Olympiad problems

2010 China Northern MO, 7

Find all positive integers $x, y, z$ that satisfy the conditions: $$[x,y,z] =(x,y)+(y,z) + (z,x), x\le y\le z, (x,y,z) = 1$$ The symbols $[m,n]$ and $(m,n)$ respectively represent positive integers, the least common multiple and the greatest common divisor of $m$ and $n$.

2006 Purple Comet Problems, 6

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We draw a radius of a circle. We draw a second radius $23$ degrees clockwise from the first radius. We draw a third radius $23$ degrees clockwise from the second. This continues until we have drawn $40$ radii each $23$ degrees clockwise from the one before it. What is the measure in degrees of the smallest angle between any two of these $40$ radii?

2023 Vietnam National Olympiad, 3

Tags: algebra , inequalities

Find the maximum value of the positive real number $k$ such that the inequality $$\frac{1}{kab+c^2} +\frac{1} {kbc+a^2} +\frac{1} {kca+b^2} \geq \frac{k+3}{a^2+b^2+c^2} $$holds for all positive real numbers $a,b,c$ such that $a^2+b^2+c^2=2(ab+bc+ca).$

PEN A Problems, 109

Tags: divisibility theory

Find all positive integers $a$ and $b$ such that \[\frac{a^{2}+b}{b^{2}-a}\text{ and }\frac{b^{2}+a}{a^{2}-b}\] are both integers.

2010 Portugal MO, 1

Tags: combinatorics

There are several candles of the same size on the Chapel of Bones. On the first day a candle is lit for a hour. On the second day two candles are lit for a hour, on the third day three candles are lit for a hour, and successively, until the last day, when all the candles are lit for a hour. On the end of that day, all the candles were completely consumed. Find all the possibilities for the number of candles.

1991 Tournament Of Towns, (319) 6

Tags: number theory , arithmetic progression , digit

An arithmetical progression (whose difference is not equal to zero) consists of natural numbers without any nines in its decimal notation. (a) Prove that the number of its terms is less than $100$. (b) Give an example of such a progression with $72$ terms. (c) Prove that the number of terms in any such progression does not exceed $72$. (V. Bugaenko and Tarasov, Moscow)

1987 IMO Longlists, 70

Tags: geometry , circumcircle , trigonometry , area of a triangle

In an acute-angled triangle $ABC$ the interior bisector of angle $A$ meets $BC$ at $L$ and meets the circumcircle of $ABC$ again at $N$. From $L$ perpendiculars are drawn to $AB$ and $AC$, with feet $K$ and $M$ respectively. Prove that the quadrilateral $AKNM$ and the triangle $ABC$ have equal areas.[i](IMO Problem 2)[/i] [i]Proposed by Soviet Union.[/i]

2004 Junior Balkan Team Selection Tests - Romania, 3

Tags: number theory , diophantine equation , diophantine

Let $p, q, r$ be primes and let $n$ be a positive integer such that $p^n + q^n = r^2$. Prove that $n = 1$. Laurentiu Panaitopol

2005 Iran MO (3rd Round), 2

Tags: limit , algebra

Suppose $\{x_n\}$ is a decreasing sequence that $\displaystyle\lim_{n \rightarrow\infty}x_n=0$. Prove that $\sum(-1)^nx_n$ is convergent

2012 Junior Balkan Team Selection Tests - Moldova, 1

Tags: inequalities

Let $ 1\leq a,b,c,d,e,f,g,h,k \leq 9 $ and $ a,b,c,d,e,f,g,h,k $ are different integers, find the minimum value of the expression $ E = a*b*c+d*e*f+g*h*k $ and prove that it is minimum.

2021-IMOC, N4

Tags: number theory

There are $m \geq 3$ positive integers, not necessarily distinct, that are arranged in a circle so that any positive integer divides the sum of its neighbours. Show that if there is exactly one $1$, then for any positive integer $n$, there are at most $\phi(n)$ copies of $n$. [i]Proposed By- (usjl, adapted from 2014 Taiwan TST)[/i]

2016 BMT Spring, 7

Tags: algebra , combinatorics

Find the coefficient of $x^2$ in the following polynomial $$(1 -x)^2(1 + 2x)^2(1 - 3x)^2... (1 -11x)^2.$$

2010 Romania National Olympiad, 4

Tags: function , calculus , derivative , integration , limit , real analysis

Let $f:[-1,1]\to\mathbb{R}$ be a continuous function having finite derivative at $0$, and \[I(h)=\int^h_{-h}f(x)\text{ d}x,\ h\in [0,1].\] Prove that a) there exists $M>0$ such that $|I(h)-2f(0)h|\le Mh^2$, for any $h\in [0,1]$. b) the sequence $(a_n)_{n\ge 1}$, defined by $a_n=\sum_{k=1}^n\sqrt{k}|I(1/k)|$, is convergent if and only if $f(0)=0$. [i]Calin Popescu[/i]

