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Difference between revisions of "2022 SSMO Relay Round 2 Problems"
 
Line 7: Line 7:
 
==Problem 2==
 
==Problem 2==
  
−
Let <math>T=</math> TNYWR. Suppose that the monic quadratic <math>f(x)</math> is tangent to the function <math>g(x)=|x+2|-T</math> at two points, when graphed on the coordinate plane. Then <math>|f(1)|</math> can be expressed as <math>\frac mn</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>10m+n</math>.
+
Let <math>T=TNYWR</math>. Suppose that the monic quadratic <math>f(x)</math> is tangent to the function <math>g(x)=|x+2|-T</math> at two points, when graphed on the coordinate plane. Then <math>|f(1)|</math> can be expressed as <math>\frac mn</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>10m+n</math>.
  
 
[[2022 SSMO Relay Round 2 Problems/Problem 2|Solution]]
 
[[2022 SSMO Relay Round 2 Problems/Problem 2|Solution]]
Line 13: Line 13:
 
==Problem 3==
 
==Problem 3==
  
−
Let <math>T =</math> TNYWR. Let <math>a+b = \lfloor \sqrt{T} \rfloor</math>. If <math>a^5 + b^5 = 15</math>, then <math>ab</math> has two possible values. The absolute difference of these values is <math>\frac{x\sqrt{y}}{z}</math>, where <math>x,y</math> and <math>z</math> are positive integers, <math>x</math> and <math>z</math> are relatively prime, and <math>y</math> is not divisible by the square of any prime. What is <math>x+y+z?</math>
+
Let <math>T =TNYWR</math>. Let <math>a+b = \lfloor \sqrt{T} \rfloor</math>. If <math>a^5 + b^5 = 15</math>, then <math>ab</math> has two possible values. The absolute difference of these values is <math>\frac{x\sqrt{y}}{z}</math>, where <math>x,y</math> and <math>z</math> are positive integers, <math>x</math> and <math>z</math> are relatively prime, and <math>y</math> is not divisible by the square of any prime. What is <math>x+y+z?</math>
  
 
[[2022 SSMO Relay Round 2 Problems/Problem 3|Solution]]
 
[[2022 SSMO Relay Round 2 Problems/Problem 3|Solution]]

Latest revision as of 19:17, 2 May 2025

Problem 1

Let $P$ be a randomly selected point on a circle, and let $A$ be a randomly selected point inside the same circle. A dilation centered at $P$ with a scale factor of $2$ sends $A$ to $A'.$ Given that the probability that $PA'$ is less than the length of the diameter of the circle can be expressed as $\frac{a\pi+b\sqrt{c}}{d\pi},$ where $a,b,c,d$ are integers such that $a$ and $d$ are positive, $c$ is squarefree, and $\gcd{(a, b, d)}=1$, find the value of $a+b+c+d.$

Solution

Problem 2

Let $T=TNYWR$. Suppose that the monic quadratic $f(x)$ is tangent to the function $g(x)=|x+2|-T$ at two points, when graphed on the coordinate plane. Then $|f(1)|$ can be expressed as $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $10m+n$.

Solution

Problem 3

Let $T =TNYWR$. Let $a+b = \lfloor \sqrt{T} \rfloor$. If $a^5 + b^5 = 15$, then $ab$ has two possible values. The absolute difference of these values is $\frac{x\sqrt{y}}{z}$, where $x,y$ and $z$ are positive integers, $x$ and $z$ are relatively prime, and $y$ is not divisible by the square of any prime. What is $x+y+z?$

Solution

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