2011 UNCO Math Contest II Problems

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University of Northern Colorado MATHEMATICS CONTEST FINAL ROUND January 29, 2011 For Colorado Students Grades 7-12

$n!$, read as n factorial, is computed as $n! = 1 \cdot 2 \cdot 3 \cdot 4 \cdots  n$

• The factorials are $1, 2, 6, 24, 120, 720,\ldots$

• The square integers are $1, 4, 9, 16, 25, 36, 49, 64, 81,\ldots$

Problem 1

The largest integer $n$ so that $3^n$ evenly divides $9! = 1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7\cdot 8\cdot 9$ is $n = 4$. Determine the largest integer $n$ so that $3^n$ evenly divides $85! = 1\cdot 2\cdot 3\cdot 4\cdots 84\cdot 85$.

Solution

Problem 2

Let $m$ and $n$ be positive integers. List all the integers in the set $\left{20 ,21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31\right}$ (Error compiling LaTeX. ! Missing delimiter (. inserted).) that $\underline{cannot}$ be written in the form $m+n+m \cdot n$. As an example, $20$ $\underline{can}$ be so expressed since $20 = 2 + 6 + 2\cdot 6$.

Solution

Problem 3

The two congruent rectangles shown have dimensions $5$ in. by $25$ in. What is the area of the shaded overlap region?

filldraw((0,0)--(13,0)--(12,5)--cycle,blue);
pair A1=(0,0),B1=(25,0),C1=(25,5),D1=(0,5);
draw(A1--B1--C1--D1--cycle,black);
pair P=.1*unit(A1-D1)+D1,R=.1*unit(C1-D1)+D1;
draw(P--(P+R-D)--R,black);
pair A2=(0,0),B2=(25/13,-60/13),C2=(25,5),D2=(25-25/13,5+60/13);
draw(A2--B2--C2--D2--cycle,black);
P=.1*unit(A2-D2)+D2;
R=.1*unit(C2-D2)+D2;
draw(P--(P+R-D)--R,black);
 (Error compiling LaTeX. draw(P--(P+R-D)--R,black);
            ^
e3e47a918c489e98fe6f1264b4357c0d6143ea05.asy: 9.13: no matching function 'operator -(pair, <overloaded>)'
draw(P--(P+R-D)--R,black);
             ^
e3e47a918c489e98fe6f1264b4357c0d6143ea05.asy: 9.14: use of variable 'D' is ambiguous)

Solution

Problem 4

Let $A = \left{ 2,5,10,17,\cdots,n^2+1,\cdots\right}$ (Error compiling LaTeX. ! Missing delimiter (. inserted).) be the set of all positive squares plus $1$ and $B = \left{101, 104, 109, 116,\cdots,m^2 + 100,\cdots\right}$ (Error compiling LaTeX. ! Missing delimiter (. inserted).) be the set of all positive squares plus $100$.

(a) What is the smallest number in both $A$ and $B$?

(b) Find all numbers that are in both $A$ and $B$.

Solution

Problem 5

Determine the area of the square $ABCD$, with the given lengths along a zigzag line connecting $B$ and $D$.

pair A=(0,4*sqrt(10)),B=(4*sqrt(10),4*sqrt(10)),C=(4*sqrt(10),0),D=(0,0);
pair F=(sqrt(10),3*sqrt(10)),E=((9/5)*sqrt(10),(3/5)*sqrt(10));
draw(A--B--C--D--cycle,black);
draw(D--E--F--B,black);
pair P=.1*unit(D-E)+E,R=.1*unit(F-E)+E;
draw(P--(P+R-E)--R,black);
P=.1*unit(B-F)+F,R=.1*unit(E-F)+F;
draw(P--(P+R-F)--R,black);
MP("A",A,NW);MP("B",B,NE);MP("C",C,SE);MP("D",D,SW);
MP("6",(D/2+E/2),NW);MP("8",(E/2+F/2),NE);MP("10",(F/2+B/2),S);

 (Error compiling LaTeX. P=.1*unit(B-F)+F,R=.1*unit(E-F)+F;
                ^
8d1eaecbb1462f3788079defda2a11d8a43b97d5.asy: 11.17: syntax error
error: could not load module '8d1eaecbb1462f3788079defda2a11d8a43b97d5.asy')

Solution

Problem 6

What is the remainder when $1! + 2! + 3! + ?+ 2011!$ is divided by $18$?

Solution

Problem 7

What is the $\underline{sum}$ of the first $999$ terms of the sequence $1, 2, 3, 6, 7, 14, 15, 30, 31, 62, 63,\cdots$ that appeared on the First Round? Recall that a term in an even numbered position is twice the previous term, while a term in an odd numbered position is one more that the previous term.

Solution

Problem 8

The integer $45$ can be expressed as a sum of two squares as $45 = 3^2 + 6^2$.

(a) Express $74$ as the sum of two squares.

(b) Express the product $45\cdot 74$ as the sum of two squares.

(c) Prove that the product of two sums of two squares, $(a^2+b^2)(c^2+d^2)$ , can be represented as the sum of two squares.

Solution

Problem 9

Let $T(n)$ be the number of ways of selecting three distinct numbers from $\left{1, 2, 3,\cdots ,n\right}$ (Error compiling LaTeX. ! Missing delimiter (. inserted).) so that they are the lengths of the sides of a triangle. As an example, $T(5) = 3$; the only possibilities are $2-3-4, 2-4-5$, and $3-4-5$.

(a) Determine a recursion for T(n).

(b) Determine a closed formula for T(n).

Solution

Problem 10

The integers $1, 2, 3,\cdots , 50$ are written on the blackboard. Select any two, call them $m$ and $n$ and replace these two with the one number $m+n+mn$". Continue doing this until only one number remains and explain, with proof, what happens. Also explain with proof what happens in general as you replace $50$ with $N$. As an example, if you select $3$ and $17$ you replace them with $3 + 17 + 51 = 71$. If you select $5$ and $7$, replace them with $47$. You now have two $47$’s in this case but that’s OK.

Solution

Problem 11

Tie breaker – Generalize problem #2, and prove your statement.

Solution

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