NMTC 2019 Mathematics Contest Examination Questions and Notes

Examination Overview and Instructions

• Examination Title: National Mathematics Talent Contest (NMTC) 2019 Previous Year Paper (Grades 9 and 10).

• Total Maximum Marks: $30\,\text{marks}$.

• Duration: $120\,\text{minutes}$ ($2\,\text{hours}$, scheduled from $2:00\,\text{PM}$ to $4:00\,\text{PM}$).

• General Rules and Guidelines:

  • Diagrams provided in the paper are visual aids only and are not drawn to scale.

  • Rough work must be performed on separate blank sheets.

  • The response sheet requires candidate details including full name, class, and institution.

• Part A Structure:

  • Question Range: Questions 1 through 15 (Objective Questions).

  • Marking Scheme: Each correct response awards $1\,\text{mark}$. Each incorrect response results in a penalty of $\frac{1}{2}\,\text{mark}$ ($-0.5\,\text{marks}$).

  • Response Format: Exactly one choice among (a), (b), (c), or (d) is correct per question.

• Part B Structure:

  • Question Range: Questions 16 through 30 (Subjective / Fill-in-the-Blank Questions).

  • Marking Scheme: Each correct response awards $1\,\text{mark}$. Each incorrect response results in a penalty of $\frac{1}{4}\,\text{mark}$ ($-0.25\,\text{marks}$).

  • Response Format: Write the exact numerical or algebraic answer directly in the designated space on the response sheet.

Part A: Multiple Choice Questions

• Question 1:

  • Problem Statement: Find the number of $6$--digit numbers of the form $ABCABC$ that are divisible by $13$, where $A$, $B$, and $C$ are distinct digits, and both $A$ and $C$ are even digits.

  • Options:

    • (a) $200$

    • (b) $250$

    • (c) $160$

    • (d) $128$

• Question 2:

  • Problem Statement: In triangle $\Delta ABC$, the medians drawn through vertices $B$ and $C$ are perpendicular to each other. Calculate the value equal to $b^2 + c^2$.

  • Options:

    • (a) $2a^2$

    • (b) $3a^2$

    • (c) $4a^2$

    • (d) $5a^2$

• Question 3:

  • Problem Statement: Given a quadrilateral $ABCD$ with side lengths $AB = AD = 10$, diagonal length $BD = 12$, and side lengths $CB = CD = 13$. Determine the circle properties of $ABCD$.

  • Options:

    • (a) $ABCD$ is a cyclic quadrilateral

    • (b) $ABCD$ has an in-circle

    • (c) $ABCD$ has both a circum-circle and an in-circle

    • (d) It has neither a circum-circle nor an in-circle

• Question 4:

  • Problem Statement: Three cubes have integer side lengths. If the total sum of the surface areas of the three cubes is $498\,\text{sq. cm}$, find the sum of the volumes of the cubes in all possible integer solutions.

  • Options:

    • (a) $731$

    • (b) $495$

    • (c) $1226$

    • (d) none of these

• Question 5:

  • Problem Statement: In a rhombus of side length $5$, the length of one diagonal is at least $6$, and the length of the other diagonal is at most $6$. Determine the maximum possible value of the sum of the two diagonals.

  • Options:

    • (a) $10\sqrt{2}$

    • (b) $14$

    • (c) $2\sqrt{35}$

    • (d) $12$

• Question 6:

  • Problem Statement: Consider the sequence $1, 4, 8, 10, 16, 21, 25, 30, 43$. Find the total number of blocks of consecutive terms whose sums are divisible by 11$.\n * **Options**:\n * (a) only one\n * (b) exactly two\n * (c) exactly three\n * (d) exactly four\n\n* **Question 7**:\n * **Problem Statement**: Let A = {1, 2, 3, \dots, 17}$. For every non-empty subset$B$of$A$, calculate the product of the reciprocals of all members of$B. Find the sum of all such products across all non-empty subsets.\n * **Options**:\n * (a) \frac{153}{191}\n * (b) \frac{191}{153}\n * (c) 18\n * (d) 17\n\n* **Question 8**:\n * **Problem Statement**: Calculate the remainder when the polynomial f(x) = x^{100} + x^{50} + x^{10} + x^2 - 6$is divided by$x^2 - 1$.

  • Options:

    • (a) $x + 1$

    • (b) $-2$

    • (c) $0$

    • (d) $2$

• Question 9:

  • Problem Statement: Calculate the number of acute-angled triangles whose vertices are chosen exclusively from the vertices of a rectangular box (cuboid).

  • Options:

    • (a) $6$

    • (b) $8$

    • (c) $12$

    • (d) $24$

• Question 10:

  • Problem Statement: Consider the subtraction problem involving repeating digits: $111\dots111\,\text{(100 digits)} - 222\dots222\,\text{(50 digits)}$. Find the sum of the digits in the final result.

  • Options:

    • (a) $375$

    • (b) $420$

    • (c) $429$

    • (d) $450$

• Question 11:

