Olympiad problems

2024 India IMOTC, 17

Fix a positive integer $a > 1$. Consider triples $(f(x), g(x), h(x))$ of polynomials with integer coefficients, such that 1. $f$ is a monic polynomial with $\deg f \ge 1$. 2. There exists a positive integer $N$ such that $g(x)>0$ for $x \ge N$ and for all positive integers $n \ge N$, we have $f(n) \mid a^{g(n)} + h(n)$. Find all such possible triples. [i]Proposed by Mainak Ghosh and Rijul Saini[/i]

2018 Nepal National Olympiad, 3a

Tags: geometry

[b]Problem Section #3 a) Circles $O_1$ and $O_2$ interest at two points $B$ and $C$, and $BC$ is the diameter of circle $O_1$. Construct a tangent line of circle $O_1$ at $C$ and intersecting circle $O_2$ at another point $A$. Join $AB$ to intersect circle $O_1$ at point $E$, then join $CE$ and extend it to intersect circle $O_2$ at point $F$. Assume $H$ is an arbitrary point on line segment $AF$. Join $HE$ and extend it to intersect circle $O_1$ at point $G$, and then join $BG$ and extend it to intersect the extend of $AC$ at point $D$. Prove: $\frac{AH}{HF}=\frac{AC}{CD}$.

2022 BMT, 6

Tags: geometry

Triangle $\vartriangle BMT$ has $BM = 4$, $BT = 6$, and $MT = 8$. Point $A$ lies on line $\overleftrightarrow{BM}$ and point $Y$ lies on line $\overleftrightarrow{BT}$ such that $\overline{AY}$ is parallel to $\overline{MT}$ and the center of the circle inscribed in triangle $\vartriangle BAY$ lies on $\overline{MT}$. Compute $AY$ .

2018 Canadian Senior Mathematics Contest, A3

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A jar contains 6 crayons, of which 3 are red, 2 are blue, and 1 is green. Jakob reaches into the jar and randomly removes 2 of the crayons. What is the probability that both of these crayons are red?

1983 IMO Longlists, 55

Tags: number theory

For every $a \in \mathbb N$ denote by $M(a)$ the number of elements of the set \[ \{ b \in \mathbb N | a + b \text{ is a divisor of } ab \}.\] Find $\max_{a\leq 1983} M(a).$

2017 Bundeswettbewerb Mathematik, 2

Tags: coloring , isosceles , polygon , combinatorics

In a convex regular $35$-gon $15$ vertices are colored in red. Are there always three red vertices that make an isosceles triangle?

1972 Canada National Olympiad, 8

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During a certain election campaign, $p$ different kinds of promises are made by the different political parties ($p>0$). While several political parties may make the same promise, any two parties have at least one promise in common; no two parties have exactly the same set of promises. Prove that there are no more than $2^{p-1}$ parties.

1993 All-Russian Olympiad, 1

Tags: number theory

For a positive integer $n$, numbers $2n+1$ and $3n+1$ are both perfect squares. Is it possible for $5n+3$ to be prime?

2017 Thailand TSTST, 3

Tags: function , functional equation , algebra

Let $f$ be a function on a set $X$. Prove that $$f(X-f(X))=f(X)-f(f(X)),$$ where for a set $S$, the notation $f(S)$ means $\{f(a) | a \in S\}$.

2012 Today's Calculation Of Integral, 829

Tags: calculus , integration , calculus computations , logarithm

Let $a$ be a positive constant. Find the value of $\ln a$ such that \[\frac{\int_1^e \ln (ax)\ dx}{\int_1^e x\ dx}=\int_1^e \frac{\ln (ax)}{x}\ dx.\]

2017 Azerbaijan BMO TST, 3

Tags: algebra , functional equation

Find all funtions $f:\mathbb R\to\mathbb R$ such that: $$f(xy-1)+f(x)f(y)=2xy-1$$ for all $x,y\in \mathbb{R}$.

2003 JHMMC 8, 4

Tags: easy , very very easy , very easy , addition , elementary contest , solve equation

A number plus $4$ is $2003$. What is the number?

