisons X ≤ 350

2nd hint

Put the solution of the last comparison Х=8k+1, k=0,1,2… into the second one:

(8k+1)mod7=kmod7+1≡4, k=3⇒X=25+56m;

Then put it back into the first one.

Let AA1 and BB1 be the altitudes in the triangle ABC, H is the point of their intersection (orthocenter of the triangle ABC), M is the midpoint of the side AB.

1st hint
ANBN is a parallelogram
2nf hint

Angle ANB = angle AHB = angle B 1 HA 1, and angle CB 1 HA 1 is inscribed.

Taxis run between the train station and the airport. Each of the cars has its own serial number: 1, 2, and so on. The traffic is organized in such a way that the first taxi leaves the station at 6:00, and after it with an interval of 7 minutes, the rest of the cars depart according to their numbers.

Once the taxi begins its shift, it runs back and forth independently of other cars, making 5-minute stops after reaching each of the destination points.

It is known that each taxi spends 2 minutes on its first kilometer and 1/9 min less for each subsequent kilometer than the previous one. The passenger left the airport at 9.44 am.

An old Indian problem.
The number of bees, equal to the square root of half of their entire swarm, sat on a jasmine bush, leaving eight-ninths of a swarm behind them. And only one bee from the same swarm is circling near the lotus, attracted by the buzzing of a friend who has inadvertently fallen into the trap of a sweet-smelling flower.

1st hint

Equation +x+2=X simulates this problem and is reduced to the form:

81х=2х² -72х+8×81

2nd hint

The problem implies that X must be an integer; it follows then that X must an even number and a multiple of 9: х=2*9k. After substitution, we see that k must be a multiplier of 4, k=4m.

A circle is inscribed in triangle ABC.
It touches its sides AC and AB at points M and N, respectively.
Another circle is inscribed in triangle AMN.

The director and their deputy come to the office at a random time every day between 9 and 10 a.m., wait for each other for 5 minutes, and then leave the office.

1st hint
Let t = 5 minutes. Then the time when the director and his deputy come into a room is subject to the condition: 0 is greater or equal to x, y is greater or equal to 12, and can be presented as the points (x,y) on the square 12*12.
2nd hint

The director and his deputy meet only if |x-y| is greater or equal to 1. The probability of their meeting is equal to the ratio of the area of this strip to the area of the whole square.

On the set Z of integers, an operation * is defined that has only the following two properties:
а) m*m=0 "mîZ и
b) m*(n*k)=(m*n)+k " m,n,k î Z

1st hint
m*0=m*(m*m)=m
2nd hint
m*1 +1=m*(1*1)=m, consequently m*1=m-1

ABCD is a parallelogram; rays BE and DF (CBE=CDF) are drawn from opposite vertices B and D inside it at equal angles to neighboring sides. From another pair of opposite vertices A and C, rays AE and CF (DAE=DCF) are drawn inside it at equal angles to neighboring sides. {E}=BEAE, {F}=DFCF.