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2023 EMCC Individual Speed Test - Exeter Math Club Competition
parmenides51   15
N 41 minutes ago by lpieleanu
20 problems for 25 minutes.


p1. Evaluate the following expression, giving your answer as a decimal: $\frac{20\times 2\times 3}{
20+2+3}$ .


p2. Given real numbers $x$ and $y$, we have that $2x + 3y = 20$ and $3x + 4y = 12$. Find the value of $x + y$.


p3. Alan, Daria, and Max want to sit in a row of three airplane seats. If Alan cannot sit in the middle, in how many ways can they sit down?


p4. Jack thinks of two distinct positive integers $a$ and $b$. He notices that neither $a$ nor $b$ is a perfect square, but $ab$ is a perfect square. What is the smallest possible value of $a + b$?


p5. What is the smallest integer greater than $2023$ whose digits sum to $4$?


p6. Triangle $ABC$ has $AB = AC$ and $\angle B = 60^o$. The altitude drawn from $C$ intersects $AB$ at $X$, where $BX = 4$. What is the area of $ABC$?


p7. Archyuta writes a program to create words with at least one letter. The probability of having $n$ letters in the word for each positive integer $n$ is $\frac{1}{2^n}$ . Each letter of the word is chosen randomly and independently from the uppercase English alphabet. The probability of Archyuta’s program outputting “EMCC” can be written as $\frac{1}{k}$ for some positive integer $k$. What is the greatest nonnegative integer $a$ such that $2^a$ divides $k$?


p8. What is the greatest whole number less than $1000$ that can be expressed as the sum of seven consecutive whole numbers, as the sum of five consecutive whole numbers, and as the sum of three consecutive whole numbers?


p9. Given a square $ABCD$ with side length $7$, square $EFGH$ is inscribed in $ABCD$ such that $E$ is on side $AB$ and $G$ is on side $CD$ such that $EA = 3$ and $GD = 4$. If square PQRS inscribed in $EFGH$ such that $PQ \parallel AB$, find the side length of $PQRS$.
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p10. Michael wants to do some exercise by going up and down a moving escalator. He first runs up the escalator, taking $30$ seconds to reach the top. Tired, he then walks at one-third of his running speed back down the escalator, taking 30 seconds to reach the bottom. Assuming his running speed and the escalator’s speed are constant, what is the ratio of his running speed to the escalator’s speed?


p11. Bob the architect has $4$ bricks shaped like rectangular prisms each of dimension $1$ foot by $1$ foot by $2$ feet which he stores inside a $2$ feet by $2$ feet by $2$ feet hollow box. In how many ways can he fit his bricks into the box? (Rotations and reflections of a configuration are considered distinct.)


p12. $P$ is a point lying inside rectangle $ABCD$. If $\angle PAB = 40^o$, $\angle PBC = 50^o$ and $\angle PCD = 60^o$, find $\angle PDA$ in degrees.


p13. Let $N$ be a positive integer. If $4$ of $N$s divisors are prime and $346$ of $N$s divisors are composite, how many of $N$s divisors are perfect squares?


p14. Two positive integers have a product of $2^{23}$. Let $S$ be the sum of all distinct possible values of their absolute difference. Find the remainder when $S$ is divided by $1000$.


p15. A rectangle has area $216$. The internal angle bisectors of each of its four vertices are drawn, bounding a square region with area $18$. Find the perimeter of the rectangle.


p16. Let $\vartriangle ABC$ be a right triangle, with a right angle at $B$. The perpendicular bisector of hypotenuse $\overline{AC}$ splits the triangle into a smaller triangle and a quadrilateral. If the triangle has an area of $5$ and the quadrilateral has an area of $13$, find the length of $\overline{AC}$.


p17. Let $N$ be the sum of the $2023$ smallest positive perfect squares minus the sum of the $2023$ smallest positive odd numbers. What is the largest prime factor of $N$?


p18. Anna designs a logo, shown below, consisting of a large square with side length $12$ and two congruent equilateral triangles placed inside the square, one in each corner and with one overlapping side. What is the distance between the marked vertices?
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p19. How many ways are there to place an isosceles right triangle with legs of length $1$ in each unit square of a two-by-two grid, such that no two isosceles triangles share an edge? One valid construction is shown on the left, followed by an invalid construction on the right.
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p20. Mr. Ibbotson and Dr. Drescher are playing a game where they write numbers on the blackboard. On the first turn, Mr. Ibbotson begins by writing $1$, followed by Dr. Drescher writing another $1$ on the second turn. Each turn afterwards, they take the two newest numbers on the board and concatenate them, writing the resulting number of the board. For instance, the first few numbers on the board are $1$, $1$, $11$, $111$, $11111$, $...$ How many turns does it take for them to write a number which is divisible by $63$?


PS. You should use hide for answers. Collected here.
15 replies
1 viewing
parmenides51
Oct 20, 2023
lpieleanu
41 minutes ago
XZ passes through the midpoint of BK, isosceles, KX = CX, angle bisector
parmenides51   5
N an hour ago by Kyj9981
Source: 1st Girls in Mathematics Tournament 2019 p5 (Brazil) / Torneio Meninas na Matematica (TM^2 )
Let $ABC$ be an isosceles triangle with $AB = AC$. Let $X$ and $K$ points over $AC$ and $AB$, respectively, such that $KX = CX$. Bisector of $\angle AKX$ intersects line $BC$ at $Z$. Show that $XZ$ passes through the midpoint of $BK$.
5 replies
parmenides51
May 25, 2020
Kyj9981
an hour ago
Solution needed ASAP
UglyScientist   7
N an hour ago by Captainscrubz
$ABC$ is acute triangle. $H$ is orthocenter, $M$ is the midpoint of $BC$, $L$ is the midpoint of smaller arc $BC$. Point $K$ is on $AH$ such that, $MK$ is perpendicular to $AL$. Prove that: $HMLK$ is paralelogram(Synthetic sol needed).
7 replies
UglyScientist
4 hours ago
Captainscrubz
an hour ago
Diophantine Equation with prime numbers and bonus conditions
p.lazarov06   10
N an hour ago by mathbetter
Source: 2023 Bulgaria JBMO TST Problem 3
Find all natural numbers $a$, $b$, $c$ and prime numbers $p$ and $q$, such that:

