Olympiad problems

2024 Vietnam National Olympiad, 5

Tags: algebra , polynomial

For each polynomial $P(x)$, define $$P_1(x)=P(x), \forall x \in \mathbb{R},$$ $$P_2(x)=P(P_1(x)), \forall x \in \mathbb{R},$$ $$...$$ $$P_{2024}(x)=P(P_{2023}(x)), \forall x \in \mathbb{R}.$$ Let $a>2$ be a real number. Is there a polynomial $P$ with real coefficients such that for all $t \in (-a, a)$, the equation $P_{2024}(x)=t$ has $2^{2024}$ distinct real roots?

2013 Gheorghe Vranceanu, 1

Tags: modular arithmetic , number theory

Find all natural numbers $ a,b $ such that $ a^3+b^3 $ a power of $3.$

2021 Bangladeshi National Mathematical Olympiad, 2

Tags: number theory , greatest common divisor , least common multiple

Let $x$ and $y$ be positive integers such that $2(x+y)=gcd(x,y)+lcm(x,y)$. Find $\frac{lcm(x,y)}{gcd(x,y)}$.

1958 AMC 12/AHSME, 22

Tags: parabola , analytic geometry , conic

A particle is placed on the parabola $ y \equal{} x^2 \minus{} x \minus{} 6$ at a point $ P$ whose $ y$-coordinate is $ 6$. It is allowed to roll along the parabola until it reaches the nearest point $ Q$ whose $ y$-coordinate is $ \minus{}6$. The horizontal distance traveled by the particle (the numerical value of the difference in the $ x$-coordinates of $ P$ and $ Q$) is: $ \textbf{(A)}\ 5\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 3\qquad \textbf{(D)}\ 2\qquad \textbf{(E)}\ 1$

2013 May Olympiad, 5

Tags: combinatorics

An $8\times 8$ square is drawn on the board divided into $64$ $1\times1$ squares by lines parallel to the sides. Gustavo erases some segments of length $ 1$ so that every $1\times 1$ square he erases $0, 1$ or $2$ sides. Gustavo states that he erased $6$ segments of length $1$ from the edge of the $8\times 8$ square and that the amount of $1\times 1$ squares that have exactly $ 1$ side erased is equal to $5$. Decide if what Gustavo said it may be true.

2024 Saint Petersburg Mathematical Olympiad, 5

Tags: number theory , combinatorics

$2 \ 000 \ 000$ points with integer coordinates are marked on the numeric axis. Segments of lengths $97$, $100$ and $103$ with ends at these points are considered. What is the largest number of such segments?

2009 China Team Selection Test, 1

Tags: inequalities , number theory

Let $ a > b > 1, b$ is an odd number, let $ n$ be a positive integer. If $ b^n|a^n\minus{}1,$ then $ a^b > \frac {3^n}{n}.$

2011 National Olympiad First Round, 18

Tags:

How many positive integer divides the expression $n(n^2-1)(n^2+3)(n^2+5)$ for every possible value of positive integer $n$? $\textbf{(A)}\ 16 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{None}$

2021 CCA Math Bonanza, T8

Tags:

Let $ABC$ be a triangle with $AB = 9$ and $AC=12$. Point $B'$ is chosen on line $AC$ such that the midpoint of $B$ and $B'$ is equidistant from $A$ and $C$. Point $C'$ is chosen similarly. Given that the circumcircle of $AB'C'$ is tangent to $BC$, compute $BC^2$. [i]2021 CCA Math Bonanza Team Round #8[/i]

2018 JBMO Shortlist, A5

Tags: algebra , inequalities

Let a$,b,c,d$ and $x,y,z,t$ be real numbers such that $0\le a,b,c,d \le 1$ , $x,y,z,t \ge 1$ and $a+b+c+d +x+y+z+t=8$. Prove that $a^2+b^2+c^2+d^2+x^2+y^2+z^2+t^2\le 28$

2024 Harvard-MIT Mathematics Tournament, 3

Tags: number theory

Compute the sum of all two-digit positive integers $x$ such that for all three-digit (base $10$) positive integers $\underline{a}\, \underline{b} \, \underline{c}$, if $\underline{a} \, \underline{b} \, \underline{c}$ is a multiple of $x$, then the three-digit (base $10$) number $\underline{b} \, \underline{c} \, \underline{a}$ is also a multiple of $x$.

2012 Brazil Team Selection Test, 3

Tags: geometry , circumcircle , symmetry , homothety , radical axis , marvio

Let $ABC$ be an acute triangle with circumcircle $\Omega$. Let $B_0$ be the midpoint of $AC$ and let $C_0$ be the midpoint of $AB$. Let $D$ be the foot of the altitude from $A$ and let $G$ be the centroid of the triangle $ABC$. Let $\omega$ be a circle through $B_0$ and $C_0$ that is tangent to the circle $\Omega$ at a point $X\not= A$. Prove that the points $D,G$ and $X$ are collinear. [i]Proposed by Ismail Isaev and Mikhail Isaev, Russia[/i]

2002 Federal Competition For Advanced Students, Part 1, 4

Tags: parallelogram , geometry

Let $A,C, P$ be three distinct points in the plane. Construct all parallelograms $ABCD$ such that point $P$ lies on the bisector of angle $DAB$ and $\angle APD = 90^\circ$.

