Practice library
Problems
1At most how many interior angles of a polygon with \(n\) vertices can be greater than \(180^\circ\)? (The polygon is simple: its sides meet only at the shared endpoints of neighbouring sides.) A \(n - 1\) …Open2Janez drew a pattern on a sheet of paper, made up of congruent squares and congruent hexagons. On top of the pattern he then drew two dashed lines perpendicular to each other, as in the figure. What is …3A regular octagon is inscribed in a square of side length \(a\) so that four sides of the octagon lie on the four sides of the square. Express the side length of the octagon in terms of \(a\).4Find all integers \(x\) and \(y\) that satisfy \[ 3xy + 2x + y = 12 . \]5Positive real numbers \(a\) and \(b\) have product \(1\), and the sum of their squares equals \(4\). Determine the exact value of \[ a^{-3} + b^{-3} . \]6Find every natural number \(n\) with the following property: there is a rectangle whose side lengths are natural numbers, whose perimeter equals \(n\), and whose area is numerically equal to \(n\) as well. …7Ana and Beno cycle clockwise along the rectangular track shown in the figure, on which the length of every section is marked. Ana rides the shorter of the two loops, Beno the longer one, and their speeds …8Jure wrote all the natural numbers from \(1\) to \(2015\) on a board. Urska then went through the numbers on the board from the smallest to the largest and rubbed out every one that was not divisible by …9In a triangle \(ABC\), the bisector of the angle \(\angle BAC\) meets the side \(BC\) at the point \(D\). The triangle \(ADC\) is isosceles with apex \(D\), that is, \(|DA| = |DC|\). Given \(|CD| = 36\) …10Find all natural numbers \(n\) and all primes \(p\) for which \[ \sqrt{\,n + \frac{p}{n}\,} \] is a natural number.11Three straight cuts divide a rectangle into four pieces, as shown in the figure; the cut meeting the top edge is perpendicular to it. The four pieces are then rearranged, without gaps or overlaps, into …12In a triangle \(ABC\), the bisector of the angle \(\angle BAC\) meets the side \(BC\) at \(D\), and the bisector of the angle \(\angle CBA\) meets the side \(AC\) at \(E\). Suppose that \(|CD| = |CE|\). …13Peter keeps \(111\) red and \(111\) blue marbles at home; they are made by his uncle. Every day Peter may visit his uncle and carry out one exchange: either he hands over \(11\) red marbles and receives …14Let \(a\) and \(b\) be real numbers satisfying \[ \frac{a^2}{1+a^2} + \frac{b^2}{1+b^2} = 1 . \] Determine all possible values of the expression \[ (a+b)\left( \frac{a}{1+a^2} + \frac{b}{1+b^2} \right) . \] …15Lara and Sara draw \(n\) straight lines on a rectangular sheet of paper, taking turns and drawing one line each time. Every line is parallel to one of the edges of the sheet and runs from edge to edge, …16Benjamin was working out the sum \(1 + 2 + 3 + \dots + 2012\). He left out a few of the terms, and the wrong total he ended up with was divisible by \(2011\). Anika was working out the sum \[ A = 1 + 2 + 3 + \dots + 2013 , \] …17Let \(ABC\) be a right triangle with its right angle at \(C\), and write \(|BC| = a\), \(|AC| = b\). Let \(D\) be a point on the opposite side of the line \(AC\) from \(B\) for which the triangle \(ACD\) …
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