Singapore Mathematical Olympiad

Conducted by: Singapore Mathematical Society (SMS)
For: Pre-university students in three sections (Junior, Senior, Open)
Subjects: Number Theory, Algebra, Combinatorics, Geometry
Rounds: Two rounds for each section
Question Types: Round 1: Short answer. Round 2: Proof-based problems.
Frequency: Held annually
Qualifies for: Selection for International Mathematical Olympiad (IMO) training

The Singapore Mathematical Olympiad (SMO) is the country's largest and most established mathematics competition for pre-university students. It is organised into three categories based on school level: the Junior Section for lower secondary, the Senior Section for upper secondary, and the Open Section for junior college students. The main purpose of the SMO is to identify students with exceptional mathematical ability and to select candidates for the national team that represents Singapore at the International Mathematical Olympiad (IMO).

The competition consists of two rounds. Round 1 is a written paper with a large number of short-answer questions that must be solved under significant time pressure. High-scoring participants from this round are then invited to sit for Round 2, which is a more challenging, proof-based paper requiring detailed, rigorous solutions.

The difficulty of the SMO lies in its focus on non-routine problems that demand creativity and a deep understanding of mathematical concepts, not just procedural knowledge. The questions, drawn from topics like number theory, combinatorics, and advanced geometry, often go far beyond the standard school syllabus. Success requires both quick thinking for Round 1 and the ability to construct logical, well-reasoned arguments for the proof-based problems in Round 2.

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From a mathematics paper rather than Singapore Mathematical Olympiad, worked the way every question on this site is. Singapore Mathematical Olympiad questions are being added.

JEE Advanced 2008

Consider three planes $P_1 : x - y + z = 1$ ; $P_2 : x + y - z = -1$ $P_3 : x - 3y + 3z = 2$ Let $L_1, L_2, L_3$ be the lines of intersection of the planes $P_2$ and $P_3$ , $P_3$ and $P_1$ , $P_1$ and $P_2$ , respectively. Statement-I : At least two of the lines $L_1, L_2$ and $L_3$ are non-parallel Statement-II : The three planes do not have a common point.

Introduction to Three Dimensional Geometry Consider three planes

P_1 : x - y + z = 1

;

P_2 : x + y - z = -1

P_3 : x - 3y + 3z = 2

Let

L_1, L_2, L_3

be the lines of intersection of the planes

P_2

and

P_3

P_3

and

P_1

P_1

and

P_2

, respectively. Statement-I : At least two of the lines

L_1, L_2

and

L_3

are non-parallel Statement-II : The three planes do not have a common point.

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Consider three planes P₁ : x - y + z = 1; P₂ : x + y - z = -1 P₃ : x - 3y + 3z = 2 Let L₁, L₂, L₃ be the lines of intersection of the planes P₂ and P₃, P₃ and P₁, P₁ and P₂, respectively. Statement-I : At least two of the lines L₁, L₂ and L₃ are non-parallel Statement-II : The three planes do not have a common point.

$P_1: x - y + z = 1, \qquad P_2: x + y - z = -1$

$P_3: x - 3y + 3z = 2$

$L_1 = P_2 \cap P_3, \quad L_2 = P_3 \cap P_1, \quad L_3 = P_1 \cap P_2$

$\text{Check: S-I (non-parallel?) and S-II (common point?)}$

$1.\ \text{normals } \vec{n}_1, \vec{n}_2, \vec{n}_3$

$2.\ \vec{d}_{L_i} = \vec{n}_j \times \vec{n}_k \Rightarrow \text{S-I}$

$3.\ \text{consistency of } P_1, P_2, P_3 \Rightarrow \text{S-II}$

$P_1: \vec{n}_1 = (1, -1, 1), \qquad P_2: \vec{n}_2 = (1, 1, -1)$

$P_3: \vec{n}_3 = (1, -3, 3)$

$\vec{d}_{L_1} = \vec{n}_2 \times \vec{n}_3 = (0, -4, -4)$

$\vec{d}_{L_2} = \vec{n}_3 \times \vec{n}_1 = (0, 2, 2)$

$\vec{d}_{L_3} = \vec{n}_1 \times \vec{n}_2 = (0, 2, 2)$

$(0, -4, -4) \parallel (0, 2, 2) \parallel (0, 2, 2) \parallel (0, 1, 1)$

$\Rightarrow L_1, L_2, L_3 \text{ all parallel} \Rightarrow \text{S-I is FALSE}$

$\vec{n}_3 = 2\vec{n}_1 - \vec{n}_2$

$2P_1 - P_2: \ 2(x-y+z) - (x+y-z) = 2(1) - (-1)$

$\Rightarrow x - 3y + 3z = 3 \ \neq \ 2$

$\text{common point} \Rightarrow x - 3y + 3z = 3$

$\text{but } P_3: x - 3y + 3z = 2 \Rightarrow \text{contradiction}$

$\Rightarrow \text{no common point} \Rightarrow \text{S-II is TRUE}$

$P_1 + P_2: \ 2x = 0 \Rightarrow x = 0$

$x = 0 \Rightarrow P_1: -y + z = 1, \quad P_3: -y + z = \tfrac{2}{3}$

$1 \neq \tfrac{2}{3} \Rightarrow \text{inconsistent}$

$\text{S-I is FALSE, S-II is TRUE}$

$\Rightarrow \text{option (4)}$

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