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The central square of a $$9 \times 9$$ chessboard is white. How many white squares are there on the board? (The squares of the chessboard are coloured alternately black and white.)
The centre is in row 5, column 5, and moving four squares left and then four squares up preserves the colour, so the top-left corner is also white. Rows $$1, 3, 5, 7, 9$$ therefore contain 5 white squares each, while rows $$2, 4, 6, 8$$ contain 4 each. Hence the total is $$5 \times 5 + 4 \times 4 = 41$$.
An integer $$M$$ is divisible by 4 but not by 256. What is the number of distinct possible remainders when $$M$$ is divided by 256?
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If $$x_1, x_2, \ldots, x_{49}$$ are non-zero integers such that $$\sum_{i=1}^{49} x_i = 0$$, then what is the minimum possible value of $$\sum_{i=1}^{49} x_i^2$$?
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All six digits of three 2-digit numbers are different. If $$N$$ is the largest possible sum of three such numbers, what is the sum of digits of $$N$$?
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In triangle $$ABC$$, we are given that $$\angle CAB = 80^\circ$$. Let the perpendicular bisector of $$BC$$ meet the circumcircle of triangle $$ABC$$ in $$N$$, where we assume that $$A$$ and $$N$$ lie on the same side of the chord $$BC$$. Then what is the measure of $$\angle NBC$$ in degrees?
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Find the number of 2-digit positive integers $$n$$ such that $$n = 26 + (a \times b)$$, where $$a$$ and $$b$$ are the two digits of $$n$$.
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Find the number of positive integers $$n$$ satisfying all the following conditions.
(a) The digits of $$n$$ lie in the set $$\{1, 2, 4, 8\}$$. (Digits may be repeated.)
(b) The sum of the digits is 14.
(c) If 1 occurs as a digit, it can occur only immediately to the right of 8.
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In trapezium $$ABCD$$, it is given that $$AB$$ is parallel to $$CD$$. Assume that $$AB = 3CD$$, $$CD = DA$$, and $$\angle CDA = 120^\circ$$. If the largest angle of $$ABCD$$ is $$x^\circ$$ and the smallest angle is $$y^\circ$$, what is the value of $$x/y$$?
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Let $$N$$ be the smallest positive integer whose digits add up to 2026. What is the leading digit of $$N + 1$$?
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What is the number of integers in the set $$\{0, \ldots, 20\}$$ which can be expressed as the sum of two square integers?
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Find the number of non-constant polynomials $$P(x)$$, with real coefficients, such that $$P(x^2) = P(P(x))$$.
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A $$7 \times 7$$ board is divided into 49 unit squares. We place checkers on the board, at most one per square. Find the largest number of checkers that can be placed on the unit squares so that each row, as well as each column, contains an even number of checkers.
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In an isosceles triangle $$ABC$$, with $$\angle ACB = 90^\circ$$, the point $$D$$ is on the side $$BC$$ such that $$\angle ADC = 75^\circ$$. If the area of triangle $$ADC$$ is 81, what is the length of segment $$BD$$?
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Let $$A$$ be a 3-digit number with distinct nonzero digits and $$B$$ be the number obtained by reversing the digits of $$A$$. Determine the largest possible prime factor of $$|A - B|$$.
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Let $$a, b, c, d$$ be positive integers such that $$a^2 + b^2 - cd^2 = 2026$$. Find the minimum possible value of $$a + b + c + d$$.
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A sequence $$a_1, a_2, a_3, \ldots$$ of real numbers satisfies
$$\frac{a_{n+3} - a_{n+2}}{a_n - a_{n+1}} = \frac{a_{n+3} + a_{n+2}}{a_n + a_{n+1}}$$
for all $$n \ge 1$$. Suppose $$a_{55} = 6$$, $$a_{66} = 2$$ and $$a_{77} = 1$$. Let $$N$$ denote the sum $$a_1^2 + a_2^2 + \cdots + a_{2026}^2$$. What is the sum of the digits of $$N$$?
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Find the number of ordered triples $$(x, y, z)$$ of positive integers such that $$1 \le x, y, z \le 8$$ and $$|x - y| + |y - z| + |z - x| = 8$$.