  • Problem Statement: Given that $m$ and $n$ are positive integers such that $\frac{m^2 + n^2}{4} + \frac{m^2 - mn + n^2}{49}$ evaluates to an integer solution, find the value of m + n$.\n * **Options**:\n * (a) 4\n * (b) 8\n * (c) 12\n * (d) 16\n\n* **Question 12**:\n * **Problem Statement**: Given a grid sheet of 16$stamps arranged in a$4 \times 4 square, find the number of ways to choose three connected stamps where two adjacent stamps must share a common edge.\n * **Options**:\n * (a) 40\n * (b) 41\n * (c) 42\n * (d) 44\n\n* **Question 13**:\n * **Problem Statement**: In an election with 320$total votes cast among five candidates, the winning candidate's margins over the other four candidates were$9$,$13$,$18$, and$25 votes. Find the lowest number of votes received by any candidate.\n * **Options**:\n * (a) 49\n * (b) 50\n * (c) 51\n * (d) 52\n\n* **Question 14**:\n * **Problem Statement**: A competition contains 25 total questions marked as follows:\n * (A) 5\,\text{marks} awarded for each correct answer to Questions 1 through 15.\n * (B) 6\,\text{marks} awarded for each correct answer to Questions 16 through 25.\n * (C) Each incorrect answer to Questions 16 through 20 loses 1\,\text{mark}.\n * (D) Each incorrect answer to Questions 21 through 25 loses 2\,\text{marks}.\n * Determine the designated score bound or target evaluation for the candidate.\n * **Options**:\n * (a) 126\n * (b) 127\n * (c) 128\n * (d) 129\n\n* **Question 15**:\n * **Problem Statement**: Let A, M, T, I$be positive integers satisfying$A + M + T + I = 10. Find the maximum possible value of the expression:\n        A M T I + A M T + A M I + A T I + M T I + A M + A T + A I + M T + M I + T I\n * **Options**:\n * (a) 109\n * (b) 121\n * (c) 133\n * (d) 144\n\n# Part B: Subjective Questions\n\n* **Question 16**:\n * **Problem Statement**: The three-digit number XYZ$when divided by$8$gives the two-digit quotient$ZX$and a remainder of$Y$. Determine the exact three-digit number$XYZ$.

• Question 17:

  • Problem Statement: The digit sum of any number is defined as the sum of its digits. $N$ is a $3$--digit number. When the digit sum of $N$ is subtracted from $N$, the result is equal to the square of the digit sum of $N$. Find the number N$.\n\n* **Question 18**:\n * **Problem Statement**: A 4 \times 4$anti-magic square is an arrangement of the numbers$1$through$16$in a$4 \times 4 grid such that the totals of each of the four rows, four columns, and two main diagonals form ten consecutive integers in some order. Given the incomplete anti-magic square:\n * Row 1: 14$,$$,$9$,$3\n * Row 2: 7$,$12$,$13$,$5\n * Row 3: 10$,$11$,$6$,$4\n        Determine the number in the grid position indicated by $.

• Question 19:

  • Problem Statement: An escalator moves upward at a constant speed. John walks up the moving escalator at a rate of $1\,\text{step/second}$ and reaches the top in $20\,\text{seconds}$. On his next attempt, John walks up at a rate of $2\,\text{steps/second}$ and reaches the top in $16\,\text{seconds}$. Calculate the total number of visible steps in the escalator.

• Question 20:

  • Problem Statement: In a stacked coin arrangement, each upper row contains exactly one coin fewer than the row immediately below it. For $9$ total coins, two distinct valid towers are possible, with the left tower being the taller one. If there are $2015$ total coins, determine the height (number of rows) of the tallest possible tower.

• Question 21:

  • Problem Statement: Circles $A$, $B$, and $C$ are externally tangent to each other and internally tangent to a larger circle $D$. Circles $A$ and $B$ are congruent. Circle $C$ has a radius of $1\,\text{unit}$ and passes through the center of circle $D$. Calculate the radius of circle $B$ in units.

• Question 22:

  • Problem Statement: Find the total number of distinct integer values of $x$ that satisfy the equation:         $(x^2 - 5x + 5)^{x^2 - 11x + 30} = 1$

• Question 23:

  • Problem Statement: In a single move, a chess King $K$ is permitted to move to any adjacent square touching its current square, including diagonal steps. Determine the number of different valid paths using exactly $7$ moves to go from starting square $A$ to destination square B$.\n\n* **Question 24**:\n * **Problem Statement**: In triangle \Delta ABC$,$AB = AC$. Point$F$is on$AB$and point$E$is on$AC$such that$AF = EF$. Point$H$is in the interior of$\Delta ABC$, point$D$is on$BC$, and point$G$is on$AB$such that$EH = CH = DH = GH = DG = BG$. Given that$\angle CHE = \angle HGF$, determine the measure of$\angle BAC in degrees.\n\n* **Question 25**:\n * **Problem Statement**: Let x$and$y be real numbers satisfying the system of equations:\n        x^4 y^5 + y^4 x^5 = 810\n        x^3 y^6 + y^3 x^6 = 945\n        Calculate the exact value of 2x^3 + x^3 y^3 + 2y^3$.

• Question 26:

  • Problem Statement: Determine the least odd prime factor of the large integer 2019^8 + 1$.\n\n* **Question 27**:\n * **Problem Statement**: Let a$,$b$, and$c$be positive integers, each strictly less than$50$, such that$a^2 - b^2 = 100c$. Find the total number of such ordered triples$(a, b, c).\n\n* **Question 28**:\n * **Problem Statement**: Find the total number of non-negative integers that can be expressed in the symmetric base-3 form:\n        b_4 \cdot 3^4 + b_3 \cdot 3^3 + b_2 \cdot 3^2 + b_1 \cdot 3^1 + b_0 \cdot 3^0\n        where each coefficient b_i \in {-1, 0, 1}$for$0 \le i \le 4$.

• Question 29:

  • Problem Statement: Let $(a_k)$ be a sequence of integers defined by $a_1 = -2$ and the functional recurrence relation $a_{m+n} = a_m + a_n + mn$ for all positive integers $m, n$. Determine the value of a_0$.\n\n* **Question 30**:\n * **Problem Statement**: Determine the coefficient of x^{90} in the polynomial product expansion:\n        (1 + x + x^2 + x^3 + \dots + x^{60})(1 + x + x^2 + \dots + x^{120})$$