MathDB

P3

Part of 2024 Israel TST

Problems(5)

Number of cool polynomials is even

Source: 2024 Israel TST Test 1 P3

8/29/2023
Let nn be a positive integer and pp be a prime number of the form 8k+58k+5. A polynomial QQ of degree at most 20232023 and nonnegative integer coefficients less than or equal to nn will be called "cool" if p∣Q(2)⋅Q(3)⋅…⋅Q(p−2)−1.p\mid Q(2)\cdot Q(3) \cdot \ldots \cdot Q(p-2)-1. Prove that the number of cool polynomials is even.
algebrapolynomialnumber theorymodular arithmeticprime numbers
Replacing two numbers by average, at most cn moves

Source: 2024 Israel TST Test 2 P3

11/7/2023
Let 0<c<10<c<1 and nn a positive integer. Alice and Bob are playing a game. Bob writes nn integers on the board, not all equal. On a player's turn, they erase two numbers from the board and write their arithmetic mean instead. Alice starts and performs at most cncn moves. After her, Bob makes moves until there are only two numbers left on the board. Alice wins if these two numbers are different, and otherwise, Bob wins. For which values of cc does Alice win for all large enough nn?
Game Theorycombinatoricsaveragesasymptotics
Concentric circles in parallelogram

Source: 2024 Israel TST Test 6 P3

3/20/2024
Let ABCDABCD be a parallelogram. Let ω1\omega_1 be the circle passing through DD tangent to ABAB at AA. Let ω2\omega_2 be the circle passing through AA tangent to CDCD at DD. The tangents from BB to ω1\omega_1 touch it at AA and PP. The tangents from CC to ω2\omega_2 touch it at DD and QQ. Lines APAP and DQDQ intersect at XX. The perpendicular bisector of BCBC intersects ADAD at RR.
Show that the circumcircles of triangles △PQX\triangle PQX, △BCR\triangle BCR are concentric.
geometryparallelogramconcentric circlesTSTperpendicular bisectorcircumcircle
Continuous from positives to &gt;1

Source: 2024 Israel TST Test 3 P3

1/29/2024
Find all continuous functions f ⁣:R>0→R≥1f\colon \mathbb{R}_{>0}\to \mathbb{R}_{\geq 1} for which the following equation holds for all positive reals xx, yy: f(f(x)y)−f(f(y)x)=xy(f(x+1)−f(y+1))f\left(\frac{f(x)}{y}\right)-f\left(\frac{f(y)}{x}\right)=xy\left(f(x+1)-f(y+1)\right)
functional equationalgebracontinuous functionPositive realsfunction
Balanced set has a big triangle

Source: 2024 Israel TST Test 8 P3

5/10/2024
For a set SS of at least 33 points in the plane, let dmind_{\text{min}} denote the minimal distance between two different points in SS and dmaxd_{\text{max}} the maximal distance between two different points in SS.
For a real c>0c>0, a set SS will be called cc-balanced if dmaxdmin≤c∣S∣\frac{d_{\text{max}}}{d_{\text{min}}}\leq c|S| Prove that there exists a real c>0c>0 so that for every cc-balanced set of points SS, there exists a triangle with vertices in SS that contains at least ∣S∣\sqrt{|S|} elements of SS in its interior or on its boundary.
combinatorial geometrydistancesSetscombinatoricsgeometry