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Difference between revisions of "2023 SSMO Accuracy Round Problems"

(Created page with "==Problem 1== Mr. Sammy proposes a Hamburger Proclamation, which has <math>500</math> lines, divided into paragraphs of <math>5</math> lines each. It takes him <math>23</math...")
 
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Suppose that <math>a, b, c</math> are real numbers such
 
Suppose that <math>a, b, c</math> are real numbers such
 +
<cmath>
 
\begin{align*}
 
\begin{align*}
 
     a + b - c &= 4 \\
 
     a + b - c &= 4 \\
Line 17: Line 18:
 
     a^3 + b^3 - c^3 &= 34 \\
 
     a^3 + b^3 - c^3 &= 34 \\
 
\end{align*}
 
\end{align*}
 +
</cmath>
 
Find the sum of all possible values of <math>a+b+c</math>.
 
Find the sum of all possible values of <math>a+b+c</math>.
  
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located in <math>P_{2023}</math> as <math>P_{2022}</math>. We repeat the process of taking smaller grids  until we eventually converge at the unit square <math>P_1.</math>  
 
located in <math>P_{2023}</math> as <math>P_{2022}</math>. We repeat the process of taking smaller grids  until we eventually converge at the unit square <math>P_1.</math>  
  
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\begin{center}
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<center><asy>
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    \begin{asy}
+
size(7cm);
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        size(7cm);
+
filldraw((0,0)--(0,10)--(10,10)--(10,0)--cycle, opacity(0.2)+lightblue, blue);
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fill((0,0)--(0,9)--(9,9)--(9,0)--cycle,  
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        fill((0,0)--(0,9)--(9,9)--(9,0)--cycle,  
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opacity(0.1)+lightblue);
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        opacity(0.1)+lightblue);
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draw((0,9)--(9,9)--(9,0),  
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        draw((0,9)--(9,9)--(9,0),  
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blue);
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        blue);
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fill((1,0)--(1,8)--(9,8)--(9,0)--cycle, opacity(0.1)+lightblue);
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        fill((1,0)--(1,8)--(9,8)--(9,0)--cycle, opacity(0.1)+lightblue);
+
draw((1,0)--(1,8)--(9,8), blue);
−
        draw((1,0)--(1,8)--(9,8), blue);
 
  
−
        label("<math>A = P_{2023}</math>", (8.3, 9.52));
+
label("$A = P_{2023}$", (8.3, 9.52));
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        label("<math>P_{2022}</math>", (6.8, 8.52));
+
label("$P_{2022}$", (6.8, 8.52));
−
        label("<math>\dots</math>", (4.78, 7.52));
+
label("$\dots$", (4.78, 7.52));
−
 
+
</asy></center>
−
    \end{asy}
 
−
\end{center}
 
  
 
Of the <math>4^{2022}</math> distinct tuples of shrinking grids <math>(P_{2023}, P_{2022}, \dots P_1)</math>, let <math>T</math> be the number of these tuples such that their last element is the center square of the original grid <math>A</math>. Find the largest integer <math>a</math> such <math>2^a \mid T.</math>
 
Of the <math>4^{2022}</math> distinct tuples of shrinking grids <math>(P_{2023}, P_{2022}, \dots P_1)</math>, let <math>T</math> be the number of these tuples such that their last element is the center square of the original grid <math>A</math>. Find the largest integer <math>a</math> such <math>2^a \mid T.</math>

Revision as of 20:19, 15 December 2023

Problem 1

Mr. Sammy proposes a Hamburger Proclamation, which has $500$ lines, divided into paragraphs of $5$ lines each. It takes him $23$ seconds to read each line. Additionally, he adds a $0.5$ second pause between two lines in a paragraph, and a $2$ second pause between paragraphs. If it takes him $S$ minutes to read the whole Hamburger Proclamation, find $\left\lfloor 10S \right\rfloor.$

Solution

Problem 2

Suppose that the average of all $n$-digit palindromes is denoted by $P_{n}$ and the average of all $n$-digit numbers is denoted by $N_{n}.$ Find $\left\lfloor\sum_{n=1}^{100}(P_{n}-N_{n})\right\rfloor.$

Solution

Problem 3

Suppose that $a, b, c$ are real numbers such \begin{align*} a + b - c &= 4 \\ a^2 + b^2 + c^2 &= 14 \\ a^3 + b^3 - c^3 &= 34 \\ \end{align*} Find the sum of all possible values of $a+b+c$.

