Olympiad problems

2012 German National Olympiad, 2

Find the maximal number of edges a connected graph $G$ with $n$ vertices may have, so that after deleting an arbitrary cycle, $G$ is not connected anymore.

2023 VN Math Olympiad For High School Students, Problem 1

Tags: geometry

Given a triangle $ABC$ with $AD$ is the $A-$symmedian $(D$ is on the side $BC).$ Prove that: $\dfrac{DB}{DC}=\dfrac{AB^2}{AC^2}.$

2004 Bulgaria Team Selection Test, 3

Tags: inradius , geometry

Find the maximum possible value of the inradius of a triangle whose vertices lie in the interior, or on the boundary, of a unit square.

2017 Canadian Open Math Challenge, B2

Tags:

Source: 2017 Canadian Open Math Challenge, Problem B2 ----- There are twenty people in a room, with $a$ men and $b$ women. Each pair of men shakes hands, and each pair of women shakes hands, but there are no handshakes between a man and a woman. The total number of handshakes is $106$. Determine the value of $a \cdot b$.

VII Soros Olympiad 2000 - 01, 9.8

Tags: geometry , locus , circumcenter , angle

Given a triangle $ABC$. On its sides $BC$ , $CA$ and $AB$ , the points $A_1$ , $B_1$ and $C_1$ are taken, respectively , such that $2 \angle B_1 A_1 C_1 + \angle BAC = 180^o$ , $2 \angle A_1 C_1 B_1 + \angle ACB = 180^o$ , $2 \angle C_1 B_1 A_1 + \angle CBA = 180^o$ . Find the locus of the centers of the circles circumscribed about the triangles $A_1 B_1 C_1$ (all possible such triangles are considered).

Kyiv City MO Juniors 2003+ geometry, 2018.9.5

Tags: geometry , bisects segment , equal angles

Given a triangle $ABC$, the perpendicular bisector of the side $AC$ intersects the angle bisector of the triangle $AK$ at the point $P$, $M$ - such a point that $\angle MAC = \angle PCB$, $\angle MPA = \angle CPK$, and points $M$ and $K$ lie on opposite sides of the line $AC$. Prove that the line $AK$ bisects the segment $BM$. (Anton Trygub)

2013 China Northern MO, 8

Tags: combinatorics , graph theory

$3n$ ($n \ge 2, n \in N$) people attend a gathering, in which any two acquaintances have exactly $n$ common acquaintances, and any two unknown people have exactly $2n$ common acquaintances. If three people know each other, it is called a [i]Taoyuan Group[/i]. (1) Find the number of all Taoyuan groups; (2) Prove that these $3n$ people can be divided into three groups, with $n$ people in each group, and the three people obtained by randomly selecting one person from each group constitute a Taoyuan group. Note: Acquaintance means that two people know each other, otherwise they are not acquaintances. Two people who know each other are called acquaintances.

1978 AMC 12/AHSME, 15

Tags: trigonometry , function , quadratic

If $\sin x+\cos x=1/5$ and $0\le x<\pi$, then $\tan x$ is $\textbf{(A) }-\frac{4}{3}\qquad\textbf{(B) }-\frac{3}{4}\qquad\textbf{(C) }\frac{3}{4}\qquad\textbf{(D) }\frac{4}{3}\qquad$ $\textbf{(E) }\text{not completely determined by the given information}$

2019 JBMO Shortlist, G6

Tags: geometry , incenter

Let $ABC$ be a non-isosceles triangle with incenter $I$. Let $D$ be a point on the segment $BC$ such that the circumcircle of $BID$ intersects the segment $AB$ at $E\neq B$, and the circumcircle of $CID$ intersects the segment $AC$ at $F\neq C$. The circumcircle of $DEF$ intersects $AB$ and $AC$ at the second points $M$ and $N$ respectively. Let $P$ be the point of intersection of $IB$ and $DE$, and let $Q$ be the point of intersection of $IC$ and $DF$. Prove that the three lines $EN, FM$ and $PQ$ are parallel. [i]Proposed by Saudi Arabia[/i]

BIMO 2022, 2

Tags: geometry

Let $ABCD$ be a circumscribed quadrilateral with incircle $\gamma$. Let $AB\cap CD=E, AD\cap BC=F, AC\cap EF=K, BD\cap EF=L$. Let a circle with diameter $KL$ intersect $\gamma$ at one of the points $X$. Prove that $(EXF)$ is tangent to $\gamma$.

2010 Junior Balkan Team Selection Tests - Romania, 4

Tags: equal segments , equal angles , isosceles , angle , geometry

Let $ABC$ be an isosceles triangle with $AB = AC$ and let $n$ be a natural number, $n>1$. On the side $AB$ we consider the point $M$ such that $n \cdot AM = AB$. On the side $BC$ we consider the points $P_1, P_2, ....., P_ {n-1}$ such that $BP_1 = P_1P_2 = .... = P_ {n-1} C = \frac{1}{n} BC$. Show that: $\angle {MP_1A} + \angle {MP_2A} + .... + \angle {MP_ {n-1} A} = \frac{1} {2} \angle {BAC}$.

2022 Kyiv City MO Round 2, Problem 4

Tags: geometry

Let $\omega$ denote the circumscribed circle of triangle $ABC$, $I$ be its incenter, and $K$ be any point on arc $AC$ of $\omega$ not containing $B$. Point $P$ is symmetric to $I$ with respect to point $K$. Point $T$ on arc $AC$ of $\omega$ containing point $B$ is such that $\angle KCT = \angle PCI$. Show that the bisectors of angles $AKC$ and $ATC$ meet on line $CI$. [i](Proposed by Anton Trygub)[/i]

2014 Junior Regional Olympiad - FBH, 2

Tags: percent

We know that raw wheat has $70\%$ moisture and dry wheat has $10\%$ moisture. One miller bought $3$ tons of raw wheat with price of $0.4 \$$ per kilo. At which price miller has to sell dry wheat, so he gets $80\%$ profit?

