Olympiad problems

1986 National High School Mathematics League, 3

Tags: inequalities

For real numbers $a,b,c$, if $$a^2-bc-8a+7=b^2+c^2+bc-6a-6=0,$$ then the range value of $a$ is $\text{(A)}(-\infty,+\infty)\qquad\text{(B)}(-\infty,1]\cup[9,+\infty)\qquad\text{(C)}(0,7)\qquad\text{(D)}[1,9]$

2015 Cono Sur Olympiad, 1

Tags: divisibility , number theory

Show that, for any integer $n$, the number $n^3 - 9n + 27$ is not divisible by $81$.

1998 Irish Math Olympiad, 1

Tags: integration , inequalities

Prove that if $ x \not\equal{} 0$ is a real number, then: $ x^8\minus{}x^5\minus{}\frac{1}{x}\plus{}\frac{1}{x^4} \ge 0$.

2003 India IMO Training Camp, 5

Tags: geometry , geometric transformation , reflection , combinatorics

On the real number line, paint red all points that correspond to integers of the form $81x+100y$, where $x$ and $y$ are positive integers. Paint the remaining integer point blue. Find a point $P$ on the line such that, for every integer point $T$, the reflection of $T$ with respect to $P$ is an integer point of a different colour than $T$.

2009 Princeton University Math Competition, 4

Tags: geometry , 3d geometry , tetrahedron , inequalities

Tetrahedron $ABCD$ has sides of lengths, in increasing order, $7, 13, 18, 27, 36, 41$. If $AB=41$, then what is the length of $CD$?

2023 Indonesia TST, 2

Tags: combinatorics , coin , invariant , monovariant

Let $n > 3$ be a positive integer. Suppose that $n$ children are arranged in a circle, and $n$ coins are distributed between them (some children may have no coins). At every step, a child with at least 2 coins may give 1 coin to each of their immediate neighbors on the right and left. Determine all initial distributions of the coins from which it is possible that, after a finite number of steps, each child has exactly one coin.

2024 PErA, P4

Tags: geometry

Let $ABC$ be a triangle, and let $E$, $F$ be the feet of the altitudes from $B$ and $C$ to sides $AC$ and $AB$, respectively. Let $P$ and $Q$ be the intersections of $EF$ with the tangents from $B$ and $C$ to $(ABC)$, respectively. If $M$ is the midpoint of $BC$, prove that $(PQM)$ is tangent to $BC$ at $M$.

2016 NIMO Summer Contest, 10

Tags: geometry

In rectangle $ABCD$, point $M$ is the midpoint of $AB$ and $P$ is a point on side $BC$. The perpendicular bisector of $MP$ intersects side $DA$ at point $X$. Given that $AB = 33$ and $BC = 56$, find the least possible value of $MX$. [i]Proposed by Michael Tang[/i]

2008 Romania National Olympiad, 2

Tags:

a) We call [i]admissible sequence[/i] a sequence of 4 even digits in which no digits appears more than two times. Find the number of admissible sequences. b) For each integer $ n\geq 2$ we denote $ d_n$ the number of possibilities of completing with even digits an array with $ n$ rows and 4 columns, such that (1) any row is an admissible sequence; (2) the sequence 2, 0, 0, 8 appears exactly ones in the array. Find the values of $ n$ for which the number $ \frac {d_{n\plus{}1}}{d_n}$ is an integer.

1991 AIME Problems, 2

Tags: geometry , rectangle , geometric transformation

Rectangle $ABCD$ has sides $\overline {AB}$ of length 4 and $\overline {CB}$ of length 3. Divide $\overline {AB}$ into 168 congruent segments with points $A=P_0, P_1, \ldots, P_{168}=B$, and divide $\overline {CB}$ into 168 congruent segments with points $C=Q_0, Q_1, \ldots, Q_{168}=B$. For $1 \le k \le 167$, draw the segments $\overline {P_kQ_k}$. Repeat this construction on the sides $\overline {AD}$ and $\overline {CD}$, and then draw the diagonal $\overline {AC}$. Find the sum of the lengths of the 335 parallel segments drawn.

1992 Tournament Of Towns, (338) 6

Tags: number theory , combinatorics

For natural numbers $n$ and $b$, let $V(n, b)$ denote the number of decompositions of $n$ into the product of integers each of which is greater than $b$: for example $$36 = 6\times 6 = 4\times 9 = 3\times 3\times 4 = 3\times 12,$$ i.e. $V(36,2) = 5$. Prove that $V(n, b) < n/b$ for all $n$ and $b$. (N.B. Vasiliev, Moscow)

1993 Romania Team Selection Test, 1

Tags: inequalities

Find max. numbers $A$ wich is true ineq.: $\frac{x}{\sqrt{y^{2}+z^{2}}}+\frac{y}{\sqrt{x^{2}+z^{2}}}+\frac{z}{\sqrt{x^{2}+y^{2}}}\geq A$ $x,y,z$ are positve reals numberes! :wink:

1999 Gauss, 4

Tags: gauss

$1+\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}$ is equal to $\textbf{(A)}\ \dfrac{15}{8} \qquad \textbf{(B)}\ 1\dfrac{3}{14} \qquad \textbf{(C)}\ \dfrac{11}{8} \qquad \textbf{(D)}\ 1\dfrac{3}{4} \qquad \textbf{(E)}\ \dfrac{7}{8}$

1972 Canada National Olympiad, 10

Tags: ratio , calculus , integration , geometric sequence , algebra

What is the maximum number of terms in a geometric progression with common ratio greater than 1 whose entries all come from the set of integers between 100 and 1000 inclusive?

