Olympiad problems

LMT Guts Rounds, 19

Tags:

Let $f(x)=x^2-2x+1.$ For some constant $k, f(x+k) = x^2+2x+1$ for all real numbers $x.$ Determine the value of $k.$

2014 Korea Junior Math Olympiad, 7

Tags: geometry , parallelogram , incenter , rhombus , projective geometry , geometry unsolved

In a parallelogram $\Box ABCD$ $(AB < BC)$ The incircle of $\triangle ABC$ meets $\overline {BC}$ and $\overline {CA}$ at $P, Q$. The incircle of $\triangle ACD$ and $\overline {CD}$ meets at $R$. Let $S$ = $PQ$ $\cap$ $AD$ $U$ = $AR$ $\cap$ $CS$ $T$, a point on $\overline {BC}$ such that $\overline {AB} = \overline {BT}$ Prove that $AT, BU, PQ$ are concurrent

2021 XVII International Zhautykov Olympiad, #1

Tags: number theory

Prove that there exists a positive integer $n$, such that the remainder of $3^n$ when divided by $2^n$ is greater than $10^{2021} $.

1974 IMO Longlists, 29

Tags: geometry

Let $A,B,C,D$ be points in space. If for every point $M$ on the segment $AB$ the sum \[S_{AMC}+S_{CMD}+S_{DMB}\] Is constant show that the points $A,B,C,D$ lie in the same plane. [hide="Note."] [i]Note. $S_X$ denotes the area of triangle $X.$[/i][/hide]

2006 Princeton University Math Competition, 5

Tags: number theory

How many pairs of positive integers $(a,b)$ are there such that $a < b$ and $a,b$ can be the legs of a right triangle with hypotenuse $340$?

2021 Iran MO (3rd Round), 2

Tags: combinatorics

Is it possible to arrange a permutation of Integers on the integer lattice infinite from both sides such that each row is increasing from left to right and each column increasing from up to bottom?

1990 IMO Longlists, 74

Tags: analytic geometry , geometry , parallelogram , vector , geometric transformation , homothety , ratio

Let $L$ be a subset in the coordinate plane defined by $L = \{(41x + 2y, 59x + 15y) | x, y \in \mathbb Z \}$, where $\mathbb Z$ is set of integers. Prove that for any parallelogram with center in the origin of coordinate and area $1990$, there exist at least two points of $L$ located in it.

2022 Indonesia TST, C

Tags: combinatorics , counting

Let $A$ be a subset of $\{1,2,\ldots,2020\}$ such that the difference of any two distinct elements in $A$ is not prime. Determine the maximum number of elements in set $A$.

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.