Olympiad problems

2020-21 KVS IOQM India, 8

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Find the largest $2$-digit number $N$ which is divisible by $4$, such that all integral powers of $N$ end with $N$.

1989 Tournament Of Towns, (207) 1

Tags: number theory

A staircase has $100$ steps. Kolya wishes to descend the staircase by alternately jumping down some steps and then up some. The possible jumps he can do are through $6$ (i.e. over $5$ and landing on the $6$th) , $7$ or $8$ steps . He also does not wish to land twice on the same step . Can he descend the staircase in this way? ( S . Fomin, Leningrad)

2011 Poland - Second Round, 2

Tags: combinatorics

$\forall n\in \mathbb{Z_{+}}-\{1,2\}$ find the maximal length of a sequence with elements from a set $\{1,2,\ldots,n\}$, such that any two consecutive elements of this sequence are different and after removing all elements except for the four we do not receive a sequence in form $x,y,x,y$ ($x\neq y$).

2011 Korea Junior Math Olympiad, 1

Tags: algebra

Real numbers $a$, $b$, $c$ which are differ from $1$ satisfies the following conditions; (1) $abc =1$ (2) $a^2+b^2+c^2 - \left( \dfrac{1}{a^2} + \dfrac{1}{b^2} + \dfrac{1}{c^2} \right) = 8(a+b+c) - 8 (ab+bc+ca)$ Find all possible values of expression $\dfrac{1}{a-1} + \dfrac{1}{b-1} + \dfrac{1}{c-1}$.

2005 JBMO Shortlist, 6

Tags: geometry , circumcircle , symmetry

Let $C_1,C_2$ be two circles intersecting at points $A,P$ with centers $O,K$ respectively. Let $B,C$ be the symmetric of $A$ wrt $O,K$ in circles $C_1,C_2 $ respectively. A random line passing through $A$ intersects circles $C_1,C_2$ at $D,E$ respectively. Prove that the center of circumcircle of triangle $DEP$ lies on the circumcircle of triangle $OKP$.

2010 Harvard-MIT Mathematics Tournament, 2

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The [i]rank[/i] of a rational number $q$ is the unique $k$ for which $q=\frac{1}{a_1}+\cdots+\frac{1}{a_k}$, where each $a_i$ is the smallest positive integer $q$ such that $q\geq \frac{1}{a_1}+\cdots+\frac{1}{a_i}$. Let $q$ be the largest rational number less than $\frac{1}{4}$ with rank $3$, and suppose the expression for $q$ is $\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}$. Find the ordered triple $(a_1,a_2,a_3)$.

2003 Austrian-Polish Competition, 9

Tags: prime number , combinatorics , pigeonhole principle , number theory

Take any 26 distinct numbers from {1, 2, ... , 100}. Show that there must be a non-empty subset of the $ 26$ whose product is a square. [hide] I think that the upper limit for such subset is 37.[/hide]

2023 Romanian Master of Mathematics, 6

Tags: graph theory , combinatorics

Let $r,g,b$ be non negative integers and $\Gamma$ be a connected graph with $r+g+b+1$ vertices. Its edges are colored in red green and blue. It turned out that $\Gamma $ contains A spanning tree with exactly $r$ red edges. A spanning tree with exactly $g$ green edges. A spanning tree with exactly $b$ blue edges. Prove that $\Gamma$ contains a spanning tree with exactly $r$ red edges, $g$ green edges and $b$ blue edges.

May Olympiad L2 - geometry, 2016.4

Tags: midline , geometry , area

In a triangle $ABC$, let $D$ and $E$ be points of the sides $ BC$ and $AC$ respectively. Segments $AD$ and $BE$ intersect at $O$. Suppose that the line connecting midpoints of the triangle and parallel to $AB$, bisects the segment $DE$. Prove that the triangle $ABO$ and the quadrilateral $ODCE$ have equal areas.

2020 CCA Math Bonanza, L1.3

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If $ABCDE$ is a regular pentagon and $X$ is a point in its interior such that $CDX$ is equilateral, compute $\angle{AXE}$ in degrees. [i]2020 CCA Math Bonanza Lightning Round #1.3[/i]

2018 MIG, 18

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How many paths are there from $A$ to $B$ in the following diagram if only moves downward are allowed? [center][img]https://cdn.artofproblemsolving.com/attachments/f/d/62a14f7959cc0461543b0f76bba51be9786847.png[/img][/center] $\textbf{(A) } 65\qquad\textbf{(B) } 67\qquad\textbf{(C) } 70\qquad\textbf{(D) } 74\qquad\textbf{(E) } 75$

1980 Miklós Schweitzer, 7

Tags: real analysis , polynomial , trigonometry , algebra

Let $ n \geq 2$ be a natural number and $ p(x)$ a real polynomial of degree at most $ n$ for which \[ \max _{ \minus{}1 \leq x \leq 1} |p(x)| \leq 1, \; p(\minus{}1)\equal{}p(1)\equal{}0 \ .\] Prove that then \[ |p'(x)| \leq \frac{n \cos \frac{\pi}{2n}}{\sqrt{1\minus{}x^2 \cos^2 \frac{\pi}{2n}}} \;\;\;\;\; \left( \minus{}\frac{1}{\cos \frac{\pi}{2n}} < x < \frac{1}{\cos \frac{\pi}{2n}} \\\\\ \right).\] [i]J. Szabados[/i]

