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Difference between revisions of "2023 WSMO Speed Round Problems"

(Created page with "==Problem 1== Find the number of square units in the area of the shaded region. <asy> size(4cm); for (int i=0; i<5; ++i) { for (int j=0; j<5; ++j) { dot((i,j)); } } pai...")
 
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There are 4 tables and 5 chairs at each table. Each chair seats 2 people. There are 10 people who are seated randomly. Andre and Emily are 2 of them, and are a couple. If the probability that Andre and Emily are in the same chair is <math>\frac{m}{n},</math> for relatively prime positive integers <math>m</math> and <math>n,</math> find <math>m+n.</math>
 
There are 4 tables and 5 chairs at each table. Each chair seats 2 people. There are 10 people who are seated randomly. Andre and Emily are 2 of them, and are a couple. If the probability that Andre and Emily are in the same chair is <math>\frac{m}{n},</math> for relatively prime positive integers <math>m</math> and <math>n,</math> find <math>m+n.</math>
  
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[[2023 WSMO Speed Round Problems/Problem 2 Solution]]
+
[[2023 WSMO Speed Round Problems/Problem 2|Solution]]
  
 
==Problem 3==
 
==Problem 3==
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There are 6 pairs of socks for each color of the rainbow (red, orange, yellow, green, blue, indigo, violet) in a sock drawer.  How many socks must be drawn from the drawer to guarantee that a pair of red socks have been drawn?
 
There are 6 pairs of socks for each color of the rainbow (red, orange, yellow, green, blue, indigo, violet) in a sock drawer.  How many socks must be drawn from the drawer to guarantee that a pair of red socks have been drawn?
  
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[[2023 WSMO Speed Round Problems/Problem 3 Solution]]
+
[[2023 WSMO Speed Round Problems/Problem 3|Solution]]
  
 
==Problem 4==
 
==Problem 4==
  
 
A right circular cone is inscribed in a right prism as shown. If the ratio of the volume of the cone to the volume of the prism is <math>\frac{m}{n}\pi,</math> for relatively prime positive integers <math>m</math> and <math>n,</math> find <math>m+n.</math>
 
A right circular cone is inscribed in a right prism as shown. If the ratio of the volume of the cone to the volume of the prism is <math>\frac{m}{n}\pi,</math> for relatively prime positive integers <math>m</math> and <math>n,</math> find <math>m+n.</math>
 +
<asy>
 +
import three; import graph3;
 +
defaultpen(linewidth(0.8));
 +
size(200);
 +
draw((0,0,0)--(1,0,0)--(1,1,0)--(0,1,0)--cycle);
 +
draw((0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle);
 +
draw((0,0,0)--(0,0,1)); draw((1,0,0)--(1,0,1)); draw((1,1,0)--(1,1,1)); draw((0,1,0)--(0,1,1));
 +
draw(Circle((0.5,0.5,0),0.5),dashed);
 +
draw((0.5,0.5,1)--(0.5,0,0),dashed); draw((0.5,0.5,1)--(0.5,1,0),dashed); draw((0.5,0.5,1)--(1,0.5,0),dashed); draw((0.5,0.5,1)--(0,0.5,0),dashed);
 +
</asy>
  
 
==Problem 5==
 
==Problem 5==
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There exists a rational polynomial <math>f(x)</math> such that for all <math>x</math> in the range <math>(0,1),</math>  <math>f(x)=\sum_{n=1}^{\infty}nx^n.</math> If the maximum of <math>f(x)</math> over <math>[6,9]</math> is <math>\frac{m}{n},</math> for relatively prime positive integers <math>m</math> and <math>n,</math> find <math>m+n.</math>
 
There exists a rational polynomial <math>f(x)</math> such that for all <math>x</math> in the range <math>(0,1),</math>  <math>f(x)=\sum_{n=1}^{\infty}nx^n.</math> If the maximum of <math>f(x)</math> over <math>[6,9]</math> is <math>\frac{m}{n},</math> for relatively prime positive integers <math>m</math> and <math>n,</math> find <math>m+n.</math>
  
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[[2023 WSMO Speed Round Problems/Problem 5 Solution]]
+
[[2023 WSMO Speed Round Problems/Problem 5|Solution]]
  
 
==Problem 6==
 
==Problem 6==
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Let <math>ABC</math> be an equilateral triangle of side length <math>6.</math> Points <math>A_1,A_2,A_3,B_1,B_2,B_3,C_1,C_2,C_3</math> are chosen such that <math>A_1,A_2,A_3</math> divide <math>BC</math> into four equal segments, <math>B_1,B_2,B_3</math> divide <math>AC</math> into four equal segments, and <math>C_1,C_2,C_3</math> divide <math>AB</math> into four equal segments. If <math>i,j,k</math> are chosen from the set <math>{1,2,3}</math> independently and randomly, the expected area of <math>A_iB_jC_k</math> is <math>\frac{a\sqrt{b}}{c},</math> for squarefree <math>b</math> and relatively prime positive integers <math>a</math> and <math>c.</math> Find <math>a+b+c.</math>
 