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Let $$P$$ be a regular polygon with 8 vertices. By a labelling of $$P$$ we mean an assignment of integers $$1, 2, \ldots, 8$$ to the vertices in some order. A labelling is good if the path consisting of line segments from 1 to 2, 2 to 3, and so on up to 7 to 8 does not self-intersect. If $$N$$ is the number of good labellings, what is the remainder when $$N$$ is divided by 100?
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Four points $$A, B, C$$ and $$D$$ lie on a straight line, in this order. A point $$E$$, not on the line, satisfies $$\angle AEB = \angle BEC = \angle CED = 45^\circ$$. Let $$F$$ and $$G$$ be the midpoints of $$AC$$ and $$BD$$, respectively. If $$\angle FEG = x^\circ$$, what is the value of $$x$$?
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Let $$M$$ be the smallest positive integer with the following two properties:
(a) The leading digit of $$M$$ is equal to 3.
(b) If $$N$$ is the number obtained by moving this leading 3 to the units place, and shifting all the other digits one place to the left, then $$N = M/4$$.
What is the sum of the digits of $$M$$?
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Complex numbers $$x, y, z$$ satisfy the following system of equations:
$$x^2 + y^2 + z = xy$$
$$x + y^2 + z^2 = yz$$
$$x^2 + y + z^2 = xz$$
Determine the sum of all distinct possible values of $$\left| (x^2 - y)(y^2 - z)(z^2 - x) \right|$$.
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Let $$N$$ be the number of distinct 8-digit numbers obtained by arranging the six numbers $$0, 1, 2, 3, 10, 23$$, where the first digit of the 8 digit number is not zero. Find the sum of the digits of $$N$$.
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A $$1 \times 5$$ rectangle is divided into five $$1 \times 1$$ squares by drawing four line segments parallel to the shorter side of the rectangle. Each of the resulting sixteen unit-length line segments is coloured red, blue or green. A $$1 \times 1$$ square is called colourful if all the three colours are used in colouring its sides. If $$N$$ is the number of ways of colouring such that all the five $$1 \times 1$$ squares are colourful, find the remainder when $$N$$ is divided by 100.
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Let $$ABCD$$ be a rectangle and let $$E$$ be a point on $$BD$$ such that $$AE$$ is perpendicular to $$BD$$. If $$AE = 12$$ and $$CE = \sqrt{193}$$, compute the area of the rectangle $$ABCD$$.
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Let $$E = \{p^4 + p^2 - 2 \mid p \text{ is a prime},\; p > 3\}$$. What is the largest positive integer that divides all the numbers in $$E$$?
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There are $$n$$ points in the plane, no three of which are collinear. Every pair of points is joined by a segment which is coloured red or blue such that the following conditions hold:
(a) If $$A, B, C$$ are three points such that $$AB$$ is red and $$BC$$ is blue, then $$AC$$ is red.
(b) For any point $$A$$, there are exactly three points $$B, C, D$$ such that $$AB, AC, AD$$ are red.
Find the sum of all possible values of $$n$$.
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The lengths of the sides of a convex quadrilateral are $$\sqrt{a}$$, $$\sqrt{a + 3}$$, $$\sqrt{a + 2}$$ and $$\sqrt{2a + 5}$$, in this order. The length of each diagonal is $$\sqrt{2a + 5}$$. If $$\theta^\circ$$ is the difference between the largest angle and the second largest angle of the quadrilateral, determine the value of $$\theta$$.
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Let $$a_1, a_2, \ldots$$ and $$b_1, b_2, \ldots$$ be strictly increasing sequences of positive integers such that
(a) $$a_{n+1} = a_n + a_{n-1}$$ for $$n \ge 2$$,
(b) $$b_n = 2b_{n-1}$$ for all $$n \ge 2$$,
(c) $$a_{10} = b_{10} < 2026$$.
Find the sum of all possible values of $$a_1 + b_1$$.
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Let $$n=\frac{4^{31}-1}{3}$$. Find the remainder when $$2^{n-1}$$ is divided by $$n$$.
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In triangle $$ABC$$, it is given that $$\angle CAB = 50^\circ$$ and $$\angle ABC = 70^\circ$$. Points $$D$$ and $$E$$ are chosen on sides $$BC$$ and $$AC$$, respectively, such that $$\angle ABE = \angle DAB = 30^\circ$$. If $$\angle DEB = x^\circ$$, what is the value of $$x$$?
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