1999 Estonia National Olympiad, 2

Tags: sum , algebra

Find the value of the expression $$f\left( \frac{1}{2000} \right)+f\left( \frac{2}{2000} \right)+...+ f\left( \frac{1999}{2000} \right)+f\left( \frac{2000}{2000} \right)+f\left( \frac{2000}{1999} \right)+...+f\left( \frac{2000}{1} \right)$$ assuming $f(x) =\frac{x^2}{1 + x^2}$ .

2018 ELMO Shortlist, 2

Tags: number theory

Call a number $n$ [i]good[/i] if it can be expressed as $2^x+y^2$ for where $x$ and $y$ are nonnegative integers. (a) Prove that there exist infinitely many sets of $4$ consecutive good numbers. (b) Find all sets of $5$ consecutive good numbers. [i]Proposed by Michael Ma[/i]

2016 Turkey Team Selection Test, 7

Tags: arithmetic sequence , combinatorics

$A_1, A_2,\dots A_k$ are different subsets of the set $\{1,2,\dots ,2016\}$. If $A_i\cap A_j$ forms an arithmetic sequence for all $1\le i <j\le k$, what is the maximum value of $k$?

2005 IMO Shortlist, 8

Tags: diagonal , combinatorics , extremal combinatorics , coloring , polygon

Suppose we have a $n$-gon. Some $n-3$ diagonals are coloured black and some other $n-3$ diagonals are coloured red (a side is not a diagonal), so that no two diagonals of the same colour can intersect strictly inside the polygon, although they can share a vertex. Find the maximum number of intersection points between diagonals coloured differently strictly inside the polygon, in terms of $n$. [i]Proposed by Alexander Ivanov, Bulgaria[/i]

2022-2023 OMMC, 17

Tags:

Let $a_1$, $a_2$, $\cdots$ be a sequence such that $a_1=a_2=\frac 15$, and for $n \ge 3$, $$a_n=\frac{a_{n-1}+a_{n-2}}{1+a_{n-1}a_{n-2}}.$$ Find the smallest integer $n$ such that $a_n>1-5^{-2022}$.

2001 Tournament Of Towns, 2

Tags: geometry , 3d geometry , number theory

The decimal expression of the natural number $a$ consists of $n$ digits, while that of $a^3$ consists of $m$ digits. Can $n + m$ be equal to 2001?

2008 District Olympiad, 3

Tags: real analysis

Let $(x_n)_{n\ge 1}$ and $(y_n)_{n\ge 1}$ a sequence of positive real numbers, such that: \[x_{n+1}\ge \frac{x_n+y_n}{2},\ y_{n+1}\ge \sqrt{\frac{x_n^2+y_n^2}{2}},\ (\forall)n\in \mathbb{N}^*\] a) Prove that the sequences $(x_n+y_n)_{n\ge 1}$ and $(x_ny_n)_{n\ge 1}$ have limit. b) Prove that the sequences $(x_n)_{n\ge 1}$ and $(y_n)_{n\ge 1}$ have limit and that their limits are equal.

Novosibirsk Oral Geo Oly IX, 2017.7

Tags: geometry , angle

A car is driving along a straight highway at a speed of $60$ km per hour. Not far from the highway there is a parallel to him a $100$-meter fence. Every second, the passenger of the car measures the angle at which the fence is visible. Prove that the sum of all the angles he measured is less than $1100^o$

2005 Slovenia National Olympiad, Problem 3

Tags: square , circumcircle , geometry

Let $T$ be a point inside a square $ABCD$. The lines $TA,TB,TC,TD$ meet the circumcircle of $ABCD$ again at $A',B',C',D'$, respectively. Prove that $A'B'\cdot C'D'=A'D'\cdot B'C'$.

2018-IMOC, G4

Tags: geometry , circumcircle , incenter , parallel

Given an acute $\vartriangle ABC$ with incenter $I$. Let $I'$ be the symmetric point $I$ with respect to the midpoint of $B,C$ and $D$ is the foot of $A$. If $DI$ and the circumcircle of vartriangle $BI'C$ intersect at $T$ and $TI' $ intersects the circumcircle of $\vartriangle ATI$ at $X$. Furthermore, $E,F$ are tangent points of the incircle and $AB,AC, P$ is the another intersection of the circumcircles of $\vartriangle ABC, \vartriangle AEF$. Show that $AX \parallel PI$. [img]https://3.bp.blogspot.com/-tj9A8HIR6Vw/XndLEPMRvnI/AAAAAAAALfk/2vw_pZbhpnkTKIc1BcKf4K7SNZ11vu4TACK4BGAYYCw/s1600/2018%2Bimoc%2Bg4.png[/img]

2014 PUMaC Team, 13

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There is a right triangle $\triangle ABC$ in which $\angle A$ is the right angle. On side $AB$, there are three points $X$, $Y$, and $Z$ that satisfy $\angle ACX=\angle XCY=\angle YCZ=\angle ZCB$ and $BZ=2AX$. The smallest angle of $\triangle ABC$ is $\tfrac ab$ degrees, where $a,b$ are positive integers such that $\gcd(a,b)=1$. Find $a+b$.

2020 Harvard-MIT Mathematics Tournament, 1

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Let $P(x)=x^3+x^2-r^2x-2020$ be a polynomial with roots $r,s,t$. What is $P(1)$? [i]Proposed by James Lin.[/i]