Olympiad problems

2023 Purple Comet Problems, 5

Tags: number theory

Positive integers $m$ and $n$ satisfy $$(m + n)(24mn + 1) = 2023.$$ Find $m + n + 12mn$.

2011 Romania National Olympiad, 1

Tags: abstract algebra , ring theory

Prove that a ring that has a prime characteristic admits nonzero nilpotent elements if and only if its characteristic divides the number of its units.

2013 Junior Balkan Team Selection Tests - Romania, 3

Tags: geometry

Let $ABCD$ be a cyclic quadrilateral and $\omega_1, \omega_2$ the incircles of triangles $ABC$ and $BCD$. Show that the common external tangent line of $\omega_1$ and $\omega_2$, the other one than $BC$, is parallel with $AD$

2005 Lithuania Team Selection Test, 2

Tags: geometric transformation , reflection , geometry

Let $ABCD$ be a convex quadrilateral, and write $\alpha=\angle DAB$; $\beta=\angle ADB$; $\gamma=\angle ACB$; $\delta= \angle DBC$; and $\epsilon=\angle DBA$. Assuming that $\alpha<\pi/2$, $\beta+\gamma=\pi /2$, and $\delta+2\epsilon=\pi$, prove that \[(DB+BC)^2=AD^2+AC^2\] [color=red][Moderator edit: Also discussed at http://www.mathlinks.ro/Forum/viewtopic.php?t=30569 .][/color]

Russian TST 2020, P2

Tags: geometry , triangle

Let $P$ be a point inside triangle $ABC$. Let $AP$ meet $BC$ at $A_1$, let $BP$ meet $CA$ at $B_1$, and let $CP$ meet $AB$ at $C_1$. Let $A_2$ be the point such that $A_1$ is the midpoint of $PA_2$, let $B_2$ be the point such that $B_1$ is the midpoint of $PB_2$, and let $C_2$ be the point such that $C_1$ is the midpoint of $PC_2$. Prove that points $A_2, B_2$, and $C_2$ cannot all lie strictly inside the circumcircle of triangle $ABC$. (Australia)

2022/2023 Tournament of Towns, P3

Tags: geometry

Let $I{}$ be the incenter of triangle $ABC{}.$ Let $N{}$ be the foot of the bisector of angle $B{}.$ The tangent line to the circumcircle of triangle $AIN$ at $A{}$ and the tangent line to the circumcircle of triangle $CIN{}$ at $C{}$ intersect at $D{}.$ Prove that lines $AC{}$ and $DI$ are perpendicular. [i]Mikhail Evdokimov[/i]

2000 All-Russian Olympiad, 8

Tags: combinatorics

All points in a $100 \times 100$ array are colored in one of four colors red, green, blue or yellow in such a way that there are $25$ points of each color in each row and in any column. Prove that there are two rows and two columns such that their four intersection points are all in different colors.

2001 National High School Mathematics League, 7

Tags: ellipse , conic

The length of minor axis of ellipse $\rho-\frac{1}{2-\cos\theta}$ is________.

2021 Regional Olympiad of Mexico West, 5

Tags: perpendicular , geometry

Let $ABC$ be a triangle such that $AC$ is its shortest side. A point $P$ is inside it and satisfies that $BP = AC$. Let $R$ be the midpoint of $BC$ and let $M$ be the midpoint of $AP$. Let $E$ be the intersection of $BP$ and $AC$. Prove that the bisector of angle $\angle BE A$ is perpendicular to segment $MR$.

2023 BMT, 2

Tags: combinatorics

Jerry has red blocks, yellow blocks, and blue blocks. He builds a tower $5$ blocks high, without any $2$ blocks of the same color touching each other. Also, if the tower is flipped upside-down, it still looks the same. Compute the number of ways Jerry could have built this tower.

2020 Azerbaijan National Olympiad, 3

Tags: algebra , inequalities

$a,b,c$ are positive numbers.$a+b+c=3$ Prove that: $\sum \frac{a^2+6}{2a^2+2b^2+2c^2+2a-1}\leq 3 $

2005 All-Russian Olympiad, 2

Tags: ceiling function , induction , combinatorics

Given 2005 distinct numbers $a_1,\,a_2,\dots,a_{2005}$. By one question, we may take three different indices $1\le i<j<k\le 2005$ and find out the set of numbers $\{a_i,\,a_j,\,a_k\}$ (unordered, of course). Find the minimal number of questions, which are necessary to find out all numbers $a_i$.

2013 Purple Comet Problems, 27

Tags: algebra , polynomial

Suppose $a,b$ and $c$ are real numbers that satisfy $a+b+c=5$ and $\tfrac{1}{a}+\tfrac{1}{b}+\tfrac{1}{c}=\tfrac15$. Find the greatest possible value of $a^3+b^3+c^3$.

1984 AIME Problems, 6

Tags: geometry , analytic geometry , graphing lines , slope , geometric transformation

Three circles, each of radius 3, are drawn with centers at $(14,92)$, $(17,76)$, and $(19,84)$. A line passing through $(17,76)$ is such that the total area of the parts of the three circles to one side of the line is equal to the total area of the parts of the three circles to the other side of it. What is the absolute value of the slope of this line?

