MathDB

1

Part of 2022 Israel TST

Problems(3)

Locus of incenter, variable point on BC

Source: 2022 Israel TST 3 P1

5/22/2022
A triangle ABCABC with orthocenter HH is given. PP is a variable point on line BCBC. The perpendicular to BCBC through PP meets BHBH, CHCH at XX, YY respectively. The line through HH parallel to BCBC meets APAP at QQ. Lines QXQX and QYQY meet BCBC at UU, VV respectively. Find the shape of the locus of the incenters of the triangles QUVQUV.
geometryincenter
Solution of system with a parameter

Source: 2022 Israel TST 8 P1

5/21/2022
Let n>1n>1 be an integer. Find all r∈Rr\in \mathbb{R} so that the system of equations in real variables x1,x2,…,xnx_1, x_2, \dots, x_n: \begin{align*} &(r\cdot x_1-x_2)(r\cdot x_1-x_3)\dots (r\cdot x_1-x_n)=\\ =&(r\cdot x_2-x_1)(r\cdot x_2-x_3)\dots (r\cdot x_2-x_n)=\\ &\qquad \qquad \qquad \qquad \vdots \\ =&(r\cdot x_n-x_1)(r\cdot x_n-x_2)\dots (r\cdot x_n-x_{n-1}) \end{align*} has a solution where the numbers x1,x2,…,xnx_1, x_2, \dots, x_n are pairwise distinct.
algebraIMO 2022 tstsystem of equationsparameterization
Bilbo, Gandalf, and Nitzan

Source: 2022 Israel TST test 10 P1

7/18/2022
Bilbo, Gandalf, and Nitzan play the following game. First, Nitzan picks a whole number between 11 and 220222^{2022} inclusive and reveals it to Bilbo. Bilbo now compiles a string of length 40444044 built from the three letters a,b,ca,b,c. Nitzan looks at the string, chooses one of the three letters a,b,ca,b,c, and removes from the string all instances of the chosen letter. Only then is the string revealed to Gandalf. He must now guess the number Nitzan chose.
Can Bilbo and Gandalf work together and come up with a strategy beforehand that will always allow Gandalf to guess Nitzan's number correctly, no matter how he acts?
combinatoricsTSTIsrael