Olympiad problems

2021 AMC 12/AHSME Spring, 2

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At a math contest, $57$ students are wearing blue shirts, and another $75$ students are wearing yellow shirts. The $132$ students are assigned into $66$ points. In exactly $23$ of these pairs, both students are wearing blue shirts. In how many pairs are both studets wearing yellow shirts? $\textbf{(A) }23 \qquad \textbf{(B) }32 \qquad \textbf{(C) }37 \qquad \textbf{(D) }41 \qquad \textbf{(E) }64$

2018 Junior Balkan MO, 3

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Let $k>1$ be a positive integer and $n>2018$ an odd positive integer. The non-zero rational numbers $x_1,x_2,\ldots,x_n$ are not all equal and: $$x_1+\frac{k}{x_2}=x_2+\frac{k}{x_3}=x_3+\frac{k}{x_4}=\ldots=x_{n-1}+\frac{k}{x_n}=x_n+\frac{k}{x_1}$$ Find the minimum value of $k$, such that the above relations hold.

2012 Mediterranean Mathematics Olympiad, 1

Tags: algebra

For a real number $\alpha>0$, consider the infinite real sequence defined by $x_1=1$ and \[ \alpha x_n = x_1+x_2+\cdots+x_{n+1} \mbox{\qquad for } n\ge1. \] Determine the smallest $\alpha$ for which all terms of this sequence are positive reals. (Proposed by Gerhard Woeginger, Austria)

2012 Today's Calculation Of Integral, 824

Tags: calculus , integration , geometry , limit , calculus computations , logarithm

In the $xy$-plane, for $a>1$ denote by $S(a)$ the area of the figure bounded by the curve $y=(a-x)\ln x$ and the $x$-axis. Find the value of integer $n$ for which $\lim_{a\rightarrow \infty} \frac{S(a)}{a^n\ln a}$ is non-zero real number.

2008 AMC 12/AHSME, 15

Tags: induction , modular arithmetic , number theory

Let $ k\equal{}2008^2\plus{}2^{2008}$. What is the units digit of $ k^2\plus{}2^k$? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 8$

PEN H Problems, 27

Tags: diophantine equation , perimeter , inradius , modular arithmetic , geometry

Prove that there exist infinitely many positive integers $n$ such that $p=nr$, where $p$ and $r$ are respectively the semi-perimeter and the inradius of a triangle with integer side lengths.

2007 Junior Balkan Team Selection Tests - Romania, 2

Tags: analytic geometry , combinatorics

There are given the integers $1 \le m < n$. Consider the set $M = \{ (x,y);x,y \in \mathbb{Z_{+}}, 1 \le x,y \le n \}$. Determine the least value $v(m,n)$ with the property that for every subset $P \subseteq M$ with $|P| = v(m,n)$ there exist $m+1$ elements $A_{i}= (x_{i},y_{i}) \in P, i = 1,2,...,m+1$, for which the $x_{i}$ are all distinct, and $y_{i}$ are also all distinct.

1979 IMO Longlists, 18

Tags: number theory

Show that for no integers $a \geq 1, n \geq 1$ is the sum \[1+\frac{1}{1+a}+\frac{1}{1+2a}+\cdots+\frac{1}{1+na}\] an integer.

2010 Contests, 3

Tags: combinatorics

Given is the set $M_n=\{0, 1, 2, \ldots, n\}$ of nonnegative integers less than or equal to $n$. A subset $S$ of $M_n$ is called [i]outstanding[/i] if it is non-empty and for every natural number $k\in S$, there exists a $k$-element subset $T_k$ of $S$. Determine the number $a(n)$ of outstanding subsets of $M_n$. [i](41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 3)[/i]

1996 Swedish Mathematical Competition, 4

Tags: pentagon , increasing , cyclic , geometry , angle

The angles at $A,B,C,D,E$ of a pentagon $ABCDE$ inscribed in a circle form an increasing sequence. Show that the angle at $C$ is greater than $\pi/2$, and that this lower bound cannot be improved.

2012 JHMT, 7

Tags: geometry , 3d geometry

What is the radius of the largest sphere that fits inside an octahedron of side length $1$?

2022 Junior Balkan Team Selection Tests - Romania, P1

Tags: number theory , squarefree

Determine all squarefree positive integers $n\geq 2$ such that \[\frac{1}{d_1}+\frac{1}{d_2}+\cdots+\frac{1}{d_k}\]is a positive integer, where $d_1,d_2,\ldots,d_k$ are all the positive divisors of $n$.

