Olympiad problems

2008 Costa Rica - Final Round, 2

Let $ ABC$ be a triangle and let $ P$ be a point on the angle bisector $ AD$, with $ D$ on $ BC$. Let $ E$, $ F$ and $ G$ be the intersections of $ AP$, $ BP$ and $ CP$ with the circumcircle of the triangle, respectively. Let $ H$ be the intersection of $ EF$ and $ AC$, and let $ I$ be the intersection of $ EG$ and $ AB$. Determine the geometric place of the intersection of $ BH$ and $ CI$ when $ P$ varies.

2018-2019 Fall SDPC, 7

Tags: geometry

The incircle of $\triangle{ABC}$ touches $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. Point $P$ is chosen on $EF$ such that $AP$ is parallel to $BC$, and $AD$ intersects the incircle of $\triangle{ABC}$ again at $G$. Show that $\angle AGP = 90^{\circ}$.

2019 All-Russian Olympiad, 2

Tags: trigonometry , algebra

Is it true, that for all pairs of non-negative integers $a$ and $b$ , the system \begin{align*} \tan{13x} \tan{ay} =& 1 \\ \tan{21x} \tan{by}= & 1 \end{align*} has at least one solution?

1961 Putnam, B5

Tags: factorial , inequalities

Let $k$ be a positive integer, and $n$ a positive integer greater than $2$. Define $$f_{1}(n)=n,\;\; f_{2}(n)=n^{f_{1}(n)},\;\ldots\;, f_{j+1}(n)=n^{f_{j}(n)}.$$ Prove either part of the inequality $$f_{k}(n) < n!! \cdots ! < f_{k+1}(n),$$ where the middle term has $k$ factorial symbols.

2009 China Team Selection Test, 3

Tags: polynomial , algebra , induction , modular arithmetic , binomial coefficient

Let $ f(x)$ be a $ n \minus{}$degree polynomial all of whose coefficients are equal to $ \pm 1$, and having $ x \equal{} 1$ as its $ m$ multiple root. If $ m\ge 2^k (k\ge 2,k\in N)$, then $ n\ge 2^{k \plus{} 1} \minus{} 1.$

2014-2015 SDML (High School), 1

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Larry always orders pizza with exactly two of his three favorite toppings: pepperoni, bacon, and sausage. If he has ordered a total of $600$ pizzas and has had each topping equally often, how many pizzas has he ordered with pepperoni? $\text{(A) }200\qquad\text{(B) }300\qquad\text{(C) }400\qquad\text{(D) }500\qquad\text{(E) }600$

2022 Francophone Mathematical Olympiad, 1

Tags: algebra , function , functional equation

find all functions $f:\mathbb{Z} \to \mathbb{Z} $ such that $f(m+n)+f(m)f(n)=n^2(f(m)+1)+m^2(f(n)+1)+mn(2-mn)$ holds for all $m,n \in \mathbb{Z}$

1999 Italy TST, 3

Tags: function , algebra

(a) Find all strictly monotone functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that \[f(x+f(y))=f(x)+y\quad\text{for all real}\ x,y. \] (b) If $n>1$ is an integer, prove that there is no strictly monotone function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that \[ f(x+f(y))=f(x)+y^n\quad \text{for all real}\ x, y.\]

1996 Putnam, 2

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Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be circles whose centers are $10$ units apart, and whose radii are $1$ and $3$. Find, with proof, the locus of all points $M$ for which there exists points $X\in \mathcal{C}_1,Y\in \mathcal{C}_2$ such that $M$ is the midpoint of $XY$.

2019 Bosnia and Herzegovina Junior BMO TST, 1

Tags: algebra

Let $x,y,z$ be real numbers ( $x \ne y$, $y\ne z$, $x\ne z$) different from $0$. If $\frac{x^2-yz}{x(1-yz)}=\frac{y^2-xz}{y(1-xz)}$, prove that the following relation holds: $$x+y+z=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$$

2013 239 Open Mathematical Olympiad, 2

Tags: number theory

For some $99$-digit number $k$, there exist two different $100$-digit numbers $n$ such that the sum of all natural numbers from $1$ to $n$ ends in the same $100$ digits as the number $kn$, but is not equal to it. Prove that $k-3$ is divisible by $5$.