Let <math>ABC</math> be an equilateral triangle of side length <math>6.</math> Points <math>A_1,A_2,A_3,B_1,B_2,B_3,C_1,C_2,C_3</math> are chosen such that <math>A_1,A_2,A_3</math> divide <math>BC</math> into four equal segments, <math>B_1,B_2,B_3</math> divide <math>AC</math> into four equal segments, and <math>C_1,C_2,C_3</math> divide <math>AB</math> into four equal segments. If <math>i,j,k</math> are chosen from the set <math>{1,2,3}</math> independently and randomly, the expected area of <math>A_iB_jC_k</math> is <math>\frac{a\sqrt{b}}{c},</math> for squarefree <math>b</math> and relatively prime positive integers <math>a</math> and <math>c.</math> Find <math>a+b+c.</math>
  
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[[2023 WSMO Speed Round Problems/Problem 6 Solution]]
+
[[2023 WSMO Speed Round Problems/Problem 6|Solution]]
  
 
==Problem 7==
 
==Problem 7==
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Let <math>e, a, j</math> be real numbers such that <math>e + a + j = 1</math> and <math>e\geq -\frac{1}{3}</math>, <math>a\geq 1</math> and <math>j\geq-\frac{5}{3}</math>. Find the maximum value of <math>\sqrt{3e+1} + \sqrt{3a+3} + \sqrt{3j+5}.</math>
 
Let <math>e, a, j</math> be real numbers such that <math>e + a + j = 1</math> and <math>e\geq -\frac{1}{3}</math>, <math>a\geq 1</math> and <math>j\geq-\frac{5}{3}</math>. Find the maximum value of <math>\sqrt{3e+1} + \sqrt{3a+3} + \sqrt{3j+5}.</math>
  
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[[2023 WSMO Speed Round Problems/Problem 7 Solution]]
+
[[2023 WSMO Speed Round Problems/Problem 7|Solution]]
  
 
==Problem 8==
 
==Problem 8==
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In regular octagon <math>ABCDEFGH</math> of sidelength <math>4,</math> quadrilaterals <math>ACEG</math> and <math>BDFH</math> are drawn. Find the square of the area of the overlap of the two quadrilaterals.
 
In regular octagon <math>ABCDEFGH</math> of sidelength <math>4,</math> quadrilaterals <math>ACEG</math> and <math>BDFH</math> are drawn. Find the square of the area of the overlap of the two quadrilaterals.
  
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[[2023 WSMO Speed Round Problems/Problem 8 Solution]]
+
[[2023 WSMO Speed Round Problems/Problem 8|Solution]]
  
 
==Problem 9==
 
==Problem 9==
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Suppose that <math>b</math> and <math>c</math> are the roots of the equation <math>x^2-\log(16)x+\log(64).</math> If <math>\sqrt{a+b}+\sqrt{a+c} = \sqrt{b+c},</math> then <math>2^a = \frac{\sqrt{m}}{n},</math> find <math>m+n.</math>
 
Suppose that <math>b</math> and <math>c</math> are the roots of the equation <math>x^2-\log(16)x+\log(64).</math> If <math>\sqrt{a+b}+\sqrt{a+c} = \sqrt{b+c},</math> then <math>2^a = \frac{\sqrt{m}}{n},</math> find <math>m+n.</math>
  
−
[[2023 WSMO Speed Round Problems/Problem 9 Solution]]
+
[[2023 WSMO Speed Round Problems/Problem 9|Solution]]
  
 
==Problem 10==
 
==Problem 10==
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Consider acute triangle <math>ABC</math>, <math>H</math> is the orthocenter. Extend <math>AH</math> to meet <math>BC</math> at <math>D</math>. The angle bisector of <math>\angle{ABH}</math> meets the midpoint of <math>AD</math>, <math>M</math>. If <math>AB=10, BH=4</math>, then the area of <math>ABC</math> is <math>\frac{a\sqrt{b}}{c},</math> for squarefree <math>b</math> and relatively prime positive integers <math>a</math> and <math>c.</math> Find <math>a+b+c.</math>
 
Consider acute triangle <math>ABC</math>, <math>H</math> is the orthocenter. Extend <math>AH</math> to meet <math>BC</math> at <math>D</math>. The angle bisector of <math>\angle{ABH}</math> meets the midpoint of <math>AD</math>, <math>M</math>. If <math>AB=10, BH=4</math>, then the area of <math>ABC</math> is <math>\frac{a\sqrt{b}}{c},</math> for squarefree <math>b</math> and relatively prime positive integers <math>a</math> and <math>c.</math> Find <math>a+b+c.</math>
  
