Olympiad problems

2006 BAMO, 3

In triangle $ABC$, choose point $A_1$ on side $BC$, point $B_1$ on side $CA$, and point $C_1$ on side $AB$ in such a way that the three segments $AA_1, BB_1$, and $CC_1$ intersect in one point $P$. Prove that $P$ is the centroid of triangle $ABC$ if and only if $P$ is the centroid of triangle $A_1B_1C_1$. Note: A median in a triangle is a segment connecting a vertex of the triangle with the midpoint of the opposite side. The centroid of a triangle is the intersection point of the three medians of the triangle. The centroid of a triangle is also known by the names ”center of mass” and ”medicenter” of the triangle.

2007 Bulgarian Autumn Math Competition, Problem 12.2

Tags: 3d geometry , geometry , pyramid , trapezoid , plane

All edges of the triangular pyramid $ABCD$ are equal in length. Let $M$ be the midpoint of $DB$, $N$ is the point on $\overline{AB}$, such that $2NA=NB$ and $N\not\in AB$ and $P$ is a point on the altitude through point $D$ in $\triangle BCD$. Find $\angle MPD$ if the intersection of the pyramid with the plane $(NMP)$ is a trapezoid.

2020 Nigerian MO round 3, #4

Tags: number theory , diophantine equation

let $p$and $q=p+2$ be twin primes. consider the diophantine equation $(+)$ given by $n!+pq^2=(mp)^2$ $m\geq1$, $n\geq1$ i. if $m=p$,find the value of $p$. ii. how many solution quadruple $(p,q,m,n)$ does $(+)$ have ?

2021 EGMO, 4

Tags: geometry , reflection , triangle

Let $ABC$ be a triangle with incenter $I$ and let $D$ be an arbitrary point on the side $BC$. Let the line through $D$ perpendicular to $BI$ intersect $CI$ at $E$. Let the line through $D$ perpendicular to $CI$ intersect $BI$ at $F$. Prove that the reflection of $A$ across the line $EF$ lies on the line $BC$.

2016 IFYM, Sozopol, 5

Tags: geometry

A convex quadrilateral is cut into smaller convex quadrilaterals so that they are adjacent to each other only by whole sides. a) Prove that if all small quadrilaterals are inscribed in a circle, then the original one is also inscribed in a circle. b) Prove that if all small quadrilaterals are cyclic, then the original one is also cyclic.

BIMO 2022, 2

Tags: number theory

Given a four digit string $ k=\overline{abcd} $, $ a, b, c, d\in \{0, 1, \cdots, 9\} $, prove that there exist a $n<20000$ such that $2^n$ contains $k$ as a substring when written in base $10$. [Extra: Can you give a better bound? Mine is $12517$]

2024 USAMTS Problems, 3

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Let $a, b$ be positive integers such that $a^2 \ge b$. Let $x = \sqrt{a+\sqrt{b}} - \sqrt{a-\sqrt{b}}$ (a) Prove that for all integers $a \ge 2$, there exists a positive integer $b$ such that $x$ is also a positive integer. (b) Prove that for all sufficiently large $a$, there are at least two $b$ such that $x$ is a positive integer. $\textbf{Note}$: We’ve received some questions about what is meant by “for all sufficiently large $a$.” To give a simple example of this phrasing, it is true that for all sufficiently large positive integers $n$, we have $n^2 \ge 100$. Specifically, this is true for all $n \ge 10$.

2005 Taiwan TST Round 2, 2

Tags: inequalities

Find all positive integers $n \ge 3$ such that there exists a positive constant $M_n$ satisfying the following inequality for any $n$ positive reals $a_1, a_2,\dots\>,a_n$: \[\displaystyle \frac{a_1+a_2+\cdots\>+a_n}{\sqrt[n]{a_1a_2\cdots\>a_n}} \le M_n \biggl( \frac{a_2}{a_1} + \frac{a_3}{a_2} +\cdots\>+ \frac{a_n}{a_{n-1}} + \frac {a_1}{a_n} \biggr).\] Moreover, find the minimum value of $M_n$ for such $n$. The difficulty is finding $M_n$...