2024 IMC, 8

Tags: recurrence relation , limit , sequence

Define the sequence $x_1,x_2,\dots$ by the initial terms $x_1=2, x_2=4$, and the recurrence relation \[x_{n+2}=3x_{n+1}-2x_n+\frac{2^n}{x_n} \quad \text{for} \quad n \ge 1.\] Prove that $\lim_{n \to \infty} \frac{x_n}{2^n}$ exists and satisfies \[\frac{1+\sqrt{3}}{2} \le \lim_{n \to \infty} \frac{x_n}{2^n} \le \frac{3}{2}.\]

2015 Ukraine Team Selection Test, 2

Tags: combinatorics

In a football tournament, $n$ teams play one round ($n \vdots 2$). In each round should play $n / 2$ pairs of teams that have not yet played. Schedule of each round takes place before its holding. For which smallest natural $k$ such that the following situation is possible: after $k$ tours, making a schedule of $k + 1$ rounds already is not possible, i.e. these $n$ teams cannot be divided into $n / 2$ pairs, in each of which there are teams that have not played in the previous $k$ rounds. PS. The 3 vertical dots notation in the first row, I do not know what it means.

2021 Abels Math Contest (Norwegian MO) Final, 4b

Tags: geometry , equal segments

The tangent at $C$ to the circumcircle of triangle $ABC$ intersects the line through $A$ and $B$ in a point $D$. Two distinct points $E$ and $F$ on the line through $B$ and $C$ satisfy $|BE| = |BF | =\frac{||CD|^2 - |BD|^2|}{|BC|}$. Show that either $|ED| = |CD|$ or $|FD| = |CD|$.

2020 USMCA, 7

Tags:

Compute the value of \[\cos \frac{2\pi}{7} + 2\cos \frac{4\pi}{7} + 3\cos \frac{6\pi}{7} + 4\cos \frac{8\pi}{7} + 5\cos \frac{10\pi}{7} + 6\cos \frac{12\pi}{7}.\]

1947 Putnam, A5

Tags: limit , sum , sequence

Let $a_1 , b_1 , c_1$ be positive real numbers whose sum is $1,$ and for $n=1, 2, \ldots$ we define $$a_{n+1}= a_{n}^{2} +2 b_n c_n, \;\;\;b_{n+1}= b_{n}^{2} +2 a_n c_n, \;\;\; c_{n+1}= c_{n}^{2} +2 a_n b_n.$$ Show that $a_n , b_n ,c_n$ approach limits as $n\to \infty$ and find those limits.

2010 AMC 12/AHSME, 9

Tags: geometry , 3d geometry

A solid cube has side length $ 3$ inches. A $ 2$-inch by $ 2$-inch square hole is cut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid? $ \textbf{(A)}\ 7\qquad \textbf{(B)}\ 8\qquad \textbf{(C)}\ 10\qquad \textbf{(D)}\ 12\qquad \textbf{(E)}\ 15$

2011 Sharygin Geometry Olympiad, 8

Tags: diagonal , geometry , convex polygon , circumcircle

A convex $n$-gon $P$, where $n > 3$, is dissected into equal triangles by diagonals non-intersecting inside it. Which values of $n$ are possible, if $P$ is circumscribed?

KoMaL A Problems 2018/2019, A. 742

Tags: geometry

Convex quadrilateral $ABCD$ is inscribed in circle $\Omega$. Its sides $AD$ and $BC$ intersect at point $E$. Let $M$ and $N$ be the midpoints of the circle arcs $AB$ and $CD$ not containing the other vertices, and let $I$, $J$, $K$, $L$ denote the incenters of triangles $ABD$, $ABC$, $BCD$, $CDA$, respectively. Suppose $\Omega$ intersects circles $IJM$ and $KLN$ for the second time at points $U \neq M$ and $V \neq N$. Show that the points $E$, $U$, and $V$ are collinear.

2023 Euler Olympiad, Round 2, 2

Tags: euler , combinatorics , construction

Let $n$ be a positive integer. The Georgian folk dance team consists of $2n$ dancers, with $n$ males and $n$ females. Each dancer, both male and female, is assigned a number from 1 to $n$. During one of their dances, all the dancers line up in a single line. Their wish is that, for every integer $k$ from 1 to $n$, there are exactly $k$ dancers positioned between the $k$th numbered male and the $k$th numbered female. Prove the following statements: a) If $n \equiv 1 \text{ or } 2 \mod{4}$, then the dancers cannot fulfill their wish. b) If $n \equiv 0 \text{ or } 3 \mod{4}$, then the dancers can fulfill their wish. [i]Proposed by Giorgi Arabidze, Georgia[/i]

2011 Today's Calculation Of Integral, 734

Tags: calculus , integration , calculus computations , logarithm

Find the extremum of $f(t)=\int_1^t \frac{\ln x}{x+t}dx\ (t>0)$.

1968 Poland - Second Round, 3

Tags: combinatorics

Show that if at least five persons are sitting at a round table, then it is possible to rearrange them so that everyone has two new neighbors.

2015 Tuymaada Olympiad, 8

Tags: combinatorics , geometry , parallelogram

There are $\frac{k(k+1)}{2}+1$ points on the planes, some are connected by disjoint segments ( also point can not lies on segment, that connects two other points). It is true, that plane is divided to some parallelograms and one infinite region. What maximum number of segments can be drawn ? [i] A.Kupavski, A. Polyanski[/i]