2021 JHMT HS, 6

Tags: angle bisector , geometry

Suppose $JHMT$ is a convex quadrilateral with perimeter $68$ and satisfies $\angle HJT = 120^\circ,$ $HM = 20,$ and $JH + JT = JM > HM.$ Furthermore, $\overrightarrow{JM}$ bisects $\angle HJT.$ Compute $JM.$

2005 USAMTS Problems, 5

Tags: geometry , 3d geometry , sphere

Sphere $S$ is inscribed in cone $C$. The height of $C$ equals its radius, and both equal $12+12\sqrt2$. Let the vertex of the cone be $A$ and the center of the sphere be $B$. Plane $P$ is tangent to $S$ and intersects $\overline{AB}$. $X$ is the point on the intersection of $P$ and $C$ closest to $A$. Given that $AX=6$, find the area of the region of $P$ enclosed by the intersection of $C$ and $P$.

2004 Greece JBMO TST, 3

Tags: number theory , digit

If in a $3$-digit number we replace with each other it's last two digits, and add the resulting number to the starting one, we find sum a $4$-digit number that starts with $173$. Which is the starting number?

2009 China Northern MO, 3

Tags: number theory , divide

Given $26$ different positive integers , in any six numbers of the $26$ integers , there are at least two numbers , one can be devided by another. Then prove : There exists six numbers , one of them can be devided by the other five numbers .

1960 Miklós Schweitzer, 1

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[b]1.[/b] Consider in the plane a set $H$ of pairwise disjoint circles of radius 1 such that, for infinitely many positive integers $n$, the circle $k_n$ with centre at the origin and of radius $n$ contains at least $cn^2$ elements of the set $H$. Prove that there exists a straight line which intersects infinitely many of the circles of $H$. Show further that if we require only that the circles $k_n$ contain o(n²) elements of $H$, the proposition will be false. [b](G. 5)[/b]

2014 AMC 8, 5

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Margie's car can go $32$ miles on a gallon of gas, and gas currently costs $\$4$ per gallon. How many miles can Margie drive on $\$20$ worth of gas? $\textbf{(A) }64\qquad\textbf{(B) }128\qquad\textbf{(C) }160\qquad\textbf{(D) }320\qquad \textbf{(E) }640$

2005 Today's Calculation Of Integral, 63

Tags: trigonometry , calculus , integration , limit , calculus computations

For a positive number $x$, let $f(x)=\lim_{n\to\infty} \sum_{k=1}^n \left|\cos \left(\frac{2k+1}{2n}x\right)-\cos \left(\frac{2k-1}{2n}x\right)\right|$ Evaluate \[\lim_{x\rightarrow\infty}\frac{f(x)}{x}\]

1977 All Soviet Union Mathematical Olympiad, 244

Tags: number theory , perfect square

Let us call "fine" the $2n$-digit number if it is exact square itself and the two numbers represented by its first $n$ digits (first digit may not be zero) and last $n$ digits (first digit may be zero, but it may not be zero itself) are exact squares also. a) Find all two- and four-digit fine numbers. b) Is there any six-digit fine number? c) Prove that there exists $20$-digit fine number. d) Prove that there exist at least ten $100$-digit fine numbers. e) Prove that there exists $30$-digit fine number.

2000 IMO Shortlist, 6

Tags: number theory , modular arithmetic , combinatorics , additive number theory , additive combinatorics , frobenius

Let $ p$ and $ q$ be relatively prime positive integers. A subset $ S$ of $ \{0, 1, 2, \ldots \}$ is called [b]ideal[/b] if $ 0 \in S$ and for each element $ n \in S,$ the integers $ n \plus{} p$ and $ n \plus{} q$ belong to $ S.$ Determine the number of ideal subsets of $ \{0, 1, 2, \ldots \}.$

2008 VJIMC, Problem 2

Tags: algebra , sequence , fe , functional equation , recurrence relation

Find all functions $f:(0,\infty)\to(0,\infty)$ such that $$f(f(f(x)))+4f(f(x))+f(x)=6x.$$

2021 Princeton University Math Competition, A1 / B3

Tags: number theory

Compute the remainder when $2^{3^5}+ 3^{5^2}+ 5^{2^3}$ is divided by $30$.

2021 China Team Selection Test, 6

Tags: geometry

Find the smallest positive real constant $a$, such that for any three points $A,B,C$ on the unit circle, there exists an equilateral triangle $PQR$ with side length $a$ such that all of $A,B,C$ lie on the interior or boundary of $\triangle PQR$.

PEN H Problems, 5

Tags: diophantine equation

Find all pairs $(x, y)$ of rational numbers such that $y^2 =x^3 -3x+2$.

2024 Harvard-MIT Mathematics Tournament, 2

Tags: combinatorics

A [i]lame king[/i] is a chess piece that can move from a cell to any cell that shares at least one vertex with it, except for the cells in the same column as the current cell. A lame king is placed in the top-left cell of a $7\times 7$ grid. Compute the maximum number of cells it can visit without visiting the same cell twice (including its starting cell).