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2025 Caucasus Mathematical Olympiad
X Caucasus Mathematical Olympiad
Juniors
Day 1
1 Anya and Vanya’s houses are located on the straight road. The distance between their houses is divided by
a shop and a school into three equal parts. If Anya and Vanya leave their houses at the same time and
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walk towards each other, they will meet near the shop. If Anya rides a scooter, then her speed will increase
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by , and they will meet near the school. Find Vanya’s speed of walking.
2 There are children standing in a circle. For each girl, it turns out that among the five people following
her clockwise, there are more boys than girls. Find the greatest number of girls that can stand in a circle.
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3 Let be a positive integer. Egor has cards with the number “ ” written on them, and cards
with the number “ ” written on them. Egor wants to paint each card red or blue so that no subset of cards
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of the same color has the sum of the numbers equal to . Find the greatest such that Egor will not view topic
be able to paint the cards in such a way.
4 In a convex quadrilateral , diagonals and are equal, and they intersect at .
Perpendicular bisectors of and intersect at point lying inside triangle , and
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perpendicular bisectors of and intersect at point lying inside triangle . Prove that view topic
.
Day 2
5 Suppose that consecutive positive integers were written on the board, where . Then some of
the written numbers were erased, and it turned out that any two of the remaining numbers are coprime.
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Find the largest possible value of .
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6 A point is chosen inside a convex quadrilateral . Could it happen that
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7 It is known that from segments of lengths , and , a triangle can be formed. Could it happen that from
segments of lengths
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a right-angled triangle can be formed?
8 [Figure unavailable]
Seniors
Day 1
1 For given positive integers and , let us consider the equation
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a. For and , find the least positive integer satisfying this equation.
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b. Prove that for any positive integers and , there exist infinitely many positive integers
satisfying this equation.
(Here, denotes the greatest common divisor of positive integers and .)
2 Let be a triangle, and let and be points on segment symmetric with respect to the
midpoint of . Let denote the circle passing through and tangent to line at . Similarly,
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let denote the circle passing through and tangent to line at . Let the circles and view topic
intersect again at point ( ). Prove that .
3 A circle is drawn on the board, and points are marked on it, dividing it into equal arcs. Petya and
Vasya are playing the following game. Petya chooses a positive integer and announces this
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number to Vasya. To win the game, Vasya needs to color all marked points using colors, such that each view topic
color is assigned to exactly two points, and for each pair of same-colored points, one of the arcs between
them contains exactly marked points. Find all for which Petya will be able to prevent Vasya
from winning.
4 Determine if there exist non-constant polynomials , and with real coefficients and
leading coefficient , such that each of the polynomials
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has at least one real root, while each of the polynomials
has no real roots.
Day 2
5 Given a board whose rows are numbered from to and whose columns are numbered
from to , Nikita wishes to place one precious stone in some cells of this board so that at least one
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stone is present and the following magical condition holds: for any and , view topic
there is a stone in the cell at the intersection of the row and the column if and only if the cross
formed by the union of the row and the column contains exactly stones. Determine
whether Nikita's wish is achievable.
6 It is known that from segments of lengths , and , a triangle can be formed. Can it happen that from
segments of lengths
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an obtuse triangle can be formed?
7 From a point lying outside the circle , two tangents are drawn touching at points and .A
point is chosen on the segment . Let points and be the midpoints of segments
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and respectively. The circumcircle of triangle intersects again at point ( ). view topic
Prove that the line passes through the centroid of triangle .
8 Determine for which integers the cells of a table can be filled with the numbers
such that the following conditions are satisfied:
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i. Each of the numbers appears exactly once.
ii. In any rectangle, one of the numbers is the arithmetic mean of the other two.
iii. The number is located in the middle cell of the table.
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