AoPS Community 2024 Assara - South Russian Girl's MO

www.artofproblemsolving.com/community/c4059821
by NO SQUARES

– Juniors

– Day 1

1 There is a set of 50 cards. Each card on both sides is colored in one of three colors — red, blue
or white, and for each card its two sides are colored in different colors. The cards were laid out
on the table. The card lies beautifully if at least one of two conditions is met: its upper side —
red; its underside is blue. It turned out that exactly 25 cards are lying beautifully. Then all the
cards were turned over. Now some of the cards are lying beautifully on the table. How many of
them can there be?
K.A.Sukhov
p p
2 Let p be a prime number. Positive integers numbers a and b are such a + b = 1 and a + b is
p take?
divisible by p. What values can an expression a+b
Yu.A.Karpenko

3 In the cells of the 4 × N table, integers are written, modulo no more than 2024 (i.e. numbers from
the set {−2024, −2023, . . . , −2, −1, 0, 1, 2, 3, . . . , 2024}) so that in each of the four lines there are
no two equal numbers. At what maximum N could it turn out that in each column the sum of
the numbers is equal to 2?
G.M.Sharafetdinova

4 Is there a described n-gon in which each side is longer than the diameter of the inscribed circle
a) at n = 4? b) when n = 7? c) when n = 6?
P.A.Kozhevnikov

– Day 2

5 Prove that 2024! is divisible by a) 20242 ; b) 20248 .

(n! = 1 · 2 · 3 · ... · n)
Z.Smysl

6 In the regular hexagon ABCDEF , a point X was marked on the diagonal AD such that ∠AEX =
65◦ . What is the degree measure of the angle ∠XCD?
A.V.Smirnov, I.A.Efremov

7 There is a chip in one of the squares on the checkered board. In one move, she can move either
1 square to the right, or diagonally 1 to the left and 1 up, or 1 to the left and 3 down (see Fig.).

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AoPS Community 2024 Assara - South Russian Girl's MO

The chip made n moves and returned to the starting square. Prove that a) n is divisible by 2, b)
n is divisible by 8.
K.A.Sukhov

8 Given a set S of 2024 natural numbers satisfying the following condition: if you select any 10
(different) numbers from S, then you can select another number from S so that the sum of all
11 selected numbers is divisible by 10. Prove that one of the numbers can be thrown out of S so
that the resulting set S ′ of 2023 numbers satisfies the condition: if you choose any 9 (different)
numbers from S ′ , then you can choose another number from S ′ so that the sum of all 10 selected
numbers is divisible by 10.
K.A.Sukhov

– Seniors

– Day 1

1 There is a set of 2024 cards. Each card on both sides is colored in one of three colors — red,
blue or white, and for each card its two sides are colored in different colors. The cards were laid
out on the table. The card lies beautifully if at least one of two conditions is met: its upper side
— red; its underside is blue. It turned out that exactly 150 cards are lying beautifully. Then all the
cards were turned over. Now some of the cards are lying beautifully on the table. How many of
them can there be?
K.A.Sukhov

2 Prove that in any described 8-gon there is a side that does not exceed the diameter of the in-
scribed circle in length.
P.A.Kozhevnikov

3 In the cells of the 4 × N table, integers are written, modulo no more than 2024 (i.e. numbers from
the set {−2024, −2023, . . . , −2, −1, 0, 1, 2, 3, . . . , 2024}) so that in each of the four lines there are
no two equal numbers. At what maximum N could it turn out that in each column the sum of
the numbers is equal to 23?
G.M.Sharafetdinova

4 A parabola p is drawn on the coordinate plane — the graph of the equation y = −x2 , and a point A
is marked that does not lie on the parabola p. All possible parabolas q of the form y = x2 +ax+b
are drawn through point A, intersecting p at two points X and Y . Prove that all possible XY
lines pass through a fixed point in the plane.
P.A.Kozhevnikov

– Day 2

5 Prove that (100!)99 > (99!)100 > (100!)98 .

K.A.Sukhov

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AoPS Community 2024 Assara - South Russian Girl's MO

6 The points A, B, C, D are marked on the straight line in this order. Circle ω1 passes through
points A and C, and the circle ω2 passes through points B and D. On the circle ω2 , the point E
is marked so that AB = BE, and on the circle ω1 , the point F is marked so that CD = CF . The
line AE intersects the circle ω2 a second time at point X, and the line DF intersects the circle
ω1 at point Y . Prove that the XY lines and AD is perpendicular.
A.D.Tereshin

7 Find all positive integers n for such the following condition holds:
”If a, b and c are positive integers such are all numbers

a2 + 2ab + b2 , b2 + 2bc + c2 , c2 + 2ca + a2

are divisible by n, then (a + b + c)2 is also divisible by n.”

G.M.Sharafetdinova

8 There are 15 boys and 15 girls in the class. The first girl is friends with 4 boys, the second with 5,
the third with 6, . . . , the 11th with 14, and each of the other four girls is friends with all the boys.
It turned out that there are exactly 3 · 225 ways to split the entire class into pairs, so that each
pair has a boy and a girl who are friends. Prove that any of the friends of the first girl are friends
with all the other girls too.
G.M.Sharafetdinova

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