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Difference between revisions of "2022 SSMO Relay Round 1 Problems"

(Created page with "==Problem 1== Suppose <math>a, b, c</math> are distinct digits where <math>a \not= 0</math> such that <math>\left(\overline{abc}\right)^2 = \overline{bad00}</math> where <mat...")
 
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[[2022 SSMO Relay Round 1 Problems/Problem 3|Solution]]
 
[[2022 SSMO Relay Round 1 Problems/Problem 3|Solution]]
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==Problem 1==
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Bobby is bored one day and flips a fair coin until it lands on tails. Bobby wins if the coin lands on heads a positive even number of times in the sequence of tosses. Then the probability that Bobby wins can be expressed in the form <math>\tfrac mn</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
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[[2022 SSMO Speed Round Problems/Problem 1|Solution]]
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==Problem 2==
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A bag is big enough to hold exactly 6 large pencils, 12 medium pencils, or 30 small pencils, with no space left over. Given that there is 1 large pencil and 3 medium pencils currently in the bag, what is the maximum number of small pencils that may be added to the bag? Note that there may still be space left over in the bag.
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[[2022 SSMO Speed Round Problems/Problem 2|Solution]]
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==Problem 3==
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Let <math>ABCD</math> be a parallelogram such that <math>E</math> is a point on <math>CD</math> such that <math>\frac{CE}{DE}=\frac{2}{3}.</math> Suppose that <math>BE</math> and <math>AC</math> intersect at <math>F.</math> If the area of triangle <math>AEF</math> is <math>15,</math> find the area of <math>ABCD</math>.
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[[2022 SSMO Speed Round Problems/Problem 3|Solution]]
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==Problem 4==
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Consider a quadrilateral <math>ABCD</math> with area <math>120</math> and satisfying <math>AB+CD=AD+BC=24</math>. There exists a point <math>P</math> in 3D space such that the distances from <math>P</math> to <math>AB</math>, <math>BC</math>, <math>CD</math>, and <math>DA</math> are all equal to <math>13</math>. Find the volume of <math>PABCD</math>.
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[[2022 SSMO Speed Round Problems/Problem 4|Solution]]
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==Problem 5==
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Let <math>ABCD</math> be a square such that <math>E</math> is on <math>AD</math> and <math>F</math> is on <math>CD.</math> If <math>AE=DF</math> and <math>\frac{[BEF]}{[ABCD]}=\frac{7}{18},</math> then the value of <math>\frac{EF^2}{BC^2}</math> can be expressed as <math>\frac mn</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
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[[2022 SSMO Speed Round Problems/Problem 5|Solution]]
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==Problem 6==
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At the beginning of day <math>1</math>, there is a single bacterium in a petri dish. During each day, each bacterium in the petri dish divides into <math>a>1</math> new bacteria, and <math>b\ge 1</math> bacteria are added to the petri dish (these bacteria do not divide on the day they were added). For example, at the end of day <math>1</math>, there are <math>a+b</math> bacteria in the petri dish. If, at the end of day <math>4</math>, the number of bacteria in the petri dish is a multiple of <math>48</math>, find the minimum possible value of <math>a+b</math>.
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[[2022 SSMO Speed Round Problems/Problem 6|Solution]]
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==Problem 7==
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Let <math>A_1=(1, 0)</math>. Define <math>A_{n+1}</math> as the image of <math>A_n</math> under a rotation of either <math>45^{\circ}</math>, <math>90^{\circ}</math>, or <math>135^{\circ}</math> clockwise about the origin, with each choice having a <math>\frac{1}{3}</math> chance of being selected. Find the expected value of the smallest positive integer <math>n>1</math> such that <math>A_n</math> coincides with <math>A_1</math>.
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[[2022 SSMO Speed Round Problems/Problem 7|Solution]]
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==Problem 8==
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How many positive integers cannot be written as <math>7a + 19b + 28c</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are positive integers (not necessarily distinct)?
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[[2022 SSMO Speed Round Problems/Problem 8|Solution]]
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==Problem 9==
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Consider a triangle <math>ABC</math> such that <math>AB=13</math>, <math>BC=14</math>, <math>CA=15</math> and a square <math>WXYZ</math> such that <math>Y</math> and <math>Z</math> lie on <math>\overleftrightarrow{BC}</math>, <math>W</math> lies on <math>\overleftrightarrow{AB}</math>, and <math>X</math> lies on <math>\overleftrightarrow{CA}</math>. Suppose further that <math>W</math>, <math>X</math>, <math>Y</math>, and <math>Z</math> are distinct from <math>A</math>, <math>B</math>, and <math>C</math>. Let <math>O</math> be the center of <math>WXYZ</math>. If <math>AO</math> intersects <math>BC</math> at <math>P</math>, then the sum of all values of <math>\frac{BP}{CP}</math> can be expressed as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
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[[2022 SSMO Speed Round Problems/Problem 9|Solution]]
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==Problem 10==
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Let <math>S = \{2,7,15,26,....\}</math> be the set of all numbers for which the <math>i^{th}</math> element in <math>S</math> is the sum of the <math>i^{th}</math> triangular number and the <math>i^{th}</math> positive perfect square. Let <math>T</math> be the set which contains all unique remainders when the elements in <math>S</math> are divided by <math>2022</math>. Find the number of elements in <math>T</math>.
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[[2022 SSMO Speed Round Problems/Problem 10|Solution]]