2007 ITest, 10

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My grandparents are Arthur, Bertha, Christoph, and Dolores. My oldest grandparent is only $4$ years older than my youngest grandparent. Each grandfather is two years older than his wife. If Bertha is younger than Dolores, what is the difference between Bertha's age and the mean of my grandparents’ ages? $\textbf{(A) }0\hspace{14em}\textbf{(B) }1\hspace{14em}\textbf{(C) }2$ $\textbf{(D) }3\hspace{14em}\textbf{(E) }4\hspace{14em}\textbf{(F) }5$ $\textbf{(G) }6\hspace{14em}\textbf{(H) }7\hspace{14em}\textbf{(I) }8$ $\textbf{(J) }2007$

1974 Putnam, A2

Tags: shortest distance , geometry

A circle stands in a plane perpendicular to the ground and a point $A$ lies in this plane exterior to the circle and higher than its bottom. A particle starting from rest at $A$ slides without friction down an inclined straight line until it reaches the circle. Which straight line allows descent in the shortest time?

1968 Miklós Schweitzer, 10

Tags: advanced fields , geometry , perimeter

Let $ h$ be a triangle of perimeter $ 1$, and let $ H$ be a triangle of perimeter $ \lambda$ homothetic to $ h$. Let $ h_1,h_2,...$ be translates of $ h$ such that , for all $ i$, $ h_i$ is different from $ h_{i\plus{}2}$ and touches $ H$ and $ h_{i\plus{}1}$ (that is, intersects without overlapping). For which values of $ \lambda$ can these triangles be chosen so that the sequence $ h_1,h_2,...$ is periodic? If $ \lambda \geq 1$ is such a value, then determine the number of different triangles in a periodic chain $ h_1,h_2,...$ and also the number of times such a chain goes around the triangle $ H$. [i]L. Fejes-Toth[/i]

1983 Poland - Second Round, 3

Tags: geometry , equal segments

The point $ P $ lies inside the triangle $ ABC $, with $ \measuredangle PAC = \measuredangle PBC $. The points $ L $ and $ M $ are the projections $ P $ onto the lines $ BC $ and $ CA $, respectively, $ D $ is the midpoint of the segment $ AB $. Prove that $ DL = DM $.

2020 BMT Fall, 8

Tags: number theory

By default, iPhone passcodes consist of four base-$10$ digits. However, Freya decided to be unconventional and use hexadecimal (base-$16$) digits instead of base-$10$ digits! (Recall that $10_{16} = 16_{10}$.) She sets her passcode such that exactly two of the hexadecimal digits are prime. How many possible passcodes could she have set?

2013 IMC, 1

Tags: inequalities , trigonometry , quadratic , function , complex analysis

Let $\displaystyle{z}$ be a complex number with $\displaystyle{\left| {z + 1} \right| > 2}$. Prove that $\displaystyle{\left| {{z^3} + 1} \right| > 1}$. [i]Proposed by Walther Janous and Gerhard Kirchner, Innsbruck.[/i]

2016 Gulf Math Olympiad, 4

Tags: combinatorics

4. Suppose that four people A, B, C and D decide to play games of tennis doubles. They might first play the team A and B against the team C and D. Next A and C might play B and D. Finally A and D might play B and C. The advantage of this arrangement is that two conditions are satisfied. (a) Each player is on the same team as each other player exactly once. (b) Each player is on the opposing team to each other player exactly twice. Is it possible to arrange a collection of tennis matches satisfying both condition (a) and condition (b) in the following circumstances? (i) There are five players. (ii) There are seven players. (iii) There are nine players.

2015 Princeton University Math Competition, A2/B3

Tags: number theory

What is the sum of all positive integers $n$ such that $\text{lcm}(2n, n^2) = 14n - 24$?

2015 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: number theory , coprime

For positive integer $n$, find all pairs of coprime integers $p$ and $q$ such that $p+q^2=(n^2+1)p^2+q$

2005 Hungary-Israel Binational, 1

Tags: number theory

Does there exist a sequence of $2005$ consecutive positive integers that contains exactly $25$ prime numbers?

1999 National Olympiad First Round, 35

Tags: pigeonhole principle

Flights are arranged between 13 countries. For $ k\ge 2$, the sequence $ A_{1} ,A_{2} ,\ldots A_{k}$ is said to a cycle if there exist a flight from $ A_{1}$ to $ A_{2}$, from $ A_{2}$ to $ A_{3}$, $ \ldots$, from $ A_{k \minus{} 1}$ to $ A_{k}$, and from $ A_{k}$ to $ A_{1}$. What is the smallest possible number of flights such that how the flights are arranged, there exist a cycle? $\textbf{(A)}\ 14 \qquad\textbf{(B)}\ 53 \qquad\textbf{(C)}\ 66 \qquad\textbf{(D)}\ 79 \qquad\textbf{(E)}\ 156$

2020 USMCA, 29

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Let $ABC$ be a triangle with circumcircle $\Gamma$ and let $D$ be the midpoint of minor arc $BC$. Let $E, F$ be on $\Gamma$ such that $DE \bot AC$ and $DF \bot AB$. Lines $BE$ and $DF$ meet at $G$, and lines $CF$ and $DE$ meet at $H$. Given that $AB = 8, AC = 10$, and $\angle BAC = 60^\circ$, find the area of $BCHG$. [i] Note: this is a modified version of Premier #2 [/i]