−
[[2023 WSMO Speed Round Problems/Problem 10 Solution]]
+
[[2023 WSMO Speed Round Problems/Problem 10|Solution]]

Revision as of 10:33, 2 May 2025

Problem 1

Find the number of square units in the area of the shaded region. [asy] size(4cm); for (int i=0; i<5; ++i) { for (int j=0; j<5; ++j) { dot((i,j)); } } pair a = (0,2); pair b = (1,4); pair c = (3,2); pair d = (4,4); pair e = (3,0); pair f = (1,2); path x = a--b--c--d--e--f--cycle; fill(x, cyan); draw(x, linewidth(2)); [/asy]

Problem 2

There are 4 tables and 5 chairs at each table. Each chair seats 2 people. There are 10 people who are seated randomly. Andre and Emily are 2 of them, and are a couple. If the probability that Andre and Emily are in the same chair is $\frac{m}{n},$ for relatively prime positive integers $m$ and $n,$ find $m+n.$

Solution

Problem 3

There are 6 pairs of socks for each color of the rainbow (red, orange, yellow, green, blue, indigo, violet) in a sock drawer. How many socks must be drawn from the drawer to guarantee that a pair of red socks have been drawn?

Solution

Problem 4

A right circular cone is inscribed in a right prism as shown. If the ratio of the volume of the cone to the volume of the prism is $\frac{m}{n}\pi,$ for relatively prime positive integers $m$ and $n,$ find $m+n.$ [asy] import three; import graph3; defaultpen(linewidth(0.8)); size(200); draw((0,0,0)--(1,0,0)--(1,1,0)--(0,1,0)--cycle); draw((0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle); draw((0,0,0)--(0,0,1)); draw((1,0,0)--(1,0,1)); draw((1,1,0)--(1,1,1)); draw((0,1,0)--(0,1,1)); draw(Circle((0.5,0.5,0),0.5),dashed); draw((0.5,0.5,1)--(0.5,0,0),dashed); draw((0.5,0.5,1)--(0.5,1,0),dashed); draw((0.5,0.5,1)--(1,0.5,0),dashed); draw((0.5,0.5,1)--(0,0.5,0),dashed); [/asy]

Problem 5

There exists a rational polynomial $f(x)$ such that for all $x$ in the range $(0,1),$ $f(x)=\sum_{n=1}^{\infty}nx^n.$ If the maximum of $f(x)$ over $[6,9]$ is $\frac{m}{n},$ for relatively prime positive integers $m$ and $n,$ find $m+n.$

Solution

Problem 6

Let $ABC$ be an equilateral triangle of side length $6.$ Points $A_1,A_2,A_3,B_1,B_2,B_3,C_1,C_2,C_3$ are chosen such that $A_1,A_2,A_3$ divide $BC$ into four equal segments, $B_1,B_2,B_3$ divide $AC$ into four equal segments, and $C_1,C_2,C_3$ divide $AB$ into four equal segments. If $i,j,k$ are chosen from the set ${1,2,3}$ independently and randomly, the expected area of $A_iB_jC_k$ is $\frac{a\sqrt{b}}{c},$ for squarefree $b$ and relatively prime positive integers $a$ and $c.$ Find $a+b+c.$

Solution

Problem 7

Let $e, a, j$ be real numbers such that $e + a + j = 1$ and $e\geq -\frac{1}{3}$, $a\geq 1$ and $j\geq-\frac{5}{3}$. Find the maximum value of $\sqrt{3e+1} + \sqrt{3a+3} + \sqrt{3j+5}.$

Solution

Problem 8

In regular octagon $ABCDEFGH$ of sidelength $4,$ quadrilaterals $ACEG$ and $BDFH$ are drawn. Find the square of the area of the overlap of the two quadrilaterals.

Solution

Problem 9

Suppose that $b$ and $c$ are the roots of the equation $x^2-\log(16)x+\log(64).$ If $\sqrt{a+b}+\sqrt{a+c} = \sqrt{b+c},$ then $2^a = \frac{\sqrt{m}}{n},$ find $m+n.$

Solution

Problem 10

Consider acute triangle $ABC$, $H$ is the orthocenter. Extend $AH$ to meet $BC$ at $D$. The angle bisector of $\angle{ABH}$ meets the midpoint of $AD$, $M$. If $AB=10, BH=4$, then the area of $ABC$ is $\frac{a\sqrt{b}}{c},$ for squarefree $b$ and relatively prime positive integers $a$ and $c.$ Find $a+b+c.$

Solution

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