Olympiad problems

1967 Czech and Slovak Olympiad III A, 2

Tags: geometry , 3D geometry , tetrahedron , acute , Triangle

Let $ABCD$ be a tetrahedron such that \[AB^2+CD^2=AC^2+BD^2=AD^2+BC^2.\] Show that at least one of its faces is an acute triangle.

2011 Junior Balkan Team Selection Tests - Romania, 5

Tags: combinatorics

Consider $n$ persons, each of them speaking at most $3$ languages. From any $3$ persons there are at least two which speak a common language. i) For $n \le 8$, exhibit an example in which no language is spoken by more than two persons. ii) For $n \ge 9$, prove that there exists a language which is spoken by at least three persons

2013 IMO Shortlist, C1

Tags: algorithm , combinatorics , Additive combinatorics , IMO Shortlist

Let $n$ be an positive integer. Find the smallest integer $k$ with the following property; Given any real numbers $a_1 , \cdots , a_d $ such that $a_1 + a_2 + \cdots + a_d = n$ and $0 \le a_i \le 1$ for $i=1,2,\cdots ,d$, it is possible to partition these numbers into $k$ groups (some of which may be empty) such that the sum of the numbers in each group is at most $1$.

2018 AMC 8, 14

Tags: AMC 8 , 2018 AMC 8

Let $N$ be the greatest five-digit number whose digits have a product of $120$. What is the sum of the digits of $N$? $\textbf{(A) }15\qquad\textbf{(B) }16\qquad\textbf{(C) }17\qquad\textbf{(D) }18\qquad\textbf{(E) }20$

1989 AMC 12/AHSME, 6

Tags: analytic geometry , geometry

If $a,b > 0$ and the triangle in the first quadrant bounded by the coordinate axes and the graph of $ax+by = 6$ has area 6, then $ab =$ $\text{(A)} \ 3 \qquad \text{(B)} \ 6 \qquad \text{(C)} \ 12 \qquad \text{(D)} \ 108 \qquad \text{(E)} \ 432$

2013 District Olympiad, 3

Tags: prism , 3D geometry , geometry , perpendicular

Let be the regular hexagonal prism $ABCDEFA'B C'D'E'F'$ with the base edge of $12$ and the height of $12 \sqrt{3}$. We denote by $N$ the middle of the edge $CC'$. a) Prove that the lines $BF'$ and $ND$ are perpendicular b) Calculate the distance between the lines $BF'$ and $ND$.

2008 Princeton University Math Competition, A4/B7

Tags: geometry

How many ordered pairs of real numbers $(x, y)$ are there such that $x^2+y^2 = 200$ and \[\sqrt{(x-5)^2+(y-5)^2}+\sqrt{(x+5)^2+(y+5)^2}\] is an integer?

1979 Chisinau City MO, 173

Tags: equal angles , pentagon , Regular , geometry

The inner angles of the pentagon inscribed in the circle are equal to each other. Prove that this pentagon is regular.

2019 Taiwan TST Round 2, 2

Tags: geometry , circumcircle

Given a triangle $ \triangle{ABC} $. Denote its incircle and circumcircle by $ \omega, \Omega $, respectively. Assume that $ \omega $ tangents the sides $ AB, AC $ at $ F, E $, respectively. Then, let the intersections of line $ EF $ and $ \Omega $ to be $ P,Q $. Let $ M $ to be the mid-point of $ BC $. Take a point $ R $ on the circumcircle of $ \triangle{MPQ} $, say $ \Gamma $, such that $ MR \perp EF $. Prove that the line $ AR $, $ \omega $ and $ \Gamma $ intersect at one point.

2011 Korea - Final Round, 3

Tags: combinatorics unsolved , combinatorics

There are $n$ boys $a_1, a_2, \ldots, a_n$ and $n$ girls $b_1, b_2, \ldots, b_n $. Some pairs of them are connected. Any two boys or two girls are not connected, and $a_i$ and $b_i$ are not connected for all $i \in \{ 1,2,\ldots,n\}$. Now all boys and girls are divided into several groups satisfying two conditions: (i) Every groups contains an equal number of boys and girls. (ii) There is no connected pair in the same group. Assume that the number of connected pairs is $m$. Show that we can make the number of groups not larger than $\max\left \{2, \dfrac{2m}{n} +1\right \}$.

2004 Junior Balkan Team Selection Tests - Romania, 4

Tags: number theory proposed , number theory

Find all positive integers $n$ for which there exist distinct positive integers $a_1,a_2,\ldots,a_n$ such that \[ \frac 1{a_1} + \frac 2{a_2} + \cdots + \frac n { a_n} = \frac { a_1 + a_2 + \cdots + a_n } n. \]

2016 Mexico National Olmypiad, 4

Tags: Mexico , number theory

We say a non-negative integer $n$ "[i]contains[/i]" another non-negative integer $m$, if the digits of its decimal expansion appear consecutively in the decimal expansion of $n$. For example, $2016$ [i]contains[/i] $2$, $0$, $1$, $6$, $20$, $16$, $201$, and $2016$. Find the largest integer $n$ that does not [i]contain[/i] a multiple of $7$.

