Olympiad problems

2022 HMNT, 10

Tags: number theory

Compute the number of distinct pairs of the form \[(\text{first three digits of }x,\text{ first three digits of }x^4)\] over all integers $x>10^{10}$. For example, one such pair is $(100,100)$ when $x=10^{10^{10}}$.

2008 National Olympiad First Round, 10

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How many pairs of positive integers $(x,y)$ are there such that $\sqrt{xy}-71\sqrt x + 30 = 0$? $ \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 18 \qquad\textbf{(C)}\ 72 \qquad\textbf{(D)}\ 2130 \qquad\textbf{(E)}\ \text{Infinitely many} $

2008 Singapore Senior Math Olympiad, 5

Tags: inequalities , algebra

Let $a,b,c \ge 0$. Prove that $$\frac{(1+a^2)(1+b^2)(1+c^2)}{(1+a)(1+b)(1+c)}\ge \frac12 (1+abc)$$

2016 Costa Rica - Final Round, G1

Tags: geometry , circumcircle

Let $\vartriangle ABC$ be isosceles with $AB = AC$. Let $\omega$ be its circumscribed circle and $O$ its circumcenter. Let $D$ be the second intersection of $CO$ with $\omega$. Take a point $E$ in $AB$ such that $DE \parallel AC$ and suppose that $AE: BE = 2: 1$. Show that $\vartriangle ABC$ is equilateral.

2024 Baltic Way, 5

Tags: algebra , inequalities

Find all positive real numbers $\lambda$ such that every sequence $a_1, a_2, \ldots$ of positive real numbers satisfying \[ a_{n+1}=\lambda\cdot\frac{a_1+a_2+\ldots+a_n}{n} \] for all $n\geq 2024^{2024}$ is bounded. [i]Remark:[/i] A sequence $a_1,a_2,\ldots$ of positive real numbers is \emph{bounded} if there exists a real number $M$ such that $a_i<M$ for all $i=1,2,\ldots$

2015 Belarus Team Selection Test, 4

Tags: polynomial , algebra

Find all pairs of polynomials $p(x),q(x)\in R[x]$ satisfying the equality $p(x^2)=p(x)q(1-x)+p(1-x)q(x)$ for all real $x$. I.Voronovich

2014 Balkan MO Shortlist, G7

Tags: geometry , incenter , orthocenter

Let $I$ be the incenter of $\triangle ABC$ and let $H_a$, $H_b$, and $H_c$ be the orthocenters of $\triangle BIC$ , $\triangle CIA$, and $\triangle AIB$, respectively. The lines $H_aH_b$ meets $AB$ at $X$ and the line $H_aH_c$ meets $AC$ at $Y$. If the midpoint $T$ of the median $AM$ of $\triangle ABC$ lies on $XY$, prove that the line $H_aT$ is perpendicular to $BC$

2024 Ecuador NMO (OMEC), 1

Tags: algebra , system of equations

Find all real solutions: $$\begin{cases}a^3=2024bc \\ b^3=2024cd \\ c^3=2024da \\ d^3=2024ab \end{cases}$$

2020 Novosibirsk Oral Olympiad in Geometry, 1

Tags: rectangle , semicircle , tangent circle , geometry

Two semicircles touch the side of the rectangle, each other and the segment drawn in it as in the figure. What part of the whole rectangle is filled? [img]https://cdn.artofproblemsolving.com/attachments/3/e/70ca8b80240a282553294a58cb3ed807d016be.png[/img]

2002 Estonia Team Selection Test, 5

Tags: trigonometry , inequalities , algebra

Let $0 < a < \frac{\pi}{2}$ and $x_1,x_2,...,x_n$ be real numbers such that $\sin x_1 + \sin x_2 +... + \sin x_n \ge n \cdot sin a $. Prove that $\sin (x_1 - a) + \sin (x_2 - a) + ... + \sin (x_n - a) \ge 0$ .

2019 Czech-Polish-Slovak Junior Match, 5

Tags: geometry , geometric transformation , rotation , combinatorial geometry , combinatorics

Let $A_1A_2 ...A_{360}$ be a regular $360$-gon with centre $S$. For each of the triangles $A_1A_{50}A_{68}$ and $A_1A_{50}A_{69}$ determine, whether its images under some $120$ rotations with centre $S$ can have (as triangles) all the $360$ points $A_1, A_2, ..., A_{360}$ as vertices.

