Olympiad problems

1971 Polish MO Finals, 4

Prove that if positive integers $x,y,z$ satisfy the equation $$x^n + y^n = z^n,$$ then $\min\, (x,y) \ge n$.

2016 South East Mathematical Olympiad, 6

Tags: combinatorics

Toss the coin $n$ times, assume that each time, only appear only head or tail Let $a(n)$ denote number of way that head appear in multiple of $3$ times among $n$ times Let $b(n)$ denote numbe of way that head appear in multiple of $6$ times among $n$ times $(1)$ Find $a(2016)$ and $b(2016)$ $(2)$ Find the number of positive integer $n\leq 2016$ that $2b(n)-a(n)\geq 0$

2013 Philippine MO, 2

Tags: geometry

2. Let P be a point in the interior of triangle ABC . Extend AP, BP, and CP to meet BC, AC, and AB at D, E, and F, respectively. If triangle APF, triangle BPD and triangle CPE have equal areas, prove that P is the centroid of triangle ABC .

2010 VTRMC, Problem 7

Tags: real analysis , sum , summation , sequence

Let $\sum_{n=1}^\infty a_n$ be a convergent series of positive terms (so $a_i>0$ for all $i$) and set $b_n=\frac1{na_n^2}$ for $n\ge1$. Prove that $\sum_{n=1}^\infty\frac n{b_1+b_2+\ldots+b_n}$ is convergent.

1997 Junior Balkan MO, 3

Tags: inequalities

Let $ABC$ be a triangle and let $I$ be the incenter. Let $N$, $M$ be the midpoints of the sides $AB$ and $CA$ respectively. The lines $BI$ and $CI$ meet $MN$ at $K$ and $L$ respectively. Prove that $AI+BI+CI>BC+KL$. [i]Greece[/i]

2014 Contests, 2

Tags: geometry

Suppose $ABCD$ is a convex quadrilateral.Points $P,Q,R$ and $S$ are four points on the line segments $AB,BC,CD$ and $DA$ respectively.The line segments $PR$ and $QS$ meet at $T$.Suppose that each of the quadrilaterals $APTS,BQTP,CRTQ$ and $DSTR$ have an incircle.Prove that the quadrilateral $ABCD$ also has an incircle.

2002 Iran MO (3rd Round), 7

Tags: trigonometry , geometry

In triangle $ABC$, $AD$ is angle bisector ($D$ is on $BC$) if $AB+AD=CD$ and $AC+AD=BC$, what are the angles of $ABC$?

2000 AIME Problems, 8

Tags: geometry , trapezoid , ratio , quadratic , trigonometry , algebra , quadratic formula

In trapezoid $ABCD,$ leg $\overline{BC}$ is perpendicular to bases $\overline{AB}$ and $\overline{CD},$ and diagonals $\overline{AC}$ and $\overline{BD}$ are perpendicular. Given that $AB=\sqrt{11}$ and $AD=\sqrt{1001},$ find $BC^2.$

2003 National High School Mathematics League, 13

Tags: inequalities

Prove that $2\sqrt{1+x}+\sqrt{2x-3}+\sqrt{15-3x}<2\sqrt{19}$, where $\frac{3}{2}\leq x\leq5$.

1997 Tournament Of Towns, (559) 4

Tags: combinatorics , chessboard

The maximum possible number of knights are placed on a $5 \times 5$ chessboard so that no two attack each other. Prove that there is only one possible placement. (A Kanel)

2019 Argentina National Olympiad Level 2, 6

Tags: combinatorics

Let $n$ be a natural number. We define $f(n)$ as the number of ways to express $n$ as a sum of powers of $2$, where the order of the terms is taken into account. For example, $f(4) = 6$, because $4$ can be written as: \begin{align*} 4;\\ 2 + 2;\\ 2 + 1 + 1;\\ 1 + 2 + 1;\\ 1 + 1 + 2;\\ 1 + 1 + 1 + 1. \end{align*} Find the smallest $n$ greater than $2019$ for which $f(n)$ is odd.

2022 Durer Math Competition Finals, 1

Tags: square , geometry , tangent , equilateral

To the exterior of side $AB$ of square $ABCD$, we have drawn the regular triangle $ABE$. Point $A$ reflected on line $BE$ is $F$, and point $E$ reflected on line $BF$ is $G$. Let the perpendicular bisector of segment $FG$ meet segment $AD$ at $X$. Show that the circle centered at $X$ with radius $XA$ touches line$ FB$.

