Olympiad problems

2000 Saint Petersburg Mathematical Olympiad, 10.4

The number $N$ is the product of $200$ distinct positive integers. Prove that it has at least 19901 distinct divisors (including 0 and itself). [I]Proposed by A. Golovanov[/i]

2008 239 Open Mathematical Olympiad, 3

Tags: graph theory , combinatorics

A connected graph has $100$ vertices, the degrees of all the vertices do not exceed $4$ and no two vertices of degree $4$ are adjacent. Prove that it is possible to remove several edges that have no common vertices from this graph such that there would be no triangles in the resulting graph.

1989 Federal Competition For Advanced Students, P2, 1

Tags: induction , polynomial , algebra

Consider the set $ S_n$ of all the $ 2^n$ numbers of the type $ 2\pm \sqrt{2 \pm \sqrt {2 \pm ...}},$ where number $ 2$ appears $ n\plus{}1$ times. $ (a)$ Show that all members of $ S_n$ are real. $ (b)$ Find the product $ P_n$ of the elements of $ S_n$.

1985 All Soviet Union Mathematical Olympiad, 395

Tags: geometry , hexagon , area

Two perpendiculars are drawn from the midpoints of each side of the acute-angle triangle to two other sides. Those six segments make hexagon. Prove that the hexagon area is a half of the triangle area.

1977 IMO Longlists, 8

Tags: geometry

A hexahedron $ABCDE$ is made of two regular congruent tetrahedra $ABCD$ and $ABCE.$ Prove that there exists only one isometry $\mathbf Z$ that maps points $A, B, C, D, E$ onto $B, C, A, E, D,$ respectively. Find all points $X$ on the surface of hexahedron whose distance from $\mathbf Z(X)$ is minimal.

2006 MOP Homework, 3

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Let $X=\{A_{1},...,A_{n}\}$ be a set of distinct 3-element subsets of the set $\{1,2,...,36\}$ such that (a) $A_{i},A_{j}$ have nonempty intersections for all $i,j$ (b) The intersection of all elements of $X$ is the empty set. Show that $n\leq 100$. Determine the number of such sets $X$ when $n=100$

2020 LMT Fall, A9

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$\triangle ABC$ has a right angle at $B$, $AB = 12$, and $BC = 16$. Let $M$ be the midpoint of $AC$. Let $\omega_1$ be the incircle of $\triangle ABM$ and $\omega_2$ be the incircle of $\triangle BCM$. The line externally tangent to $\omega_1$ and $\omega_2$ that is not $AC$ intersects $AB$ and $BC$ at $X$ and $Y$, respectively. If the area of $\triangle BXY$ can be expressed as $\frac{m}{n}$, compute is $m+n$. [i]Proposed by Alex Li[/i]

2023 Grosman Mathematical Olympiad, 1

Tags: number theory , divisibility , arithmetic sequence

An arithmetic progression of natural numbers of length $10$ and with difference $11$ is given. Prove that the product of the numbers in this progression is divisible by $10!$.

2013 Moldova Team Selection Test, 4

Tags: inequalities

Prove that for any positive real numbers $a_i,b_i,c_i$ with $i=1,2,3$, $(a_1^3+b_1^3+c_1^3+1)(a_2^3+b_2^3+c_2^3+1)(a_3^3+b_3^3+c_3^3+1)\geq \frac{3}{4} (a_1+b_1+c_1)(a_2+b_2+c_2)(a_3+b_3+c_3)$

2022 HMNT, 1

Tags: clock

Emily’s broken clock runs backwards at five times the speed of a regular clock. Right now, it is displaying the wrong time. How many times will it display the correct time in the next 24 hours? It is an analog clock (i.e. a clock with hands), so it only displays the numerical time, not AM or PM. Emily’s clock also does not tick, but rather updates continuously.

2013 Hanoi Open Mathematics Competitions, 13

Tags: algebra , system of equations

Solve the system of equations $\begin{cases} \frac{1}{x}+\frac{1}{y}=\frac{1}{6} \\ \frac{3}{x}+\frac{2}{y}=\frac{5}{6} \end{cases}$

2010 AMC 12/AHSME, 20

Tags: number theory , greatest common divisor , modular arithmetic

Arithmetic sequences $ (a_n)$ and $ (b_n)$ have integer terms with $ a_1 \equal{} b_1 \equal{} 1 < a_2 \le b_2$ and $ a_nb_n \equal{} 2010$ for some $ n$. What is the largest possible value of $ n$? $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 288 \qquad \textbf{(E)}\ 2009$

