Olympiad problems

2006 Finnish National High School Mathematics Competition, 2

Tags: inequalities

Show that the inequality \[3(1 + a^2 + a^4)\geq (1 + a + a^2)^2\] holds for all real numbers $a.$

1996 Iran MO (2nd round), 4

Tags: inequalities , combinatorics

Let $n$ blue points $A_i$ and $n$ red points $B_i \ (i = 1, 2, \ldots , n)$ be situated on a line. Prove that \[\sum_{i,j} A_i B_j \geq \sum_{i<j} A_iA_j + \sum_{i<j} B_iB_j.\]

2014 PUMaC Team, 1

Tags:

The evilest number $666^{666}$ has $1881$ digits. Let $a$ be the sum of digits of $66^{666}$ and let $b$ be the sum of digits of $a$ and let $c$ be the sum of digits of $b$. Find $c$.

2008 China Team Selection Test, 3

Tags: gauss , polynomial , geometry , complex number , combinatorial geometry , algebra

Let $ z_{1},z_{2},z_{3}$ be three complex numbers of moduli less than or equal to $ 1$. $ w_{1},w_{2}$ are two roots of the equation $ (z \minus{} z_{1})(z \minus{} z_{2}) \plus{} (z \minus{} z_{2})(z \minus{} z_{3}) \plus{} (z \minus{} z_{3})(z \minus{} z_{1}) \equal{} 0$. Prove that, for $ j \equal{} 1,2,3$, $\min\{|z_{j} \minus{} w_{1}|,|z_{j} \minus{} w_{2}|\}\leq 1$ holds.

2005 Taiwan National Olympiad, 1

Tags: inequalities , function

Let $a,b,c$ be three positive real numbers such that $abc=1$. Prove that: \[ 1+\frac{3}{a+b+c}\ge{\frac{6}{ab+bc+ca}} . \]

1966 IMO Shortlist, 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 $g.$

2016 NIMO Problems, 6

Tags:

Let $ABC$ be a triangle with $AB=20$, $AC=34$, and $BC=42$. Let $\omega_1$ and $\omega_2$ be the semicircles with diameters $\overline{AB}$ and $\overline{AC}$ erected outwards of $\triangle ABC$ and denote by $\ell$ the common external tangent to $\omega_1$ and $\omega_2$. The line through $A$ perpendicular to $\overline{BC}$ intersects $\ell$ at $X$ and $BC$ at $Y$. The length of $\overline{XY}$ can be written in the form $m+\sqrt n$ where $m$ and $n$ are positive integers. Find $100m+n$. [i]Proposed by David Altizio[/i]

2016 ASDAN Math Tournament, 3

Tags: algebra test

Real numbers $x,y,z$ form an arithmetic sequence satisfying \begin{align*} x+y+z&=6\\ xy+yz+zx&=10. \end{align*} What is the absolute value of their common difference?

2001 Greece Junior Math Olympiad, 4

Tags: geometry , equal segments , angle chasing

Let $ABC$ be a triangle with altitude $AD$ , angle bisectors $AE$ and $BZ$ that intersecting at point $I$. From point $I$ let $IT$ be a perpendicular on $AC$. Also let line $(e)$ be perpendicular on $AC$ at point $A$. Extension of $ET$ intersects line $(e)$ at point $K$. Prove that $AK=AD$.

1996 Baltic Way, 17

Tags: combinatorics

Using each of the eight digits $1,3,4,5,6,7,8$ and $9$ exactly once, a three-digit number $A$, two two-digit numbers $B$ and $C$, $B<C$, and a one digit number $D$ are formed. The numbers are such that $A+D=B+C=143$. In how many ways can this be done?

1984 Tournament Of Towns, (065) A3

Tags: combinatorics

An infinite (in both directions) sequence of rooms is situated on one side of an infinite hallway. The rooms are numbered by consecutive integers and each contains a grand piano. A finite number of pianists live in these rooms. (There may be more than one of them in some of the rooms.) Every day some two pianists living in adjacent rooms (the Arth and ($k +1$)st) decide that they interfere with each other’s practice, and they move to the ($k - 1$)st and ($k + 2$)nd rooms, respectively. Prove that these moves will cease after a finite number of days. (VG Ilichev)

2022 Kyiv City MO Round 1, Problem 3

Tags: geometry , orthocenter , circumcenter , circumcircle

Let $H$ and $O$ be the orthocenter and the circumcenter of the triangle $ABC$. Line $OH$ intersects the sides $AB, AC$ at points $X, Y$ correspondingly, so that $H$ belongs to the segment $OX$. It turned out that $XH = HO = OY$. Find $\angle BAC$. [i](Proposed by Oleksii Masalitin)[/i]

2011 Federal Competition For Advanced Students, Part 2, 1

Tags: geometry , geometric transformation , combinatorics

Every brick has $5$ holes in a line. The holes can be filled with bolts (fitting in one hole) and braces (fitting into two neighboring holes). No hole may remain free. One puts $n$ of these bricks in a line to form a pattern from left to right. In this line no two braces and no three bolts may be adjacent. How many different such patterns can be produced with $n$ bricks?

