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If four different positive integers $$m, n, p, q$$ satisfy the equation $$(7-m)(7-n)(7-p)(7-q) = 4$$ then the sum $$m+n+p+q$$ is equal to
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If four different positive integers $$m, n, p, q$$ satisfy the equation $$(7-m)(7-n)(7-p)(7-q) = 4$$ then the sum $$m+n+p+q$$ is equal to
The four numbers $$7-m, 7-n, 7-p, 7-q$$ are distinct integers whose product is 4, so they must be $$1, -1, 2, -2$$ in some order. This gives $$m, n, p, q$$ equal to $$6, 8, 5, 9$$ in some order, and all four are different positive integers as required. Hence $$m+n+p+q = 6+8+5+9 = 28$$.
A three member sequence $$a, b, c$$ is said to be an up-down sequence if $$a c$$. For example $$1, 3, 2$$ is an up-down sequence. The sequence 1342 contains three up-down sequences, namely $$(1,3,2)$$, $$(1,4,2)$$ and $$(3,4,2)$$. How many up-down sequences are contained in the sequence 132597684?
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For a positive integer $$n$$, let $$P(n)$$ denote the product of the digits of $$n$$ when $$n$$ is written in base 10. For example, $$P(123) = 6$$ and $$P(788) = 448$$. If $$N$$ is the smallest positive integer such that $$P(N) > 1000$$, and $$N$$ is written as $$100x + y$$ where $$x, y$$ are integers with $$0 \le x, y < 100$$, then $$x+y$$ equals
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The sum of 2025 consecutive odd integers is $$2025^{2025}$$. The largest of these odd numbers is
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$$ABC$$ is an equilateral triangle with side length 6. $$P, Q, R$$ are points on the sides $$AB, BC, CA$$ respectively such that $$AP = BQ = CR = 1$$. The ratio of the area of the triangle $$ABC$$ to the area of the triangle $$PQR$$ is
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How many three-digit positive integers are there if the digits are the side lengths of some isosceles or equilateral triangle?
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All the positive integers whose sum of digits is 7 are written in the increasing order. The first few are $$7, 16, 25, 34, 43, \ldots$$ What is the 125 th number in this list?
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The bisectors of the angles $$A, B, C$$ of the triangle $$ABC$$ meet the circum circle of the triangle again at the points $$D, E, F$$ respectively. What is the value of $$\frac{AD \cos \frac{A}{2} + BE \cos \frac{B}{2} + CF \cos \frac{C}{2}}{\sin A + \sin B + \sin C}$$ if the circum radius of $$ABC$$ is 1 ?
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For a real number $$x$$, let $$\lfloor x \rfloor$$ be the greatest integer less than or equal to $$x$$. For example, $$\lfloor 1.7 \rfloor = 1$$ and $$\lfloor \sqrt{2} \rfloor = 1$$. Let $$N = \left\lfloor \frac{10^{93}}{10^{31}+3} \right\rfloor$$. Find the remainder when $$N$$ is divided by 100.
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A point $$(x, y)$$ in the plane is called a lattice point if both its coordinates $$x, y$$ are integers. The number of lattice points that lie on the circle with center at $$(199, 0)$$ and radius 199 is
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The sum of all real numbers $$p$$ such that the equation $$5x^3 - 5(p+1)x^2 + (71p-1)x - (66p-1) = 0$$ has all its three roots positive integers.
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If $$1 - x + x^2 - x^3 + \cdots + x^{20}$$ is rewritten in the form $$a_0 + a_1(x-4) + a_2(x-4)^2 + \cdots + a_{20}(x-4)^{20}$$, where $$a_0, a_1, \ldots, a_{20}$$ are all real numbers, the value of $$a_0 + a_1 + a_2 + \cdots + a_{20}$$ is
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For a positive integer $$n$$, a distinct 3-partition of $$n$$ is a triple $$(a, b, c)$$ of positive integers such that $$a < b < c$$ and $$a+b+c = n$$. For example, $$(1, 2, 4)$$ is a distinct 3-partition of 7. The number of distinct 3-partitions of 15 is
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If $$m$$ and $$n$$ are positive integers such that $$30mn - 6m - 5n = 2019$$, what is the value of $$30mn - 5m - 6n$$?
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A class of 100 students takes a six question exam. For the first question, a student receives 1 point for answering correctly, -1 point for answering incorrectly or not answering at all. For the second question, the student receives 2 points for answering correctly and -2 points for answering incorrectly or not answering at all and so on. What is the minimum number of students having the same scores?
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The value of $$\frac{1}{2} + \frac{1^2+2^2}{6} + \frac{1^2+2^2+3^2}{12} + \frac{1^2+2^2+3^2+4^2}{20} + \cdots + \frac{1^2+2^2+\cdots+60^2}{3660}$$ is
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The largest prime divisor of $$3^{21} + 1$$ is
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A circular garden divided into 10 equal sectors needs to be planted with flower plants that yield flowers of 3 different colors, in such a way that no two adjacent sectors will have flowers of the same color. The number of ways in which this can be done is
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We call an integer special if it is positive and we do not need to use the digit 0 to write it down in base 10. For example, 2126 is special whereas 2025 is not. The first 10 special numbers are $$1, 2, 3, 4, 5, 6, 7, 8, 9, 11$$. The 2025th special number is
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Let $$a, b, c$$ be non zero real numbers such that $$a+b+c = 0$$ and $$a^3+b^3+c^3 = a^5+b^5+c^5$$. The value of $$\frac{5}{a^2+b^2+c^2}$$ is
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The equation $$x^3 - \frac{1}{x} = 4$$ has two real roots $$\alpha, \beta$$. The value of $$(\alpha + \beta)^2$$ is
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If $$x, y, z$$ are positive integers satisfying the system of equations $$xy + yz + zx = 2024$$ and $$xyz + x + y + z = 2025$$, find $$\max(x, y, z)$$.
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If $$p, q, r$$ are primes such that $$pq + qr + rp = pqr - 2025$$, find $$p+q+r$$.
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A cyclic quadrilateral has side lengths $$3, 5, 5, 8$$ in this order. If $$R$$ is its circumradius, find $$3R^2$$.
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Consider the sequence of numbers $$24, 2534, 253534, 25353534, \ldots$$ Let $$N$$ be the first number in the sequence that is divisible by 99. Find the number of digits in the base 10 representation of $$N$$.
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An isosceles triangle has integer sides and has perimeter 16. Find the largest possible area of the triangle.
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Suppose that $$a, b, c$$ are positive real numbers such that $$a^2 + b^2 = c^2$$ and $$ab = c$$. Find the value of $$\frac{(a+b+c)(a-b+c)(a+b-c)(a-b-c)}{c^2}$$
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In a right angled triangle with integer sides, the radius of the inscribed circle is 12. Compute the largest possible length of the hypotenuse.
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Points $$C$$ and $$D$$ lie on opposite sides of the line $$AB$$. Let $$M$$ and $$N$$ be the centroids of the triangles $$ABC$$ and $$ABD$$ respectively. If $$AB = 25$$, $$BC = 24$$, $$AC = 7$$, $$AD = 20$$ and $$BD = 15$$, find $$MN$$.
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Let $$a_0 = 1$$ and for $$n \ge 1$$, define $$a_n = 3a_{n-1} + 1$$. Find the remainder when $$a_{11}$$ is divided by 97.
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