2010 Bosnia and Herzegovina Junior BMO TST, 2

Tags: polynomial , algebra , minimization

Let us consider every third degree polynomial $P(x)$ with coefficients as nonnegative positive integers such that $P(1)=20$. Among them determine polynomial for which is: $a)$ Minimal value of $P(4)$ $b)$ Maximal value of $P(3)/P(2)$

2000 Polish MO Finals, 1

Tags: trigonometry , function , algebra

Find number of solutions in non-negative reals to the following equations: \begin{eqnarray*}x_1 + x_n ^2 = 4x_n \\ x_2 + x_1 ^2 = 4x_1 \\ ... \\ x_n + x_{n-1}^2 = 4x_{n-1} \end{eqnarray*}

2004 National Olympiad First Round, 16

Tags:

What is the sum of real roots of the equation $x^4-4x^3+5x^2-4x+1 = 0$? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 1 $

2020-2021 OMMC, 2

Tags:

Sequences $a_n$ and $b_n$ are defined for all positive integers $n$ such that $a_1 = 5,$ $b_1 = 7,$ $$a_{n+1} = \frac{\sqrt{(a_n+b_n-1)^2+(a_n-b_n+1)^2}}{2},$$ and $$b_{n+1} = \frac{\sqrt{(a_n+b_n+1)^2+(a_n-b_n-1)^2}}{2}.$$ $ $ \\ How many integers $n$ from 1 to 1000 satisfy the property that $a_n, b_n$ form the legs of a right triangle with a hypotenuse that has integer length?

2014 Dutch IMO TST, 4

Tags: geometric transformation , reflection , angle bisector , geometry

Let $\triangle ABC$ be a triangle with $|AC|=2|AB|$ and let $O$ be its circumcenter. Let $D$ be the intersection of the bisector of $\angle A$ with $BC$. Let $E$ be the orthogonal projection of $O$ to $AD$ and let $F\ne D$ be the point on $AD$ satisfying $|CD|=|CF|$. Prove that $\angle EBF=\angle ECF$.

2016 Harvard-MIT Mathematics Tournament, 8

Tags:

Define $\phi^{!}(n)$ as the product of all positive integers less than or equal to $n$ and relatively prime to $n$. Compute the number of integers $2 \le n \le 50$ such that $n$ divides $\phi^{!}(n)+1$.

2016 AIME Problems, 1

Tags: geometric series

For $-1 < r < 1$, let $S(r)$ denote the sum of the geometric series \[12 + 12r + 12r^2 + 12r^3 + \ldots.\] Let $a$ between $-1$ and $1$ satisfy $S(a)S(-a)=2016$. Find $S(a) + S(-a)$.

2016 Switzerland Team Selection Test, Problem 6

Tags: induction , symmetry , algebra , polynomial

Prove that for every nonnegative integer $n$, the number $7^{7^{n}}+1$ is the product of at least $2n+3$ (not necessarily distinct) primes.

2018 Iran MO (1st Round), 8

Tags: combinatorics

The license plate of each automobile in Iran consists of a two-digit and a three-digit number as well as a letter of the Persian alphabet. The digit $0$ is not used in the two numbers. To each license plate, we assign the product of the two numbers on it. For example, if the two numbers are $12$ and $365$ on a license plate, the assigned number would be $12 \times 365 = 4380$. What is the average of all the assigned numbers to all possible license plates?

2022 IMO Shortlist, G7

Tags: euler , geometry , circumcircle

Two triangles $ABC, A’B’C’$ have the same orthocenter $H$ and the same circumcircle with center $O$. Letting $PQR$ be the triangle formed by $AA’, BB’, CC’$, prove that the circumcenter of $PQR$ lies on $OH$.

2023/2024 Tournament of Towns, 2

Tags: number theory

For which maximal $N$ there exists an $N$-digit number with the following property: among any sequence of its consecutive decimal digits some digit is present once only? Alexey Glebov

2006 Stanford Mathematics Tournament, 18

Tags: probability

Alex and Brian take turns shooting free throws until they each shoot twice. Alex and Brian have 80% and 60% chances of making their free throws, respectively. What is the probability that after each free throw they take, Alex has made at least as many free throws as Brian if Brian shoots first?

2022 Austrian Junior Regional Competition, 2

Tags: tiles , tiling , combinatorics

You are given a rectangular playing field of size $13 \times 2$ and any number of dominoes of sizes $2\times 1$ and $3\times 1$. The playing field should be seamless with such dominoes and without overlapping, with no domino protruding beyond the playing field may. Furthermore, all dominoes must be aligned in the same way, i. e. their long sides must be parallel to each other. How many such coverings are possible? (Walther Janous)