1960 AMC 12/AHSME, 11

Tags: vieta , calculus , integration

For a given value of $k$ the product of the roots of \[ x^2-3kx+2k^2-1=0 \] is $7$. The roots may be characterized as: $ \textbf{(A) }\text{integral and positive} \qquad\textbf{(B) }\text{integral and negative} \qquad$ $\textbf{(C) }\text{rational, but not integral} \qquad\textbf{(D) }\text{irrational} \qquad\textbf{(E) } \text{imaginary} $

2006 Greece JBMO TST, 4

Tags: algebra

Find the minimum value of $$K(x,y)=16\frac{x^3}{y}+\frac{y^3}{x}-\sqrt{xy}$$ where $x,y$ are the real allowed values

2009 India IMO Training Camp, 3

Tags: number theory

Let $ a,b$ be two distinct odd natural numbers.Define a Sequence $ { < a_n > }_{n\ge 0}$ like following: $ a_1 \equal{} a \\ a_2 \equal{} b \\ a_n \equal{} \text{largest odd divisor of }(a_{n \minus{} 1} \plus{} a_{n \minus{} 2})$. Prove that there exists a natural number $ N$ such that $ a_n \equal{} gcd(a,b) \forall n\ge N$.

2008 Oral Moscow Geometry Olympiad, 1

Tags: cut , geometry , similar triangles

Each of two similar triangles was cut into two triangles so that one of the resulting parts of one triangle is similar to one of the parts of the other triangle. Is it true that the remaining parts are also similar? (D. Shnol)

1995 May Olympiad, 1

Tags: game strategy , game , number theory , divisor

Veronica, Ana and Gabriela are forming a round and have fun with the following game. One of them chooses a number and says out loud, the one to its left divides it by its largest prime divisor and says the result out loud and so on. The one who says the number out loud $1$ wins , at which point the game ends. Ana chose a number greater than $50$ and less than $100$ and won. Veronica chose the number following the one chosen by Ana and also won. Determine all the numbers that could have been chosen by Ana.

2017 AMC 10, 1

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Mary thought of a positive two-digit number. She multiplied it by $3$ and added $11.$ Then she switched the digits of the result, obtaining a number between $71$ and $75$, inclusive. What was Mary's number? $\textbf{(A)} \text{ 11} \qquad \textbf{(B)} \text{ 12} \qquad \textbf{(C)} \text{ 13} \qquad \textbf{(D)} \text{ 14} \qquad \textbf{(E)} \text{ 15}$

2020-2021 Winter SDPC, #4

Tags: algebra , polynomial

Find all polynomials $P(x)$ with integer coefficients such that for all positive integers $n$, we have that $P(n)$ is not zero and $\frac{P(\overline{nn})}{P(n)}$ is an integer, where $\overline{nn}$ is the integer obtained upon concatenating $n$ with itself.

2011 AMC 10, 7

Tags: angle , geometry

The sum of two angles of a triangle is $\frac{6}{5}$ of a right angle, and one of these two angles is $30 ^\circ$ larger than the other. What is the degree measure of the largest angle in the triangle? $ \textbf{(A)}\ 69 \qquad \textbf{(B)}\ 72 \qquad \textbf{(C)}\ 90 \qquad \textbf{(D)}\ 102 \qquad \textbf{(E)}\ 108 $

2014 Argentina National Olympiad Level 2, 4

Tags: combinatorics

There is a number written in each square of a $13\times13$ board such that any two numbers in squares with a common side differ by exactly $1$. Each of the numbers $2$ and $24$ is written twice. How many times is the number $13$ written? Find all possibilities.

2008 Tournament Of Towns, 1

Tags: combinatorics , geometry , rectangle , ratio

A square board is divided by lines parallel to the board sides ($7$ lines in each direction, not necessarily equidistant ) into $64$ rectangles. Rectangles are colored into white and black in alternating order. Assume that for any pair of white and black rectangles the ratio between area of white rectangle and area of black rectangle does not exceed $2.$ Determine the maximal ratio between area of white and black part of the board. White (black) part of the board is the total sum of area of all white (black) rectangles.

2016 Baltic Way, 2

Tags: number theory

Prove or disprove the following hypotheses. a) For all $k \geq 2,$ each sequence of $k$ consecutive positive integers contains a number that is not divisible by any prime number less than $k.$ b) For all $k\geq 2,$ each sequence of $k$ consecutive positive integers contains a number that is relatively prime to all other members of the sequence.

2005 Junior Balkan Team Selection Tests - Romania, 5

Tags: rhombus , geometry

On the sides $AD$ and $BC$ of a rhombus $ABCD$ we consider the points $M$ and $N$ respectively. The line $MC$ intersects the segment $BD$ in the point $T$, and the line $MN$ intersects the segment $BD$ in the point $U$. We denote by $Q$ the intersection between the line $CU$ and the side $AB$ and with $P$ the intersection point between the line $QT$ and the side $CD$. Prove that the triangles $QCP$ and $MCN$ have the same area.

1946 Moscow Mathematical Olympiad, 108

Tags: 4-digit , remainder , number theory

Find a four-digit number such that the remainders after its division by $131$ and $132$ are $112$ and $98$, respectively.