© Abramson Math, 2022

Hints are better than ready-made answers!

Primary school

Problems that first-graders easily solve

Who broke the cup? Could Oksana have broken the cup? What about Lena or Sonya?
Oksana said that it was Sonya who had broken the cup. Lena and Sonya both said who had broken the cup, but so quietly that no one heard them. It is known that only the girl who had broken the cup told the truth.

1st hint
Could Oksana break the cup?
2nd hint
Could Oksana tell the truth?
How to cook three pancakes?
How to cook three pancakes (both sides) as fast as possible on a pan that can fit only two pancakes simultaneously? It takes 2 minutes to cook one side of one pancake.

Every pancake has 2 sides, there are 6 sides in total. In one minute, you can cook 2 bottom sides of 2 pancakes. If you cooked a maximum of two sides in a 2 minutes, you would be able to cook all 6 sides in 3 minutes.

There is one restriction: you can’t simultaneously cook two sides of the same pancake.
1st hint
You have to cook six sides of three pancakes, and you can’t cook both sides of one pancake simultaneously.
2nd hint

You can achieve minimal time if there are always two pancakes in the pan. Then you can cook them all in six minutes.

Problems for the second grade

Which player would have a winning strategy: the first one or the second one? What is the winning strategy?
There is a box with 120 chocolates on the table. Two players take chocolates from the box in turn. There are two rules: you can’t take all chocolates in your first turn, and you can’t take more chocolates in your turn than the other player took in theirs. The winner is the player who took the last chocolate. The first player is the one who makes the first move.

1st hint
If the first player takes an odd number of chocolates, the second player will take one and win.
2nd hint
If the first player takes 2, 6, or 10 chocolates, the second player will take two and win because number 120 has an even number of twos.
How many knights and knaves are there in the cabinet of ministers?
Is it possible to assert anything about the first minister? About the last one? About the second one?
There are 16 people in the cabinet of ministers. All ministers are either knights or knaves. Knights always tell the truth, and knaves always lie.

The first minister said: “There are no knights among us!”
The second minister said: “There is no more than one knight among us!”
The third minister said: “There are no more than two knights among us!”
The last minister said: “There are no more than 15 knights among us!”
1st hint
Was the first minister a knight or a knave?
2nd hint

Who was the last minister, a knight or a knave?

Problems for the third grade

Find out to which barrel an evil pirate has added sleeping pills.
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.

Prove that there is a regular triangle in the plane (possibly of another size and position) with vertices of the same color.
The plane is divided into regular triangles by three equidistant families of parallel lines. The vertices of these triangles are randomly painted in two colors.
1st hint
If there is no such triangle, then in any small triangle with side length 1, three vertices are painted in two colors, for example, as in triangle 123. What color should be vertex 4, so there would be no equilateral triangle with vertices of the same color?
2nd hint
What color should vertex 5 be? Vertices 6 and 7? Vertex 8? Think where you should put vertex 9.

Problems for the fourth grade

How many markers were produced?
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.

Find the measure of angle A.
In a triangle ABC, the bisector from vertex A, the altitude from vertex B, and the perpendicular bisector to side AB all intersect at one point.
1st hint
Triangle AOB is an isosceles triangle.
2nd hint

Triangle ABD is the right triangle.

Middle school

Problems that are easily solved by the fifth graders who study with Yakov Abramson

Prove that the point N, which is symmetric to the point H with respect to M, lies on the circle circumscribed about the triangle ABC.
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.

How long did the candles burn?
Two candles of the same length were lit at the same time: the thicker candle, burning down in 4 hours, and the thinner candle, burning down in 2 hours. After a while, both candles were extinguished. The stub of a thick candle turned out to be 3 times longer than the stub of a thin candle.
1st hint
Let us build graphs of two candles burning and mark candle stubs on them
2nd hint

Based on the similarity of the corresponding pairs of triangles BA2 and OL2, CA4 and OL4, we formulate a system of equations.

Problems for the sixth graders

Find the taxi’s serial number, given that the distance between the train station and the airport is 10 km.
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.
1st hint
Find the time it takes for a taxi to get from the airport to the train station. When did the taxi driver leave the train station for the last time?
2nd hint

The total amount of time between two drives is 40 minutes, and the time when the car should leave the train station for the first time must be a multiple of 7.

Problems for the seventh grade

How many bees were in the swarm?
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.

Problems for the eighth grade

Prove that the center of the circle lies on the first circle.
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.
1st hint
The center O of the incircle of triangle AMN also lies on the bisector AI. The only thing left to proof is that IK = IN = IM = IO.
2nd hint

Angle ION is external to triangle AON, and AMIN is inscribed.

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Senior school

A problem that students solve in the ninth grade

How often do they meet there on average, say, during the academic year?
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.

The problem that is given in the freshman year at the Faculty of Mechanics and Mathematics of the Lomonosov Moscow State University and easily solved by tenth graders

Solve 2021*1976
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
2nd hint
m*1 +1=m*(1*1)=m, consequently m*1=m-1

The problem that is given in the sophomore year at the Faculty of Mechanics and Mathematics of the Lomonosov Moscow State University and easily solved by eleventh graders who study Abramson Math

Prove that EF||AD
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.
1st hint
Triangles ABF and DCK are equal as having pairwise parallel sides, and AB = CD, so BCKF is a parallelogram, hence angle CKF = a.
2nd hint

FGHE and HCKD are inscribed, hence angle CKH=a, hence F, H, and K lie on the same line.

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