Solution

Problem 4

In square $ABCD,$ point $E$ is selected on diagonal $AC.$ Let $F$ be the intersection of the circumcircles of triangles $ABE$ and $CDE.$ Given that $AB = 10$ and $EF = 6,$ find the maximum possible area of triangle $BEC.$ (A circumcircle of some triangle $\triangle ABC$ is the circle containing $A$, $B$, and $C$)

Solution

Problem 5

Define the $\textit{relationship}$ between two numbers $a$ and $b$ to be $\frac{\sigma(ab)}{\sigma(a)\sigma(b)}$ where $\sigma(x)$ is the number of divisors of $x$. Find the sum of integers $1 \le n \le 100$ which have a relationship of $\frac{3}{4}$ with $360$.

Solution

Problem 6

Let the roots of $P(x) = x^3 - 2023x^2 + 2023^{2023}$ be $p, q, r$. Find \[ 

   \frac{p^2 + q^2}{p + q} + \frac{q^2 + r^2}{q + r} + \frac{r^2 + p^2}{r + p}

\]

Solution

Problem 7

Concentric circles $\omega$ and $\omega_1$ are drawn, with radii $3$ and $5,$ respectively. Chords $AB$ and $CD$ of $\omega_1$ are both tangent to $\omega$ and intersect at $P.$ If $PA=PC = 3,$ then the sum of all possible distinct values of $[PAD]$ can be expressed as $\frac{m}{n},$ for relatively prime positive integers $m$ and $n.$ Find $m+n.$

Solution

Problem 8

There is a quadrilateral $ABCD$ inscribed in a circle $\omega$ with center $O$. In quadrilateral $ABCD$, diagonal $AC$ is a diameter of the circle, $\angle BAC = 30^\circ,$ and $\angle DAC = 15^\circ.$ Let $E$ be the base of the altitude from $O$ onto side $BA$. Let $F$ be the base of the altitude from $E$ onto $BO$. Given that $EF = 3,$ and that the product of the lengths of the diagonals of $ABCD$ is $a\sqrt{b},$ for some squarefree $b,$ find $a+b.$

Solution

Problem 9

Consider a $2023 \times 2023$ grid called $A = P_{2023}$. We take one of the four smaller $2022 \times 2022$ grids located in $P_{2023}$ as $P_{2022}$. We repeat the process of taking smaller grids until we eventually converge at the unit square $P_1.$

[asy] size(7cm); filldraw((0,0)--(0,10)--(10,10)--(10,0)--cycle, opacity(0.2)+lightblue, blue); fill((0,0)--(0,9)--(9,9)--(9,0)--cycle, opacity(0.1)+lightblue); draw((0,9)--(9,9)--(9,0), blue); fill((1,0)--(1,8)--(9,8)--(9,0)--cycle, opacity(0.1)+lightblue); draw((1,0)--(1,8)--(9,8), blue); label("$A = P_{2023}$", (8.3, 9.52)); label("$P_{2022}$", (6.8, 8.52)); label("$\dots$", (4.78, 7.52)); [/asy]

Of the $4^{2022}$ distinct tuples of shrinking grids $(P_{2023}, P_{2022}, \dots P_1)$, let $T$ be the number of these tuples such that their last element is the center square of the original grid $A$. Find the largest integer $a$ such $2^a \mid T.$ Solution

Problem 10

Let $\triangle ABC$ be a triangle such $AB = 13$, $BC = 14$, $CA = 15$. Let the incircle of $\triangle ABC$ touch $BC$ at $D$, $AC$ at $E$, and $AB$ at $F$. Let $\ell_A$ be the line through the midpoints of $AE$ and $EF$. Define $\ell_B$ and $\ell_C$ similarily. Let the area of the star created by the union of $\triangle ABC$ and the triangle bound by $\ell_A$, $\ell_B$, and $\ell_C$ be $\frac{p}{q}$ for relatively prime $p$ and $q$. Find $p + q$.

Solution

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