$\blacksquare$ $4\nmid c$
$\blacksquare$ $p\not\equiv 11\pmod{16}$
$\blacksquare$ $p^aq^b-1=(p+4)^c$
10 replies
1 viewing
p.lazarov06
May 7, 2023
mathbetter
an hour ago
Concurrence in Cyclic Quadrilateral
GrantStar   39
N an hour ago by ItsBesi
Source: IMO Shortlist 2023 G3
Let $ABCD$ be a cyclic quadrilateral with $\angle BAD < \angle ADC$. Let $M$ be the midpoint of the arc $CD$ not containing $A$. Suppose there is a point $P$ inside $ABCD$ such that $\angle ADB = \angle CPD$ and $\angle ADP = \angle PCB$.

Prove that lines $AD, PM$, and $BC$ are concurrent.
39 replies
GrantStar
Jul 17, 2024
ItsBesi
an hour ago
IMO Genre Predictions
ohiorizzler1434   22
N an hour ago by rhydon516
Everybody, with IMO upcoming, what are you predictions for the problem genres?


Personally I predict: predict
22 replies
ohiorizzler1434
Today at 6:51 AM
rhydon516
an hour ago
Inequality
MathsII-enjoy   1
N 2 hours ago by arqady
A interesting problem generalized :-D
1 reply
MathsII-enjoy
3 hours ago
arqady
2 hours ago
Inequality
lgx57   2
N 2 hours ago by mashumaro
Source: Own
$a,b,c>0,ab+bc+ca=1$. Prove that

$$\sum \sqrt{8ab+1} \ge 5$$
(I don't know whether the equality holds)
2 replies
lgx57
2 hours ago
mashumaro
2 hours ago
Find min
lgx57   1
N 2 hours ago by arqady
Source: Own
Find min of $\dfrac{a^2}{ab+1}+\dfrac{b^2+2}{a+b}$
1 reply
lgx57
2 hours ago
arqady
2 hours ago
Product is a perfect square( very easy)
Nuran2010   1
N 2 hours ago by SomeonecoolLovesMaths
Source: Azerbaijan Junior National Olympiad 2021 P1
At least how many numbers must be deleted from the product $1 \times 2 \times \dots \times 22 \times 23$ in order to make it a perfect square?
1 reply
Nuran2010
4 hours ago
SomeonecoolLovesMaths
2 hours ago
smo 2018 open 2nd round q2
dominicleejun   7
N 2 hours ago by Kyj9981
Let O be a point inside triangle ABC such that $\angle BOC$ is $90^\circ$ and $\angle BAO = \angle BCO$. Prove that $\angle OMN$ is $90$ degrees, where $M$ and $N$ are the midpoints of $\overline{AC}$ and $\overline{BC}$, respectively.
7 replies
dominicleejun
Aug 15, 2019
Kyj9981
2 hours ago
Geometry Transformation Problems
ReticulatedPython   7
N Apr 22, 2025 by ReticulatedPython
Problem 1:
A regular hexagon of side length $1$ is rotated $360$ degrees about one side. The space through which the hexagon travels during the rotation forms a solid. Find the volume of this solid.

Problem 2:

A regular octagon of side length $1$ is rotated $360$ degrees about one side. The space through which the octagon travels through during the rotation forms a solid. Find the volume of this solid.

Source:Own

Hint

Useful Formulas
7 replies
ReticulatedPython
Apr 17, 2025
ReticulatedPython
Apr 22, 2025
Geometry Transformation Problems
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G H BBookmark kLocked kLocked NReply
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ReticulatedPython
617 posts
#1
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Problem 1:
A regular hexagon of side length $1$ is rotated $360$ degrees about one side. The space through which the hexagon travels during the rotation forms a solid. Find the volume of this solid.

Problem 2:

A regular octagon of side length $1$ is rotated $360$ degrees about one side. The space through which the octagon travels through during the rotation forms a solid. Find the volume of this solid.

Source:Own

Hint

Useful Formulas
This post has been edited 11 times. Last edited by ReticulatedPython, Apr 21, 2025, 3:33 PM
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jb2015007
1931 posts
#2
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ack i hate transformation problems and i suck at them so ill do this lol ill do it when i get home from school
@reticulated python check out the problem i posted in the intro to geo message board its real interesting
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ReticulatedPython
617 posts
#3
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Ok I will.
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ReticulatedPython
617 posts
#4
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Anyone want to give it a try?
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cheltstudent
619 posts
#5
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wait which intro to geo class is this
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ReticulatedPython
617 posts
#6
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\bump :)
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ReticulatedPython
617 posts
#7
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I will present the solution to problem 1 and see if anyone can use that to solve problem 2. Solution
This post has been edited 2 times. Last edited by ReticulatedPython, Apr 21, 2025, 7:26 PM
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ReticulatedPython
617 posts
#8
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Alright I will give the answer to problem 2; the method is similar to that of problem 1. Answer
This post has been edited 1 time. Last edited by ReticulatedPython, Apr 22, 2025, 3:31 PM
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