INMO Day 1 Practice
Soham Dutta
December 2023
1
Problems:
1. Let n be a natural number exceeding 1, and let An be the set of all natural numbers that are not relatively
prime with n (i.e., An = {x ∈ N : gcd(x, n) ̸= 1}). Let us call the number n cazz if for each two numbers
x, y ∈ An , their sum x + y is also an element of An (i.e., x + y ∈ An for x, y ∈ An ).
(a) Prove that 67 is a cazz number.
(b) Prove that 2001 is not a cazz number.
(c) Find all cazz numbers.
2. Let a, b, c be the sides of a triangle. Show that
√
X1
3
≤
a
2r
cyc
where r is the inradius of the triangle.
3. Let n be a positive integer. Show that the fractional part of the decimal representation of (5 +
starts with n identical digits.
√
26)n
4. An integer sequence {an }n≥1 is defined by
a1 = 2, an+1 =
3
an .
2
Show that it has infinitely many even and infinitely many odd integers.
5. Let ABC be a triangle. Circle Γ passes through A, meets segments AB and AC again at points D and E
respectively, and intersects segment BC at F and G such that F lies between B and G. The tangent to
circle BDF at F and the tangent to circle CEG at G meet at point T . Suppose that points A and T are
distinct. Prove that line AT is parallel to BC.
6. If a, b, c be strictly positive reals such that ab + bc + ca + abc = 4, then show that
√
√
√
ab + bc + ca ≤ 3 ≤ a + b + c.
7. Find all f : R → R such that
f (x + y) = f (x)f (y)f (xy)
8. Let a1 , a2 , . . . be a sequence of real numbers satisfying ai+j ≤ ai + aj for all i, j = 1, 2, . . . .
Prove that
a1 +
a2
a3
an
+
+ ··· +
≥ an
2
3
n
for each positive integer n.
9. Consider the natural number N . It is known that there are 2041 ordered pairs (x, y) such that
1
1
1
+ =
x y
N
Show that N is a perfect square. Moreover, find the least value of N .
10. Solve for f : R → R
f (xf (z) + f (y)) = zf (x) + y
1