2019 India IMO Training Camp, P2

Tags: geometry

Let $ABC$ be an acute-angled scalene triangle with circumcircle $\Gamma$ and circumcenter $O$. Suppose $AB < AC$. Let $H$ be the orthocenter and $I$ be the incenter of triangle $ABC$. Let $F$ be the midpoint of the arc $BC$ of the circumcircle of triangle $BHC$, containing $H$. Let $X$ be a point on the arc $AB$ of $\Gamma$ not containing $C$, such that $\angle AXH = \angle AFH$. Let $K$ be the circumcenter of triangle $XIA$. Prove that the lines $AO$ and $KI$ meet on $\Gamma$. [i]Proposed by Anant Mudgal[/i]

2008 ITest, 22

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Tony plays a game in which he takes $40$ nickels out of a roll and tosses them one at a time toward his desk where his change jar sits. He awards himself $5$ points for each nickel that lands in the jar, and takes away $2$ points from his score for each nickel that hits the ground. After Tony is done tossing all $40$ nickels, he computes $88$ as his score. Find the greatest number of nickels he could have successfully tossed into the jar.

2022 Argentina National Olympiad Level 2, 6

Tags: combinatorics

In a hockey tournament, there is an odd number $n$ of teams. Each team plays exactly one match against each of the other teams. In this tournament, each team receives $2$ points for a win, $1$ point for a draw, and $0$ points for a loss. At the end of the tournament, it was observed that all the points obtained by the $n$ teams were different. For each $n$, determine the maximum possible number of draws that could have occurred in this tournament.

2017 CMIMC Algebra, 10

Tags: algebra , polynomial

Let $c$ denote the largest possible real number such that there exists a nonconstant polynomial $P$ with \[P(z^2)=P(z-c)P(z+c)\] for all $z$. Compute the sum of all values of $P(\tfrac13)$ over all nonconstant polynomials $P$ satisfying the above constraint for this $c$.

2003 Vietnam National Olympiad, 3

Tags: function , induction , limit , algebra

Let $\mathcal{F}$ be the set of all functions $f : (0,\infty)\to (0,\infty)$ such that $f(3x) \geq f( f(2x) )+x$ for all $x$. Find the largest $A$ such that $f(x) \geq A x$ for all $f\in\mathcal{F}$ and all $x$.

1997 Moscow Mathematical Olympiad, 4

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Along a circular railroad, $n$ trains circulate in the same direction at equal distances between them. Stations $A, B$ and $C$ on this railroad (denoted as the trains pass them) form an equilateral triangle. Ira enters station $A$ at the same time as Alex enters station $B$ in order to take the nearest train. It is knows that if they enter the stations at the same time as the driver Roma passes a forest, then Ira takes her train earlier than Alex; otherwise Alex takes the train earlier than or simultaneously with Ira. What part of the railroad goes through the forest (between which stations)?

2002 Irish Math Olympiad, 3

Tags: number theory

Find all triples of positive integers $ (p,q,n)$, with $ p$ and $ q$ primes, satisfying: $ p(p\plus{}3)\plus{}q(q\plus{}3)\equal{}n(n\plus{}3)$.

2021 CMIMC, 1.5

Tags: algebra

Suppose $f$ is a degree 42 polynomial such that for all integers $0\le i\le 42$, $$f(i)+f(43+i)+f(2\cdot43+i)+\cdots+f(46\cdot43+i)=(-2)^i$$ Find $f(2021)-f(0)$. [i]Proposed by Adam Bertelli[/i]