Revision as of 20:54, 31 May 2023

Problem 1

Suppose $a, b, c$ are distinct digits where $a \not= 0$ such that $\left(\overline{abc}\right)^2 = \overline{bad00}$ where $d = a+b$. Find $a+2b$.

Solution

Problem 2

Let $T=$ TNYWR. Now, let $\ell$ and $m$ have equations $y=(2+\sqrt{3})x+16$ and $y=\frac{x\sqrt{3}}{3}+20,$ respectively. Suppose that $A$ is a point on $\ell,$ such that the shortest distance from $A$ to $m$ is $T$. Given that $O$ is a point on $m$ such that $\overline{AO}\perp m,$ and $P$ is a point on $\ell$ such that $PO\perp \ell$, find $PO^2.$

Solution

Problem 3

Let $T=$ TNYWR. Now, let $ABC$ a triangle such that $AB=T,$ $AC=100$, and $\angle{ABC}=36^{\circ}.$ Find the remainder when the product of all possible values of $BC$ is divided by $1000$.

Solution


Problem 1

Bobby is bored one day and flips a fair coin until it lands on tails. Bobby wins if the coin lands on heads a positive even number of times in the sequence of tosses. Then the probability that Bobby wins can be expressed in the form $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 2

A bag is big enough to hold exactly 6 large pencils, 12 medium pencils, or 30 small pencils, with no space left over. Given that there is 1 large pencil and 3 medium pencils currently in the bag, what is the maximum number of small pencils that may be added to the bag? Note that there may still be space left over in the bag.

Solution

Problem 3

Let $ABCD$ be a parallelogram such that $E$ is a point on $CD$ such that $\frac{CE}{DE}=\frac{2}{3}.$ Suppose that $BE$ and $AC$ intersect at $F.$ If the area of triangle $AEF$ is $15,$ find the area of $ABCD$.

Solution

Problem 4

Consider a quadrilateral $ABCD$ with area $120$ and satisfying $AB+CD=AD+BC=24$. There exists a point $P$ in 3D space such that the distances from $P$ to $AB$, $BC$, $CD$, and $DA$ are all equal to $13$. Find the volume of $PABCD$.

Solution

Problem 5

Let $ABCD$ be a square such that $E$ is on $AD$ and $F$ is on $CD.$ If $AE=DF$ and $\frac{[BEF]}{[ABCD]}=\frac{7}{18},$ then the value of $\frac{EF^2}{BC^2}$ can be expressed as $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 6

At the beginning of day $1$, there is a single bacterium in a petri dish. During each day, each bacterium in the petri dish divides into $a>1$ new bacteria, and $b\ge 1$ bacteria are added to the petri dish (these bacteria do not divide on the day they were added). For example, at the end of day $1$, there are $a+b$ bacteria in the petri dish. If, at the end of day $4$, the number of bacteria in the petri dish is a multiple of $48$, find the minimum possible value of $a+b$.

Solution

Problem 7

Let $A_1=(1, 0)$. Define $A_{n+1}$ as the image of $A_n$ under a rotation of either $45^{\circ}$, $90^{\circ}$, or $135^{\circ}$ clockwise about the origin, with each choice having a $\frac{1}{3}$ chance of being selected. Find the expected value of the smallest positive integer $n>1$ such that $A_n$ coincides with $A_1$.

Solution

Problem 8

How many positive integers cannot be written as $7a + 19b + 28c$, where $a$, $b$, and $c$ are positive integers (not necessarily distinct)?

Solution

Problem 9

Consider a triangle $ABC$ such that $AB=13$, $BC=14$, $CA=15$ and a square $WXYZ$ such that $Y$ and $Z$ lie on $\overleftrightarrow{BC}$, $W$ lies on $\overleftrightarrow{AB}$, and $X$ lies on $\overleftrightarrow{CA}$. Suppose further that $W$, $X$, $Y$, and $Z$ are distinct from $A$, $B$, and $C$. Let $O$ be the center of $WXYZ$. If $AO$ intersects $BC$ at $P$, then the sum of all values of $\frac{BP}{CP}$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 10

Let $S = \{2,7,15,26,....\}$ be the set of all numbers for which the $i^{th}$ element in $S$ is the sum of the $i^{th}$ triangular number and the $i^{th}$ positive perfect square. Let $T$ be the set which contains all unique remainders when the elements in $S$ are divided by $2022$. Find the number of elements in $T$.

Solution

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