1966 IMO Longlists, 27

Tags: geometry , construction

Given a point $P$ lying on a line $g,$ and given a circle $K.$ Construct a circle passing through the point $P$ and touching the circle $K$ and the line $g.$

1992 Tournament Of Towns, (320) 1

Tags: algebra , inequalities

At the beginning of a month a shop has $10$ different products for sale, each with equal prices. Every day the price of each product is either doubled or trebled. By the beginning of the following month all the prices have become different. Prove that the ratio (the maximal price) /(the minimal price) is greater than $27$. (D. Fomin and Stanislav Smirnov, St Petersburg)

2012 Tournament of Towns, 4

Tags: combinatorics , square table , table , signs

Each entry in an $n\times n$ table is either $+$ or $-$. At each step, one can choose a row or a column and reverse all signs in it. From the initial position, it is possible to obtain the table in which all signs are $+$. Prove that this can be accomplished in at most $n$ steps.

2016 AIME Problems, 15

Tags: circles

Circles $\omega_1$ and $\omega_2$ intersect at points $X$ and $Y$. Line $\ell$ is tangent to $\omega_1$ and $\omega_2$ at $A$ and $B$, respectively, with line $AB$ closer to point $X$ than to $Y$. Circle $\omega$ passes through $A$ and $B$ intersecting $\omega_1$ again at $D \neq A$ and intersecting $\omega_2$ again at $C \neq B$. The three points $C$, $Y$, $D$ are collinear, $XC = 67$, $XY = 47$, and $XD = 37$. Find $AB^2$.

2017 China Team Selection Test, 2

Tags: graph theory , combinatorics

$2017$ engineers attend a conference. Any two engineers if they converse, converse with each other in either Chinese or English. No two engineers converse with each other more than once. It is known that within any four engineers, there was an even number of conversations and furthermore within this even number of conversations: i) At least one conversation is in Chinese. ii) Either no conversations are in English or the number of English conversations is at least that of Chinese conversations. Show that there exists $673$ engineers such that any two of them conversed with each other in Chinese.

2021 AMC 10 Spring, 19

Tags:

Suppose that $S$ is a finite set of positive integers. If the greatest integer in $S$ is removed from $S$, then the average value (arithmetic mean) of the integers remaining is $32$. If the least integer is $S$ is [i]also[/i] removed, then the average value of the integers remaining is $35$. If the greatest integer is then returned to the set, the average value of the integers rises to $40$. The greatest integer in the original set $S$ is $72$ greater than the least integer in $S$. What is the average value of all the integers in the set $S$? $\textbf{(A)} ~36.2 \qquad\textbf{(B)} ~36.4 \qquad\textbf{(C)} ~36.6 \qquad\textbf{(D)} ~36.8 \qquad\textbf{(E)} ~37$

2016 NIMO Problems, 4

Tags: geometry , rhombus

In rhombus $ABCD$, let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $AD$. If $CN = 7$ and $DM = 24$, compute $AB^2$. [i]Proposed by Andy Liu[/i]

2020 Dutch BxMO TST, 2

Tags: geometry , concyclic , cyclic quadrilateral

In an acute-angled triangle $ABC, D$ is the foot of the altitude from $A$. Let $D_1$ and $D_2$ be the symmetric points of $D$ wrt $AB$ and $AC$, respectively. Let $E_1$ be the intersection of $BC$ and the line through $D_1$ parallel to $AB$ . Let $E_2$ be the intersection of$ BC$ and the line through $D_2$ parallel to $AC$. Prove that $D_1, D_2, E_1$ and $E_2$ on one circle whose center lies on the circumscribed circle of $\vartriangle ABC$.

2007 All-Russian Olympiad, 2

Tags: number theory

$100$ fractions are written on a board, their numerators are numbers from $1$ to $100$ (each once) and denominators are also numbers from $1$ to $100$ (also each once). It appears that the sum of these fractions equals to $a/2$ for some odd $a$. Prove that it is possible to interchange numerators of two fractions so that sum becomes a fraction with odd denominator. [i]N. Agakhanov, I. Bogdanov [/i]

2005 Baltic Way, 1

Tags: ceiling function , algebra

Let $a_0$ be a positive integer. Define the sequence $\{a_n\}_{n \geq 0}$ as follows: if \[ a_n = \sum_{i = 0}^jc_i10^i \] where $c_i \in \{0,1,2,\cdots,9\}$, then \[ a_{n + 1} = c_0^{2005} + c_1^{2005} + \cdots + c_j^{2005}. \] Is it possible to choose $a_0$ such that all terms in the sequence are distinct?

2023 BMT, 8

Tags: geometry

Circle $\omega_1$ is centered at $O_1$ with radius $3$, and circle $\omega_2$ is centered at $O_2$ with radius $2$. Line $\ell$ is tangent to $\omega_1$ and $\omega_2$ at $X$, $Z$, respectively, and intersects segment $\overline{O_1O_2}$ at $Y$ . The circle through $O_1$, $X$, $Y$ has center $O_3$, and the circle through $O_2$, $Y$ , $Z$ has center $O_4$. Given that $O_1O_2 = 13$, find $O_3O_4$.