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Difference between revisions of "2023 SSMO Team Round Problems"

(Created page with "==Problem 1== Let <math>(a, b, c, d)</math> be a permutation of <math>(2, 0, 2, 3)</math>. Find the largest possible value of <math>a^b + b^c + c^d + d^a</math> 2022 SSMO...")
 
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==Problem 6==
 
==Problem 6==
  
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  Suppose that <math>a,b,c</math> are positive reals satisfying<cmath>(a^3+4)(b^3+6)(c^3+8) = 8(a+b+c)^3.</cmath> Find the sum of all possible values of <math>\frac{bc}{a^2}.</math> If you believe there are no solutions, put <math>0</math> as your answer. If you believe the sum is infinity, put <math>1000</math> as your answer.
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Suppose that <math>a,b,c</math> are positive reals satisfying<cmath>(a^3+4)(b^3+6)(c^3+8) = 8(a+b+c)^3.</cmath> Find the sum of all possible values of <math>\frac{bc}{a^2}.</math> If you believe there are no solutions, put <math>0</math> as your answer. If you believe the sum is infinity, put <math>1000</math> as your answer.
  
 
[[2022 SSMO Team Round Problems/Problem 6|Solution]]
 
[[2022 SSMO Team Round Problems/Problem 6|Solution]]
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Consider a piece of paper in the shape of a regular pentagon with sidelength <math>2.</math> We fold it in half. We then fold it such that the vertices of the longest side become the same side. The area of the folded figure can be expressed as <math>\frac{1}{a}\sqrt{b + c\sqrt{5}}</math> where <math>a, b, c</math> are integers and <math>\gcd(b, c)</math> is squarefree. Find <math>a + b + c.</math> (For convenience, note that <math>\cos(36^\circ) = \frac{1 + \sqrt{5}}{4}</math>)
 
Consider a piece of paper in the shape of a regular pentagon with sidelength <math>2.</math> We fold it in half. We then fold it such that the vertices of the longest side become the same side. The area of the folded figure can be expressed as <math>\frac{1}{a}\sqrt{b + c\sqrt{5}}</math> where <math>a, b, c</math> are integers and <math>\gcd(b, c)</math> is squarefree. Find <math>a + b + c.</math> (For convenience, note that <math>\cos(36^\circ) = \frac{1 + \sqrt{5}}{4}</math>)
  
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<center><asy>[width=\the\linewidth/4,inline=true]
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<center>
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<asy>
 
  /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
 
  /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
 
real xmin = -2.464957433726653, xmax = 3.328270917519617, ymin = -1.0176780156925203, ymax = 3.975434729820146;  /* image dimensions */
 
real xmin = -2.464957433726653, xmax = 3.328270917519617, ymin = -1.0176780156925203, ymax = 3.975434729820146;  /* image dimensions */
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clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);  
 
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);  
 
  /* end of picture */
 
  /* end of picture */
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</asy></center>
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</asy>
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</center>
 
[[2022 SSMO Team Round Problems/Problem 15|Solution]]
 
[[2022 SSMO Team Round Problems/Problem 15|Solution]]

Revision as of 20:30, 15 December 2023

Problem 1

Let $(a, b, c, d)$ be a permutation of $(2, 0, 2, 3)$. Find the largest possible value of $a^b + b^c + c^d + d^a$

Solution

Problem 2

A plane and a car start both move northward. The car moves northbound at 60 miles per hour. The plane moves northeast and increases in altitude at an angle of $30^{\circ}.$ Let $s$ the speed in feet per second that the plane must fly at to move north at the same speed as the car. Find $3s^2$.

Solution

Problem 3

Let $ABC$ be a triangle such that $AB=4\sqrt{2}, BC=5\sqrt{2},$ and $AC=\sqrt{82}.$ Let $\omega$ be the circumcircle of $\triangle ABC$. Let $D$ be on the circle such that $\overline{BD} \perp \overline{AC}.$ Let $E$ be the point diametrically opposite of $B$. Let $F$ be the point diametrically opposite $D$. Find the area of the quadrilateral $ADEF$ in terms of a mixed number $a\frac{b}{c}$. Find $a+b+c$.

Solution

Problem 4

Find the sum of values for prime $p$ such that $p \mid (2023^{p^2}+(p-1)!+2^{p^4}).$

Solution

Problem 5

Joshy is playing a game with a dartboard that has two sections. If Joshy hits the first section, he gets $20$ points, and if he hits the second section, he gets $23$ points. Assume Joshy always hits one of the two sections. Let $a$ be the maximum value that Joshy cannot achieve. Let $b$ be the number of positive integer scores Joshy cannot achieve. Let $c$ be the number of ways for Joshy to achieve $2023$ points. Find $(a-b)c$.

Solution

Problem 6

Suppose that $a,b,c$ are positive reals satisfying\[(a^3+4)(b^3+6)(c^3+8) = 8(a+b+c)^3.\] Find the sum of all possible values of $\frac{bc}{a^2}.$ If you believe there are no solutions, put $0$ as your answer. If you believe the sum is infinity, put $1000$ as your answer.