2012 Mathcenter Contest + Longlist, 6 sl14

Tags: algebra , inequalities , max

For a real number $a,b,c>0$ where $bc-ca-ab=1$ find the maximum value of $$P=\frac{4024}{1+a^2}-\frac{4024}{1+b^2}-\frac{2555}{1+c^2}$$ and find out when that holds . [i](PP-nine)[/i]

1966 Swedish Mathematical Competition, 4

Tags: composition , sequence , algebra

Let $f(x) = 1 + \frac{2}{x}$. Put $f_1(x) = f(x)$, $f_2(x) = f(f_1(x))$, $f_3(x) = f(f_2(x))$, $... $. Find the solutions to $x = f_n(x)$ for $n > 0$.

2013 Junior Balkan Team Selection Tests - Romania, 5

Tags: integer , number theory , set

a) Prove that for every positive integer n, there exist $a, b \in R - Z$ such that the set $A_n = \{a - b, a^2 - b^2, a^3 - b^3,...,a^n - b^n\}$ contains only positive integers. b) Let $a$ and $b$ be two real numbers such that the set $A = \{a^k - b^k | k \in N*\}$ contains only positive integers. Prove that $a$ and $b$ are integers.

2005 National Olympiad First Round, 18

Tags: modular arithmetic , quadratic

How many integers $0\leq x < 121$ are there such that $x^5+5x^2 + x + 1 \equiv 0 \pmod{121}$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5 $

2020 AMC 8 -, 13

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Jamal has a drawer containing $6$ green socks, $18$ purple socks, and $12$ orange socks. After adding more purple socks, Jamal noticed that there is now a $60\%$ chance that a sock randomly selected from the drawer is purple. How many purple socks did Jamal add? $\textbf{(A)}\ 6\qquad~~\textbf{(B)}\ 9\qquad~~\textbf{(C)}\ 12\qquad~~\textbf{(D)}\ 18\qquad~~\textbf{(E)}\ 24$

2008 IMO Shortlist, 1

Tags: geometry , circumcircle , trigonometry , radical axis

Let $ H$ be the orthocenter of an acute-angled triangle $ ABC$. The circle $ \Gamma_{A}$ centered at the midpoint of $ BC$ and passing through $ H$ intersects the sideline $ BC$ at points $ A_{1}$ and $ A_{2}$. Similarly, define the points $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$. Prove that the six points $ A_{1}$, $ A_{2}$, $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$ are concyclic. [i]Author: Andrey Gavrilyuk, Russia[/i]

2020 Tournament Of Towns, 4

Tags: diophantine equation , diophantine , number theory

For some integer n the equation $x^2 + y^2 + z^2 -xy -yz - zx = n$ has an integer solution $x, y, z$. Prove that the equation$ x^2 + y^2 - xy = n$ also has an integer solution $x, y$. Alexandr Yuran

2011 Cono Sur Olympiad, 1

Tags: number theory

Find all triplets of positive integers $(x,y,z)$ such that $x^{2}+y^{2}+z^{2}=2011$.

1983 AMC 12/AHSME, 17

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The diagram to the right shows several numbers in the complex plane. The circle is the unit circle centered at the origin. One of these numbers is the reciprocal of $F$. Which one? $\text{(A)} \ A \qquad \text{(B)} \ B \qquad \text{(C)} \ C \qquad \text{(D)} \ D \qquad \text{(E)} \ E$

2000 Vietnam Team Selection Test, 3

Tags: combinatorics

A collection of $2000$ congruent circles is given on the plane such that no two circles are tangent and each circle meets at least two other circles. Let $N$ be the number of points that belong to at least two of the circles. Find the smallest possible value of $N$.

2000 Stanford Mathematics Tournament, 14

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The author of this question was born on April 24, 1977. What day of the week was that?

2005 AMC 10, 6

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The average (mean) of $ 20$ numbers is $ 30$, and the average of $ 30$ other numbers is $ 20$. What is the average of all $ 50$ numbers? $ \textbf{(A)}\ 23 \qquad \textbf{(B)}\ 24 \qquad \textbf{(C)}\ 25 \qquad \textbf{(D)}\ 26 \qquad \textbf{(E)}\ 27$

1932 Eotvos Mathematical Competition, 1

Tags: number theory , divisible

Let $a, b$ and $n$ be positive integers such that $ b$ is divisible by $a^n$. Prove that $(a+1)^b-1$ is divisible by $a^{n+1}$.