2004 Regional Olympiad - Republic of Srpska, 2

Tags: inequalities

The positive real numbers $x,y,z$ satisfy $x+y+z=1$. Show that \[\sqrt{3xyz}\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{1-x}+\frac{1}{1-y}+\frac{1}{1-z}\right)\geq4+ \frac{4xyz}{(1-x)(1-y)(1-z)}.\]

1964 Poland - Second Round, 2

Tags: perpendicular , arc midpoint , geometry

The circle is divided into four non-overlapping gaps $ AB $, $ BC $, $ CD $ and $ DA $. Prove that the segment joining the midpoints of the arcs $AB$ and $CD$ is perpendicular to the segment joining the midpoints of the arcs $BC$ and $DA$.

2019 CCA Math Bonanza, L1.1

Tags:

How many integers divide either $2018$ or $2019$? Note: $673$ and $1009$ are both prime. [i]2019 CCA Math Bonanza Lightning Round #1.1[/i]

1989 USAMO, 2

Tags: combinatorics

The 20 members of a local tennis club have scheduled exactly 14 two-person games among themselves, with each member playing in at least one game. Prove that within this schedule there must be a set of 6 games with 12 distinct players.

2009 IMO Shortlist, 7

Tags: induction , probability , algebra , combinatorics

Let $ a_1, a_2, \ldots , a_n$ be distinct positive integers and let $ M$ be a set of $ n \minus{} 1$ positive integers not containing $ s \equal{} a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n.$ A grasshopper is to jump along the real axis, starting at the point $ 0$ and making $ n$ jumps to the right with lengths $ a_1, a_2, \ldots , a_n$ in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in $ M.$ [i]Proposed by Dmitry Khramtsov, Russia[/i]

2019 Iran Team Selection Test, 3

Tags: geometry , circumcircle

In triangle $ABC$, $M,N$ and $P$ are midpoints of sides $BC,CA$ and $AB$. Point $K$ lies on segment $NP$ so that $AK$ bisects $\angle BKC$. Lines $MN,BK$ intersects at $E$ and lines $MP,CK$ intersects at $F$. Suppose that $H$ be the foot of perpendicular line from $A$ to $BC$ and $L$ the second intersection of circumcircle of triangles $AKH, HEF$. Prove that $MK,EF$ and $HL$ are concurrent. [i]Proposed by Alireza Dadgarnia[/i]

1999 Harvard-MIT Mathematics Tournament, 12

Tags: combinatorics

A fair coin is flipped every second and the results are recorded with $1$ meaning heads and $0$ meaning tails. What is the probability that the sequence $10101$ occurs before the first occurance of the sequence $010101$?

2019 USMCA, 6

Tags:

Seven two-digit integers form a strictly increasing arithmetic sequence. If the first and last terms of this sequence have the same set of digits, what is the sum of all possible medians of the sequence?

2018 Saudi Arabia JBMO TST, 3

Tags: geometry

Let $ABC$ be a triangle inscribed in the circle $K_1$ and $I$ be center of the inscribed in $ABC$ circle. The lines $IB$ and $IC$ intersect circle $K_1$ again in $J$ and $L$. Circle $K_2$, circumscribed to $IBC$, intersects again $CA$ and $AB$ in $E$ and $F$. Show that $EL$ and $FJ$ intersects on the circle $K_2$.

2013 NIMO Problems, 6

Tags: induction

A strictly increasing sequence $\{x_i\}_{i=1}^{\infty}$ of positive integers is said to be [i]large[/i] if, for every real number $L$, there exists an integer $n$ such that $\frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n} > L$. Do there exist large sequences $\{a_i\}_{i=1}^\infty$ and $\{b_i\}_{i=1}^{\infty}$ such that the sequence $\{a_i+b_i\}_{i=1}^{\infty}$ is not large? [i]Proposed by Lewis Chen[/i]

2023 Junior Balkan Team Selection Tests - Romania, P3

Tags: algebra

Initially the numbers $i^3-i$ for $i=2,3 \ldots 2n+1$ are written on a blackboard, where $n\geq 2$ is a positive integer. On one move we can delete three numbers $a, b, c$ and write the number $\frac{abc} {ab+bc+ca}$. Prove that when two numbers remain on the blackboard, their sum will be greater than $16$.