1953 AMC 12/AHSME, 28

Tags: ratio , geometry , angle bisector

In triangle $ ABC$, sides $ a,b$ and $ c$ are opposite angles $ A,B$ and $ C$ respectively. $ AD$ bisects angle $ A$ and meets $ BC$ at $ D$. Then if $ x \equal{} \overline{CD}$ and $ y \equal{} \overline{BD}$ the correct proportion is: $ \textbf{(A)}\ \frac {x}{a} \equal{} \frac {a}{b \plus{} c} \qquad\textbf{(B)}\ \frac {x}{b} \equal{} \frac {a}{a \plus{} c} \qquad\textbf{(C)}\ \frac {y}{c} \equal{} \frac {c}{b \plus{} c} \\ \textbf{(D)}\ \frac {y}{c} \equal{} \frac {a}{b \plus{} c} \qquad\textbf{(E)}\ \frac {x}{y} \equal{} \frac {c}{b}$

2010 Saudi Arabia IMO TST, 2

Tags: functional equation , functional , algebra

Find all functions $f,g : N \to N$ such that for all $m ,n \in N$ the following relation holds: $$f(m ) - f(n) = (m - n)(g(m) + g(n))$$. Note: $N = \{0,1,2,...\}$

2017 AIME Problems, 13

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For each integer $n\ge 3$, let $f(n)$ be the number of 3-element subsets of the vertices of a regular $n$-gon that are the vertices of an isosceles triangle (including equilateral triangles). Find the sum of all values of $n$ such that $f(n+1)=f(n)+78$.

III Soros Olympiad 1996 - 97 (Russia), 11.6

Tags: inequalities , analytic geometry , logarithm

On the coordinate plane, draw a set of points $M(x,y)$, the coordinates of which satisfy the inequality $$\log_{x+y} (x^2+y^2) \le 1.$$

Croatia MO (HMO) - geometry, 2013.3

Tags: geometry , parallelogram , bisects , cyclic quadrilateral , concyclic , bisects segment

Given a pointed triangle $ABC$ with orthocenter $H$. Let $D$ be the point such that the quadrilateral $AHCD$ is parallelogram. Let $p$ be the perpendicular to the direction $AB$ through the midpoint $A_1$ of the side $BC$. Denote the intersection of the lines $p$ and $AB$ with $E$, and the midpoint of the length $A_1E$ with $F$. The point where the parallel to the line $BD$ through point $A$ intersects $p$ denote by $G$. Prove that the quadrilateral $AFA_1C$ is cyclic if and only if the lines $BF$ passes through the midpoint of the length $CG$.

1991 AMC 12/AHSME, 17

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A positive integer $N$ is a [i]palindrome[/i] if the integer obtained by reversing the sequence of digits of $N$ is equal to $N$. The year 1991 is the only year in the current century with the following two properties: (a) It is a palindrome (b) It factors as a product of a 2-digit prime palindrome and a 3-digit prime palindrome. How many years in the millennium between 1000 and 2000 (including the year 1991) have properties (a) and (b)? $ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 $

2009 BAMO, 1

Tags: combinatorial geometry , combinatorics

A square grid of $16$ dots (see the figure) contains the corners of nine $1\times1$ squares, four $2\times 2$ squares, and one $3\times3$ square, for a total of $14$ squares whose sides are parallel to the sides of the grid. What is the smallest possible number of dots you can remove so that, after removing those dots, each of the $14$ squares is missing at least one corner? Justify your answer by showing both that the number of dots you claim is sufficient and by explaining why no smaller number of dots will work. [img]https://cdn.artofproblemsolving.com/attachments/0/9/bf091a769dbec40eceb655f5588f843d4941d6.png[/img]

2012 Today's Calculation Of Integral, 855

Tags: calculus , integration , function , limit , calculus computations

Let $f(x)$ be a function which is differentiable twice and $f''(x)>0$ on $[0,\ 1]$. For a positive integer $n$, find $\lim_{n\to\infty} n\left\{\int_0^1 f(x)\ dx-\frac{1}{n}\sum_{k=0}^{n-1} f\left(\frac{k}{n}\right)\right\}.$

MBMT Guts Rounds, 2015.20

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How many lattice points are exactly twice as close to $(0,0)$ as they are to $(15,0)$? (A lattice point is a point $(a,b)$ such that both $a$ and $b$ are integers.)