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Let $$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}=\frac{m}{n}$$, where $$m$$ and $$n$$ are positive integers with no common divisors other than $$1$$. The highest power of $$7$$ that divides $$m$$ is
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Let $$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}=\frac{m}{n}$$, where $$m$$ and $$n$$ are positive integers with no common divisors other than $$1$$. The highest power of $$7$$ that divides $$m$$ is
Using denominator $$60$$, the sum is $$\frac{60+30+20+15+12+10}{60}=\frac{147}{60}=\frac{49}{20}$$. Thus $$m=49=7^2$$. Therefore, the highest power of $$7$$ dividing $$m$$ is $$2$$.
Five spherical balls of diameter $$10\text{ cm}$$ each fit inside a closed cylindrical tin with internal diameter $$16\text{ cm}$$. What is the smallest possible height of the tin can be?
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The expression $$\frac{7n+18}{2n+3}$$ takes integer values for certain integer values of $$n$$. The sum of such values of the expression is
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$$P$$ is a point inside $$ABCD$$ such that $$PA=2$$, $$PB=4$$, $$PC=5$$ and $$PD=6$$. The maximum area of quadrilateral $$ABCD$$ is
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The value of $$(3^{4/3}-3^{1/3})^3+(3^{5/3}-3^{2/3})^3+(3^{6/3}-3^{3/3})^3+\cdots+(3^{10/3}-3^{7/3})^3$$ is
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One hundred people are standing in a line and they are required to count off in fives as "one, two, three, four, five" and so on
from the first person in the line. Anyone who counts "five" walks out of the line. Those remaining repeat this procedure until only
four people remain in the line. What was the original position in the line of the last person to leave?
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The number of values positive integers $$n$$ for which $$1!+2!+\cdots+n!$$ is a perfect square is
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When $$2025^{2026}-2025$$ is divided by $$2025^2+2026$$, the remainder is
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For a real number $$x$$, let $$[x]$$ denote the largest integer $$ \leq x$$. For example, $$ [3,4] = 2 $$ and $$ [4,9] = 4 $$ Let $$N=[(\sqrt{27}+\sqrt{23})^6]$$. The remainder when $$N$$ is divided by $$1000$$ is
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$$ABC$$ is a triangle. Point $$D$$ lies on $$AC$$ such that $$AB=BD=CD$$. All the angles in the diagram are positive whole numbers of degrees. The largest possible size, in degrees of $$\angle ABC$$ is
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The side lengths of a right-angled triangle are in geometric progression, and the smallest side has length $$2$$ units. The length of the hypotenuse is
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In a regular polygon, there are two diagonals intersect inside the polygon at an angle of $$50^\circ$$. The least number of sides of the polygon for which this is possible is
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In triangle $$PQR$$, $$\angle R=2\angle P$$, $$PR=5$$ and $$QR=4$$. The length of $$PQ$$ is
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Each of ten people around a circle chooses a number and tells it to the neighbor on each side. Thus each person gives out one
number and receives two numbers. The players then announce the average of the two numbers they received. The announced
numbers, in order around the circle were $$1,2,3,4,5,6,7,8,9,10$$. The number chosen by the person who announced the number $$6$$ is
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A regular octagon is formed by cutting four equal isosceles right-angled triangles from the corners of a square of side length $$1$$. The area of the octagon is
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The diagram shows the net of a cube, that is, we can fold along the edges of the squares to
make a cube from this net. On each face there is an integer written - $$1,a,b,c,d,2026$$. Each of the four numbers $$a,b,c,d$$ equals the average of the numbers on the four faces of the cube adjacent to it, The value of $$a$$ is
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Let $$S=\{1,2,3,\ldots,15\}$$. The number of subsets $$A$$ of $$S$$ containing four elements such that any two elements of $$A$$ differ by at least $$2$$ is
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Given a deck of $$52$$ cards with numbers $$1,2,\ldots,52$$ written on them, one number per card. The deck is shuffled and $$13$$ cards are chosen at random from the shuffled deck and thrown away, without noting the numbers on them. From the remaining $$39$$ cards, one card is chosen at random. If the probability that the number on this card is a multiple of $$13$$ is $$\frac{m}{n}$$, where $$m,n$$ are integers with no common divisor other than $$1$$, then $$m+n$$ equals $$\underline{\qquad}$$.
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Suppose $$10$$ objects are placed along a circle at equal distances. The number of ways can three objects be chosen from among them so that no two of the chosen objects are adjacent or diametrically opposite is $$\underline{\qquad}$$.
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For a positive integer $$n$$, let $$n\bmod13$$ denote the remainder $$r$$, $$0\leq r<13$$ when divided by $$13$$. If $$a,b,c$$ are integers such that $$4a+5b+6c = 1 mod13 a-b-7c=3 mod13 3a-4b+5c=9 mod13$$, then $$a+b+c\bmod13$$ is $$\underline{\qquad}$$.
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The sum of all positive integers $$N<2024$$ such that $$N$$ equals $$13$$ times the sum of digits of N (when N is written in base 10) is
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Six identical regular hexagons are arranged inside a larger regular hexagon as shown in the Figure. The outer hexagon has area $$900$$ square units. The area of the shaded region (the total area contained in the smaller
hexagons) is _____ square units.
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A positive integer is said to be special if the sum of the remainders obtained when it is divided by five consecutive positive integers is $$32$$. For example, $$24$$ is special since when divided by $$11,12,13,14,15$$ the remainders are $$2,0,11,10,9$$ and these add up to $$32$$. The smallest positive integer that is special is $$\underline{\qquad}$$.
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The largest three digit number with the property that the number is equal to the sum of the hundreds digit, the square of its tens digit and the cube of its units digit is $$\underline{\qquad}$$.
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It is a surprising fact that $$1\times2\times3\times4\times5\times6=8\times9\times10$$. It is more surprising that $$8\times9\times10\times11\times12\times13\times14$$ can be written as a product of consecutive positive integers. The smallest number in the product is $$\underline{\qquad}$$.
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There are $$100$$ points $$P_1,P_2,\ldots,P_{100}$$ placed on a line such that the distance between $$P_i$$ and $$P_{i+1}$$ is $$\frac{1}{i}$$ for $$1\leq i\leq99$$. The sum of the distances between every pair of these points is
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For a positive integer $$n$$, let $$d(n)$$ denote the number of positive divisors of $$n$$. For example, $$d(4) = 3$$, The smallest positive integer $$n$$ for which $$d(n-2)+d(n)+d(n+2)=21$$ is
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Two bugs sit at vertices $$A$$ and $$H$$ of a cube $$ABCDEFGH$$ with edge length $$4\sqrt{110}$$ units. The bugs start moving simultaneously along $$AC$$ and $$HF$$, with the speed of the first bug twice that of the other one. The shortest distance between the bugs is
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The smallest positive integer $$n$$ for which it is possible to draw an $$n$$-gon whose vertex angles all measure either $$163^\circ$$ or $$171^\circ$$ is
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Let $$P(x)=ax^3+bx^2+cx+d$$ be a cubic polynomial such that $$P(2)=7$$, $$P(3)=13$$ and $$P(5)=7$$. If the sum of the three roots of $$P(x)=0$$ is $$40$$, the value of $$P(35)$$ is
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