Olympiad problems

STEMS 2023 Math Cat A, 5

Consider a polynomial $P(x) \in \mathbb{R}[x]$, with degree $2023$, such that $P(\sin^2(x))+P(\cos^2(x)) =1$ for all $x \in \mathbb{R}$. If the sum of all roots of $P$ is equal to $\dfrac{p}{q}$ with $p, q$ coprime, then what is the product $pq$?

2003 Greece JBMO TST, 1

If point $M(x,y)$ lies on the line with equation $y=x+2$ and $1<y<3$, calculate the value of $A=\sqrt{y^2-8x}+\sqrt{y^2+2x+5}$

computational algebra

STEMS 2023 Math Cat A, 4

Let $f : \mathbb{N} \to \mathbb{N}$ be a function such that the following conditions hold: $\qquad\ (1) \; f(1) = 1.$ $\qquad\ (2) \; \dfrac{(x + y)}{2} < f(x + y) \le f(x) + f(y) \; \forall \; x, y \in \mathbb{N}.$ $\qquad\ (3) \; f(4n + 1) < 2f(2n + 1) \; \forall \; n \ge 0.$ $\qquad\ (4) \; f(4n + 3) \le 2f(2n + 1) \; \forall \; n \ge 0.$ Find the sum of all possible values of $f(2023)$.

function computational algebra functional equations mathematical induction

2018 May Olympiad, 3

Let $ABCDEFGHIJ$ be a regular $10$-sided polygon that has all its vertices in one circle with center $O$ and radius $5$. The diagonals $AD$ and $BE$ intersect at $P$ and the diagonals $AH$ and $BI$ intersect at $Q$. Calculate the measure of the segment $PQ$.

computational geometry polygon polygons

STEMS 2023 Math Cat A, 7

Suppose a biased coin gives head with probability $\dfrac{2}{3}$. The coin is tossed repeatedly, if it shows heads then player $A$ rolls a fair die, otherwise player $B$ rolls the same die. The process ends when one of the players get a $6$, and that player is declared the winner. If the probability that $A$ will win is given by $\dfrac{m}{n}$ where $m,n$ are coprime, then what is the value of $m^2n$?

computational coin toss probability algebra number theory

2018 Iranian Geometry Olympiad, 1

There are three rectangles in the following figure. The lengths of some segments are shown. Find the length of the segment $XY$ . [img]https://2.bp.blogspot.com/-x7GQfMFHzAQ/W6K57utTEkI/AAAAAAAAJFQ/1-5WhhuerMEJwDnWB09sTemNLdJX7_OOQCK4BGAYYCw/s320/igo%2B2018%2Bintermediate%2Bp1.png[/img] Proposed by Hirad Aalipanah

computational rectangle geometry

May Olympiad L1 - geometry, 2018.3

Let $ABCDEFGHIJ$ be a regular $10$-sided polygon that has all its vertices in one circle with center $O$ and radius $5$. The diagonals $AD$ and $BE$ intersect at $P$ and the diagonals $AH$ and $BI$ intersect at $Q$. Calculate the measure of the segment $PQ$.

computational geometry polygon polygons

STEMS 2023 Math Cat A, 8

For how many pairs of primes $(p, q)$, is $p^2 + 2pq^2 + 1$ also a prime?

computational algebra prime number modular arithmetic prime numbers

2018 Yasinsky Geometry Olympiad, 5

In the trapezium $ABCD$ ($AD // BC$), the point $M$ lies on the side of $CD$, with $CM:MD=2:3$, $AB=AD$, $BC:AD=1:3$. Prove that $BD \perp AM$.

computational geometry trapezoid

STEMS 2023 Math Cat A, 6

There are $5$ vertices labelled $1,2,3,4,5$. For any two pairs of vertices $u, v$, the edge $uv$ is drawn with probability $1/2$. If the probability that the resulting graph is a tree is given by $\dfrac{p}{q}$ where $p, q$ are coprime, then find the value of $q^{1/10} + p$.

computational probability graph theory algebra combinatorics number theory

2017 Yasinsky Geometry Olympiad, 6

Given a trapezoid $ABCD$ with bases $BC$ and $AD$, with $AD=2 BC$. Let $M$ be the midpoint of $AD, E$ be the intersection point of the sides $AB$ and $CD$, $O$ be the intersection point of $BM$ and $AC, N$ be the intersection point of $EO$ and $BC$. In what ratio, point $N$ divides the segment $BC$?

ratio computational geometry trapezoid