2018 Pan-African Shortlist, C1

Tags: combinatorics , ratio

A chess tournament is held with the participation of boys and girls. The girls are twice as many as boys. Each player plays against each other player exactly once. By the end of the tournament, there were no draws and the ratio of girl winnings to boy winnings was $\frac{7}{9}$. How many players took part at the tournament?

Russian TST 2019, P2

Tags: algebra , maximum value , sequence

Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.

2004 IMO Shortlist, 7

Tags: geometry , inradius , incenter , triangle

For a given triangle $ ABC$, let $ X$ be a variable point on the line $ BC$ such that $ C$ lies between $ B$ and $ X$ and the incircles of the triangles $ ABX$ and $ ACX$ intersect at two distinct points $ P$ and $ Q.$ Prove that the line $ PQ$ passes through a point independent of $ X$.

2011 Brazil Team Selection Test, 3

Tags: algebra , polynomial , inequalities

Let the real numbers $a,b,c,d$ satisfy the relations $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12.$ Prove that \[36 \leq 4 \left(a^3+b^3+c^3+d^3\right) - \left(a^4+b^4+c^4+d^4 \right) \leq 48.\] [i]Proposed by Nazar Serdyuk, Ukraine[/i]

2024 Vietnam National Olympiad, 6

Tags: number theory

For each positive integer $n$, let $\tau (n)$ be the number of positive divisors of $n$. a) Find all positive integers $n$ such that $\tau(n)+2023=n$. b) Prove that there exist infinitely many positive integers $k$ such that there are exactly two positive integers $n$ satisfying $\tau(kn)+2023=n$.

2008 Moldova National Olympiad, 12.5

Tags: polynomial , algebra

Find the least positive integer $ n$ so that the polynomial $ P(X)\equal{}\sqrt3\cdot X^{n\plus{}1}\minus{}X^n\minus{}1$ has at least one root of modulus $ 1$.

2024 Polish MO Finals, 3

Tags: prime number , system of equations , diophantine equation , number theory

Determine all pairs $(p,q)$ of prime numbers with the following property: There are positive integers $a,b,c$ satisfying \[\frac{p}{a}+\frac{p}{b}+\frac{p}{c}=1 \quad \text{and} \quad \frac{a}{p}+\frac{b}{p}+\frac{c}{p}=q+1.\]

2021 Estonia Team Selection Test, 2

Tags: algebra , inequalities

Positive real numbers $a, b, c$ satisfy $abc = 1$. Prove that $$\frac{a}{1+b}+\frac{b}{1+c}+\frac{c}{1+a} \ge \frac32$$

2018 BMT Spring, 3

Tags:

$2018$ people (call them $A, B, C, \ldots$) stand in a line with each permutation equally likely. Given that $A$ stands before $B$, what is the probability that $C$ stands after $B$?

2022 Thailand Online MO, 6

Tags: combinatorics

Let $n$ and $k$ be positive integers. Chef Kao cuts a circular pizza through $k$ diameters, dividing the pizza into $2k$ equal pieces. Then, he dresses the pizza with $n$ toppings. For each topping, he chooses $k$ consecutive pieces of pizza and puts that topping on all of the chosen pieces. Then, for each piece of pizza, Chef Kao counts the number of distinct toppings on it, yielding $2k$ numbers. Among these numbers, let $m$ and $M$ being the minimum and maximum, respectively. Prove that $m + M = n$.

MOAA Team Rounds, 2019.2

Tags: geometry , team , algebra

The lengths of the two legs of a right triangle are the two distinct roots of the quadratic $x^2 - 36x + 70$. What is the length of the triangle’s hypotenuse?

2024 Harvard-MIT Mathematics Tournament, 16

Tags: guts

Let $ABC$ be an isosceles triangle with orthocenter $H.$ Let $M$ and $N$ be the midpoints of sides $\overline{AB}$ and $\overline{AC},$ respectively. The circumcircle of triangle $MHN$ intersects line $BC$ at two points $X$ and $Y.$ Given $XY=AB=AC=2,$ compute $BC^2.$

1982 IMO Longlists, 23

Tags: number theory

Determine the sum of all positive integers whose digits (in base ten) form either a strictly increasing or a strictly decreasing sequence.