2007 China Team Selection Test, 2

Tags: combinatorics proposed , combinatorics

Given an integer $ k > 1.$ We call a $ k \minus{}$digits decimal integer $ a_{1}a_{2}\cdots a_{k}$ is $ p \minus{}$monotonic, if for each of integers $ i$ satisfying $ 1\le i\le k \minus{} 1,$ when $ a_{i}$ is an odd number, $ a_{i} > a_{i \plus{} 1};$ when $ a_{i}$ is an even number, $ a_{i}<a_{i \plus{} 1}.$ Find the number of $ p \minus{}$monotonic $ k \minus{}$digits integers.

2004 Purple Comet Problems, 11

Tags:

Find the sum of all integers $x$ satisfying $1 + 8x \le 358 - 2x \le 6x + 94$.

2014 ELMO Shortlist, 4

Tags: function , algebra , Elmo

Find all triples $(f,g,h)$ of injective functions from the set of real numbers to itself satisfying \begin{align*} f(x+f(y)) &= g(x) + h(y) \\ g(x+g(y)) &= h(x) + f(y) \\ h(x+h(y)) &= f(x) + g(y) \end{align*} for all real numbers $x$ and $y$. (We say a function $F$ is [i]injective[/i] if $F(a)\neq F(b)$ for any distinct real numbers $a$ and $b$.) [i]Proposed by Evan Chen[/i]

2020 CMIMC Algebra & Number Theory, 3

Tags: algebra , number theory , 2020

Call a number ``Sam-azing" if it is equal to the sum of its digits times the product of its digits. The only two three-digit Sam-azing numbers are $n$ and $n + 9$. Find $n$.

2018 Peru IMO TST, 3

Tags: geometry , IMO Shortlist , pentagon , Inscribed circle

Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.

2003 China Western Mathematical Olympiad, 2

Tags: vector , inequalities

Let $ a_1, a_2, \ldots, a_{2n}$ be $ 2n$ real numbers satisfying the condition $ \sum_{i \equal{} 1}^{2n \minus{} 1} (a_{i \plus{} 1} \minus{} a_i)^2 \equal{} 1$. Find the greatest possible value of $ (a_{n \plus{} 1} \plus{} a_{n \plus{} 2} \plus{} \ldots \plus{} a_{2n}) \minus{} (a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n)$.

1977 Dutch Mathematical Olympiad, 4

Tags: geometry , combinatorics , combinatorial geometry

There are an even number of points in a plane. No three of them lie on one straight line. Half of the points are red, the other half are blue. Prove that there exists a connecting line of a red and a blue point such that in each of the half-planes bounded by that line the number of red points is equal to the number of blue points.

2010 Benelux, 2

Tags: algebra , polynomial , functional equation , algebra proposed

Find all polynomials $p(x)$ with real coeffcients such that \[p(a + b - 2c) + p(b + c - 2a) + p(c + a - 2b) = 3p(a - b) + 3p(b - c) + 3p(c - a)\] for all $a, b, c\in\mathbb{R}$. [i](2nd Benelux Mathematical Olympiad 2010, Problem 2)[/i]

2017 European Mathematical Cup, 2

Tags: combinatorics

A friendly football match lasts 90 minutes. In this problem, we consider one of the teams, coached by Sir Alex, which plays with 11 players at all times. a) Sir Alex wants for each of his players to play the same integer number of minutes, but each player has to play less than 60 minutes in total. What is the minimum number of players required? b) For the number of players found in a), what is the minimum number of substitutions required, so that each player plays the same number of minutes? [i]Remark:[/i] Substitutions can only take place after a positive integer number of minutes, and players who have come off earlier can return to the game as many times as needed. There is no limit to the number of substitutions allowed. Proposed by Athanasios Kontogeorgis and Demetres Christofides.

1955 Moscow Mathematical Olympiad, 305

Tags: minimum , combinatorics , Tournament

$25$ chess players are going to participate in a chess tournament. All are on distinct skill levels, and of the two players the one who plays better always wins. What is the least number of games needed to select the two best players?

2001 National High School Mathematics League, 15

Tags:

The schematic wiring diagram below is made of six electric resistances $a_1,a_2,a_3,a_4,a_5,a_6(a_1>a_2>a_3>a_4>a_5>a_6)$. How can we choose the electric resistances, so that the all-in resistance takes its minimum value? [center][img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNy81LzBiZGVjZjczN2Y4MWY0YmNhMTg1YmQxNGEzZWMwZDc2OTE1NzUwLnBuZw==&rn=MjExMTExMTExMTExMTFnaGoucG5n[/img][/center]

2019 Thailand Mathematical Olympiad, 3

Tags: algebra

Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that $f(x+yf(x)+y^2) = f(x)+2y$ for every $x,y\in\mathbb{R}^+$.

2017 Purple Comet Problems, 20

Tags: Purple Comet

A right circular cone has a height equal to three times its base radius and has volume 1. The cone is inscribed inside a sphere as shown. The volume of the sphere is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [center][img]https://snag.gy/92ikv3.jpg[/img][/center]