Category Archives: MATHCOUNTS

MATHCOUNTS Exercises – 18

Posted on January 30, 2020 by kevin

How many 4-digit integers between 1000 and 9999 have their digits in non-decreasing order, such as 3447? Click here for solutions. Solution 1 – Brute Force Denote $f(m, n)$ as the number of $m$-digit integers with their digits in non-decreasing … Continue reading →

Posted in Combinatorics, MATHCOUNTS | Comments Off on MATHCOUNTS Exercises – 18

MATHCOUNTS Exercises – 17

Posted on January 28, 2020 by kevin

In rectangle $ABCD$, shown here, point $M$ is the midpoint of side $BC$, and point $N$ lies on $CD$ such that $DN:NC=1:4$. Segment $BN$ intersects $AM$ and $AC$ at points $R$ and $S$, respectively. If $NS:SR:RB=x:y:z$, where $x$, $y$ and … Continue reading →

Posted in Geometry, MATHCOUNTS | Comments Off on MATHCOUNTS Exercises – 17

MATHCOUNTS Exercises – 16

Posted on January 26, 2020 by kevin

A semicircle and a circle are placed inside a square with sides of length 4 cm as shown. The circle is tangent to two adjacent sides of the square and to the semicircle. The diameter of the semicircle is a … Continue reading →

Posted in Circles in a Square, Geometry, MATHCOUNTS | Comments Off on MATHCOUNTS Exercises – 16

MATHCOUNTS Exercises – 15

Posted on January 22, 2020 by kevin

How many distinct unit cubes are there with two faces painted red, two faces painted green and two faces painted blue? Two unit cubes are considered distinct if one unit cube cannot be obtained by rotating the other. SOURCE: 2014 … Continue reading →

Posted in Combinatorics, Geometry, MATHCOUNTS | Comments Off on MATHCOUNTS Exercises – 15

MATHCOUNTS Exercises – 14

Posted on January 19, 2020 by kevin

Lisa wants to use her calculator to square a two-digit positive integer, but she accidentally enters the tens digit incorrectly. When she squares the number entered, the result is 2340 greater than the result she would have gotten had she … Continue reading →

Posted in Algebra, MATHCOUNTS | Comments Off on MATHCOUNTS Exercises – 14