Rmo 1993 Solutions Jun 2026

( P(x) = 1 ) or ( P(x) = -1 ).

For ( n \leq 4 ), test directly:

Let $f(x) = x^2 + 2x + 1$. Find the range of $f(x)$ for $x \in [-2, 2]$. rmo 1993 solutions

The original RMO 1993 consisted of 6 problems, to be solved in 3 hours. The problems spanned: ( P(x) = 1 ) or ( P(x) = -1 )

d=42+32=16+9=5d equals the square root of 4 squared plus 3 squared end-root equals the square root of 16 plus 9 end-root equals 5 ✅ The distance between the midpoints is . 2. Geometry: Rectangle and Touching Circles Problem: In a rectangle ABCDcap A cap B cap C cap D with sides R1cap R sub 1 is the radius of a circle passing through and touching CDcap C cap D R2cap R sub 2 is the radius of a circle passing through and touching ADcap A cap D . Prove that Solution Step-by-Step: Derive R1cap R sub 1 : For a circle passing through ) and touching the opposite side CDcap C cap D ), the radius is given by Derive R2cap R sub 2 : By symmetry, for a circle passing through ) and touching ADcap A cap D Apply Titu's Lemma: The sum is . Using the inequality derived from Cauchy-Schwarz, we find: The original RMO 1993 consisted of 6 problems,

So from Menelaus: ( \fracBEEA \cdot \fracAFFC = \fracDBCD ).

$$\binom10 - 1n - 1 = \binom9n - 1$$