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2022 Puerto Rico Team Selection Test

Part of Puerto Rico Team Selection Test

Subcontests

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3 colors for 6-point star - Puerto Rico TST 2022.1.4

The six-pointed star in the figure is regular: all interior angles of the small triangles are equal. To each of the thirteen points marked are assigned a color: green or red. Prove that there will always be three points of the same color that are vertices of an equilateral triangle. https://cdn.artofproblemsolving.com/attachments/b/f/c50a1f8cb81ea861f16a6a47c3b758c5993213.png

gluing sequence of hexagons

Let's construct a family {Kn}\{K_n\} of geometric figures following the pattern shown in pictures: https://cdn.artofproblemsolving.com/attachments/4/1/76d6cf2b7ec3bd69de7bf33e2a382885f744a0.png where each hexagon (like the starting one) is constructed by cutting the two corners tops of a square, in such a way that the two figures removed are identical isosceles triangles, and the three resulting upper sides have the same length. Continuing like this, a pattern is produced with which we can build the figures KnK_n, for integer n≥0n \ge 0 . Then, we denote by PnP_n and AnA_n the perimeter and area of the figure KnK_n, respectively. If the side of square to build K0K_0 measures xx:
(a) Calculate P0P_0 and A0A_0 (in terms of the length xx).
(b) Find an explicit formula for PnP_n, and for AnA_n, in terms of xx and of nn. Simplify your answers.
(c) If P2022=A2022P_{2022} = A_{2022}, find the measure of the six sides of the figure K0K_0, in its simplest form.
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2 right isosceles triangles - Puerto Rico TST 2022.1.3

In the figure, ABCABC and CDECDE are right-angled and isosceles triangles. Segments ADAD and BCBC intersect at PP, and segments CDCD and BEBE intersect at QQ. (a) Show that segmentPQ PQ is parallel to segment AEAE. (b) If BP=4BP = 4 and DQ=9DQ = 9, find the measure of segment BDBD. https://cdn.artofproblemsolving.com/attachments/d/3/4c2c7514d71bbac68d58fc6de9ec2649e58957.png

isosceles and ratio wanted, Puerto Rico TST 2022.2.3

Let ω\omega be a circle with center OO and diameter ABAB. A circle with center at BB intersects ω\omega at C and ABAB at DD. The line CDCD intersects ω\omega at a point EE (E≠CE\ne C). The intersection of lines OEOE and BCBC is FF. (a) Prove that triangle OBFOBF is isosceles. (b) If DD is the midpoint of OBOB, find the value of the ratio FBBD\frac{FB}{BD}.
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a^n+1/a^n is integer when a+1/a is integer - Puerto Rico TST 2022.1.2

Suppose aa is a non-zero real number such that a+1aa +\frac{1}{a} is a whole number. (a) Prove that a2+1a2a^2 +\frac{1}{a^2} is also an integer. (b) Prove that an+1ana^n+\frac{1}{a^n} is also an integer, for any integer value positive of nn.

max sum by 9 bars of border of an equilateral 3x3 grid

There are1 1 cm long bars with a number1 1, 22 or 33 written on each one from them. There is an unlimited supply of bars with each number. Two triangles formed by three bars are considered different if none of them can be built with the bars of the other triangle. (a) How many different triangles formed by three bars are possible? (b) An equilateral triangle of side length 33 cm is formed using 1818 bars, , divided into 99 equilateral triangles, different by pairs, 11 cm long on each side. Find the largest sum possible from the numbers written on the 99 bars of the border of the big triangle. https://cdn.artofproblemsolving.com/attachments/1/1/c2f7edeea3c70ba4689d7b12fe8ac8be72f115.png
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