Solution

Problem 7

Let $S = \{1, 2, 3, 4, \cdots, 23\}$ and let there be randomly chosen sets $A, B, C$ where $A, B, C \subseteq S$. The probability that $|A| + |B| = |C|$ can be expressed as $\frac{m}{n}$. Let $2^a$ be the largest power of $2$ such $2^a \mid n$. Find $a$.

Solution

Problem 8

Three rabbits run away from the origin at the same speed and constant velocity such that the angle between any two rabbits' directions is $120^\circ$. After $12$ seconds, a hunter with a speed $\sqrt{7}$ times that of the rabbits runs from the origin. Let the minimal time in seconds needed for her to meet (and subsequently) catch all three rabbits be $a + b\sqrt{c}$. Find $a + b + c$.

Solution

Problem 9

Let $B$, $K$, and $R$ be the total number of possible moves for a bishop, knight, or rook from any position of a $9$ by $9$ grid. Find $B + K + R$.

(A bishop moves along diagonals, a rook moves along rows, and a knight moves in the form of a $2 \times 1$ "L" shape)

Solution

Problem 10

There exists a lane of infinite cars. Each car has a $\frac{1}{3}$ chance of being high quality and a $\frac{2}{3}$ chance of being low quality. John goes down the row of cars buying high-quality cars. However, after John sees 3 low-quality cars, he gives up on buying additional cars. Let the probability that he buys at least $5$ cars before giving up as $\frac{m}{n}$. Find $m+n$.

Solution

Problem 11

Let $ABCD$ be a cyclic quadrilateral such that $AC$ is the diameter. Let $P$ be the orthocenter of $ABD$. Define $X = AB\cup CD$, and $Y = AD\cup BC$. If $AB = 8$, $BC = 1$, and $CD = 4$, suppose $\frac{[CBPD]}{[AXY]}=\frac{m}{n}.$ Find $m+n$.

Solution

Problem 12

Let $T$ be the set of rationals of the form $\frac{a}{2^b}$ for nonnegative $a$ and $b$. Define the function $f \colon T \to \mathbb{Z}$ such that, for $t = \frac{a}{2^b}$ such $b$ is minimal, we have that \[f\left(\frac{a}{2^b}\right) = \begin{cases} 1 & b = 0 \\ f\left(\frac{a-1}{2^b}\right) + f\left(\frac{a+1}{2^b}\right) & b \ne 0 \\ \end{cases}\]

Suppose \[\sum_{i=0}^{2^{10} - 1} \frac{f\left(\frac{i+1}{2^{10}}\right)}{f\left(\frac{i}{2^{10}}\right)}\] equals $\frac{m}{n}$. Find $m+n$.

Solution

Problem 13

Let $D(n)$ denote the product of all divisors of $n$ Let $P(i,j)$ denote the set of all integers that are both a multiple of $i$ and a factor of $j.$ Let \[ -F(a) = \sqrt{\left|\log_{10}\left(\frac{D(10^{a})}{\prod_{\omega\in P(10^2,10^{a+2})}\omega}\right)\right|}\text{ and }G(n) = \sqrt[n-1]{\prod_{i=2}^{n}10^{F(i)}}. \] Suppose $\sum_{k=2}^{\infty}G(k)$ is $\frac{a+b\sqrt{c}}{d}$. Find the value of $a+b+c+d$.

Solution

Problem 14

Find the sum of all perfect squares of the form $2p^3 - 5p^2q + q^2$ where $p$ and $q$ are positive integers such $p$ is prime and $p \nmid q$.

Solution

Problem 15

Consider a piece of paper in the shape of a regular pentagon with sidelength $2.$ We fold it in half. We then fold it such that the vertices of the longest side become the same side. The area of the folded figure can be expressed as $\frac{1}{a}\sqrt{b + c\sqrt{5}}$ where $a, b, c$ are integers and $\gcd(b, c)$ is squarefree. Find $a + b + c.$ (For convenience, note that $\cos(36^\circ) = \frac{1 + \sqrt{5}}{4}$)

[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ real xmin = -2.464957433726653, xmax = 3.328270917519617, ymin = -1.0176780156925203, ymax = 3.975434729820146; /* image dimensions */ /* draw figures */ filldraw((0,0)--(2,0)--(2.618033988749895,1.9021130325903064)--(1,3.077683537175253)--(-0.6180339887498947,1.9021130325903073)--cycle, opacity(0.4)+lightblue,lightblue); filldraw((0.3090169943749479,2.853169548885461)--(0.690983005625053,2.853169548885461)--(1.1545084971874735,1.4265847744427302)--(-0.30901699437494734,0.9510565162951536)--(-0.6180339887498947,1.9021130325903073)--(0.190983005625053,2.48989828488278)--cycle, opacity(0.4)+palegreen,green); draw((xmin, 0.32491969623290623*xmin + 1.0514622242382672)--(xmax, 0.32491969623290623*xmax + 1.0514622242382672), dashed); /* line */ draw((xmin, -3.0776835371752562*xmin + 4.979796569765563)--(xmax, -3.0776835371752562*xmax + 4.979796569765563), dashed); /* line */ /* dots and labels */ clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]

Solution

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