2007 Cuba MO, 1

Tags: algebra , system of equations , system

Find all the real numbers $x, y$ such that $x^3 - y^3 = 7(x - y)$ and $x^3 + y^3 = 5(x + y).$

1986 Dutch Mathematical Olympiad, 2

Tags: algebra , sum

Prove that for all positive integers $n$ holds that $$\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+...+\frac{1}{(2n-1) \cdot 2n}=\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{2n}$$

2023 Bosnia and Herzegovina Junior BMO TST, 3.

Tags: incenter , geometry

Let ABC be an acute triangle with an incenter $I$.The Incircle touches sides $AC$ and $AB$ in $E$ and $F$ ,respectively. Lines CI and EF intersect at $S$. The point $T$≠$I$ is on the line AI so that $EI$=$ET$.If $K$ is the foot of the altitude from $C$ in triangle $ABC$,prove that points $K$,$S$ and $T$ are colinear.

2014 China Northern MO, 2

Tags: inequalities , algebra

Define a positive number sequence sequence $\{a_n\}$ by \[a_{1}=1,(n^2+1)a^2_{n-1}=(n-1)^2a^2_{n}.\]Prove that\[\frac{1}{a^2_1}+\frac{1}{a^2_2}+\cdots +\frac{1}{a^2_n}\le 1+\sqrt{1-\frac{1}{a^2_n}} .\]

1989 IMO Longlists, 75

Tags: floor function

Solve in the set of real numbers the equation \[ 3x^3 \minus{} [x] \equal{} 3,\] where $ [x]$ denotes the integer part of $ x.$

2020/2021 Tournament of Towns, P3

Tags: game , combinatorics

Alice and Bob are playing the following game. Each turn Alice suggests an integer and Bob writes down either that number or the sum of that number with all previously written numbers. Is it always possible for Alice to ensure that at some moment among the written numbers there are [list=a] [*]at least a hundred copies of number 5? [*]at least a hundred copies of number 10? [/list] [i]Andrey Arzhantsev[/i]

2017 Purple Comet Problems, 9

Tags: geometry

The diagram below shows $\vartriangle ABC$ with point $D$ on side $\overline{BC}$. Three lines parallel to side $\overline{BC}$ divide segment $\overline{AD}$ into four equal segments. In the triangle, the ratio of the area of the shaded region to the area of the unshaded region is $\frac{49}{33}$ and $\frac{BD}{CD} = \frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/2/8/77239633b68f073f4193aa75cdfb9238461cae.png[/img]

2005 Korea Junior Math Olympiad, 8

Tags: combinatorics

A group of $6$ students decided to make study groups and service activity groups according to the following principle: Each group must have exactly $3$ members. For any pair of students, there are same number of study groups and service activity groups that both of the students are members. Supposing there are at least one group and no three students belong to the same study group and service activity group, prove that the minimum number of groups is $8$.

2024 Yasinsky Geometry Olympiad, 3

Tags: geometry , concurrency

Let $ W $ be the midpoint of the arc $ BC $ of the circumcircle of triangle $ ABC $, such that $ W $ and $ A $ lie on opposite sides of line $ BC $. On sides $ AB $ and $ AC $, points $ P $ and $ Q $ are chosen respectively so that $ APWQ $ is a parallelogram, and on side $ BC $, points $ K $ and $ L $ are chosen such that $ BK = KW $ and $ CL = LW $. Prove that the lines $ AW $, $ KQ $, and $ LP $ are concurrent. [i]Proposed by Matthew Kurskyi[/i]

1999 All-Russian Olympiad, 7

Tags: radical axis , geometry

A circle through vertices $A$ and $B$ of triangle $ABC$ meets side $BC$ again at $D$. A circle through $B$ and $C$ meets side $AB$ at $E$ and the first circle again at $F$. Prove that if points $A$, $E$, $D$, $C$ lie on a circle with center $O$ then $\angle BFO$ is right.

1969 IMO Longlists, 50

Tags: geometry , pentagon , construction , angle bisector

$(NET 5)$ The bisectors of the exterior angles of a pentagon $B_1B_2B_3B_4B_5$ form another pentagon $A_1A_2A_3A_4A_5.$ Construct $B_1B_2B_3B_4B_5$ from the given pentagon $A_1A_2A_3A_4A_5.$

2019 MIG, 7

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In one peculiar family, the mother and the three children have exactly the same birthday. Currently, the mother is $37$ years old while each of children are $9$ years old. How old will the mother be when the sum of the ages of the three children equals her age? $\textbf{(A) }14\qquad\textbf{(B) }27\qquad\textbf{(C) }42\qquad\textbf{(D) }57\qquad\textbf{(E) }66$

2024 Thailand TSTST, 9

Tags: geometry , spiral similarity , cartesian coordinates , complex bash , inversion

Let triangle $ ABC $ be an acute-angled triangle. Square $ AEFB $ and $ ADGC $ lie outside triangle $ ABC $. $ BD $ intersects $ CE $ at point $ H $, and $ BG $ intersects $ CF $ at point $ I $. The circumcircle of triangle $ BFI $ intersects the circumcircle of triangle $ CGI $ again at point $ K $. Prove that line segment $ HK $ bisects $ BC $.