2022 CCA Math Bonanza, L2.2

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A rectangle $ABCD$ has side lengths $AB=6 \text{ miles}$ and $BC=9\text{ miles}.$ A pigeon hovers at point $P$, which is 5 miles above some randomly chosen point inside $ABCD$. Given that the expected value of \[AP^2+CP^2-BP^2-DP^2\] can be expressed as $\tfrac{a}{b}$, what is $ab$? [i]2022 CCA Math Bonanza Lightning Round 2.2[/i]

2017 Math Prize for Girls Olympiad, 4

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A [i]lattice point[/i] is a point in the plane whose two coordinates are both integers. A [i]lattice line[/i] is a line in the plane that contains at least two lattice points. Is it possible to color every lattice point red or blue such that every lattice line contains exactly 2017 red lattice points? Prove that your answer is correct.

2002 Bosnia Herzegovina Team Selection Test, 1

Tags: algebra

Let $x,y,z$ be real numbers that satisfy \[x+y+z= 3 \ \ \text{ and } \ \ xy+yz+zx= a\]where $a$ is a real parameter. Find the value of $a$ for which the difference between the maximum and minimum possible values of $x$ equals $8$.

2017 India PRMO, 1

Tags: number theory , sum of digits , divide , divisible

How many positive integers less than $1000$ have the property that the sum of the digits of each such number is divisible by $7$ and the number itself is divisible by $3$?

2023 AMC 10, 14

Tags: integer , algebra

How many ordered pairs of integers $(m, n)$ satisfy the equation $m^2+mn+n^2=m^2n^2$? $\textbf{(A) }7\qquad\textbf{(B) }1\qquad\textbf{(C) }3\qquad\textbf{(D) }6\qquad\textbf{(E) }5$

2012 Online Math Open Problems, 5

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Two circles have radius 5 and 26. The smaller circle passes through center of the larger one. What is the difference between the lengths of the longest and shortest chords of the larger circle that are tangent to the smaller circle? [i]Ray Li.[/i]

1985 IMO, 2

Tags: number theory , greatest common divisor , relatively prime , coloring , combinatorics

Let $n$ and $k$ be relatively prime positive integers with $k<n$. Each number in the set $M=\{1,2,3,\ldots,n-1\}$ is colored either blue or white. For each $i$ in $M$, both $i$ and $n-i$ have the same color. For each $i\ne k$ in $M$ both $i$ and $|i-k|$ have the same color. Prove that all numbers in $M$ must have the same color.

2010 Contests, 1

Tags: digit , number theory , sum of squares , algebra

Nine positive integers $a_1,a_2,...,a_9$ have their last $2$-digit part equal to $11,12,13,14,15,16,17,18$ and $19$ respectively. Find the last $2$-digit part of the sum of their squares.

1983 Vietnam National Olympiad, 3

Tags: geometry

A triangle $ABC$ and a positive number $k$ are given. Find the locus of a point $M$ inside the triangle such that the projections of $M$ on the sides of $\Delta ABC$ form a triangle of area $k$.

LMT Team Rounds 2021+, A 24

Tags: combinatorics

A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five. Using the four words “Hi”, “hey”, “hello”, and “haiku”, How many haikus Can somebody make? (Repetition is allowed, Order does matter.) [i]Proposed by Jeff Lin[/i]

1965 IMO, 4

Tags: system of equations , algebra

Find all sets of four real numbers $x_1, x_2, x_3, x_4$ such that the sum of any one and the product of the other three is equal to 2.

1994 APMO, 3

Tags: relatively prime , prime number , number theory

Let $n$ be an integer of the form $a^2 + b^2$, where $a$ and $b$ are relatively prime integers and such that if $p$ is a prime, $p \leq \sqrt{n}$, then $p$ divides $ab$. Determine all such $n$.

2000 Vietnam National Olympiad, 3

Tags: algebra , polynomial , number theory

Consider the polynomial $ P(x) \equal{} x^3 \plus{} 153x^2 \minus{} 111x \plus{} 38$. (a) Prove that there are at least nine integers $ a$ in the interval $ [1, 3^{2000}]$ for which $ P(a)$ is divisible by $ 3^{2000}$. (b) Find the number of integers $ a$ in $ [1, 3^{2000}]$ with the property from (a).

2003 National Olympiad First Round, 28

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Let $a$, $x$, $y$, $z$ be real numbers such that $ax-y+z=3a-1$ ve $x-ay+z=a^2-1$, which of the followings cannot be equal to $x^2+y^2+z^2$? $ \textbf{(A)}\ \sqrt 2 \qquad\textbf{(B)}\ \sqrt 3 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ \sqrt[3]{4} \qquad\textbf{(E)}\ \text{None of the preceding} $