1951 AMC 12/AHSME, 32

Tags: geometry , perimeter

If $ \triangle ABC$ is inscribed in a semicircle whose diameter is $ AB$, then $ AC \plus{} BC$ must be $ \textbf{(A)}\ \text{equal to }AB \qquad\textbf{(B)}\ \text{equal to }AB\sqrt {2} \qquad\textbf{(C)}\ \geq AB\sqrt {2}$ $ \textbf{(D)}\ \leq AB\sqrt {2} \qquad\textbf{(E)}\ AB^2$

2021 Iran MO (2nd Round), 1

Tags: combinatorics , geometry

There are two distinct Points $A$ and $B$ on a line. We color a point $P$ on segment $AB$, distinct from $A,B$ and midpoint of segment $AB$ to red. In each move , we can reflect one of the red point wrt $A$ or $B$ and color the midpoint of the resulting point and the point we reflected from ( which is one of $A$ or $B$ ) to red. For example , if we choose $P$ and the reflection of $P$ wrt to $A$ is $P'$ , then midpoint of $AP'$ would be red. Is it possible to make the midpoint of $AB$ red after a finite number of moves?

2011 Math Prize For Girls Problems, 11

Tags: linear algebra , matrix , algebra , polynomial , trigonometry , mewing

The sequence $a_0$, $a_1$, $a_2$, $\ldots\,$ satisfies the recurrence equation \[ a_n = 2 a_{n-1} - 2 a_{n - 2} + a_{n - 3} \] for every integer $n \ge 3$. If $a_{20} = 1$, $a_{25} = 10$, and $a_{30} = 100$, what is the value of $a_{1331}$?

2015 Tuymaada Olympiad, 7

Tags: geometry , circumcircle

In $\triangle ABC$ points $M,O$ are midpoint of $AB$ and circumcenter. It is true, that $OM=R-r$. Bisector of external $\angle A$ intersect $BC$ at $D$ and bisector of external $\angle C$ intersect $AB$ at $E$. Find possible values of $\angle CED$ [i]D. Shiryaev [/i]

2023 Nordic, P4

Tags: geometry

Let $ABC$ be a triangle, and $M$ the midpoint of the side $BC$. Let $E$ and $F$ be points on the sides $AC$ and $AB$, respectively, so that $ME=MF$. Let $D$ be the second intersection of the circumcircle of $MEF$ and the side $BC$. Consider the lines $\ell_D$, $\ell_E$ and $\ell_F$ through $D, E$ and $F$, respectively, such that $\ell_D \perp BC$, $\ell_E \perp AC$ and $\ell_F \perp AB$. Show that $\ell_D, \ell_E$ and $\ell_F$ are concurrent.

1967 IMO, 4

Tags: geometry , triangle , similar triangles , area of a triangle

$A_0B_0C_0$ and $A_1B_1C_1$ are acute-angled triangles. Describe, and prove, how to construct the triangle $ABC$ with the largest possible area which is circumscribed about $A_0B_0C_0$ (so $BC$ contains $B_0, CA$ contains $B_0$, and $AB$ contains $C_0$) and similar to $A_1B_1C_1.$

Kyiv City MO Seniors Round2 2010+ geometry, 2010.10.4

Tags: fixed , geometry , fixed point , length , circumcircle , angle

The points $A \ne B$ are given on the plane. The point $C$ moves along the plane in such a way that $\angle ACB = \alpha$ , where $\alpha$ is the fixed angle from the interval ($0^o, 180^o$). The circle inscribed in triangle $ABC$ has center the point $I$ and touches the sides $AB, BC, CA$ at points $D, E, F$ accordingly. Rays $AI$ and $BI$ intersect the line $EF$ at points $M$ and $N$, respectively. Show that: a) the segment $MN$ has a constant length, b) all circles circumscribed around triangle $DMN$ have a common point

2020 Germany Team Selection Test, 3

Tags: ceiling function , number theory , perfect square

Let $a$ and $b$ be two positive integers. Prove that the integer \[a^2+\left\lceil\frac{4a^2}b\right\rceil\] is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.) [i]Russia[/i]

2013 Romania National Olympiad, 1

Tags: linear algebra , matrix , vector , algebra , polynomial

Given A, non-inverted matrices of order n with real elements, $n\ge 2$ and given ${{A}^{*}}$adjoin matrix A. Prove that $tr({{A}^{*}})\ne -1$ if and only if the matrix ${{I}_{n}}+{{A}^{*}}$ is invertible.

2017 India PRMO, 6

Tags: sum , algebra

Let the sum $\sum_{n=1}^{9} \frac{1}{n(n+1)(n+2)}$ written in its lowest terms be $\frac{p}{q}$ . Find the value of $q - p$.

2007 AMC 10, 1

Tags:

One ticket to a show costs $ \$20$ at full price. Susan buys 4 tickets using a coupon that gives her a $25\%$ discount. Pam buys 5 tickets using a coupon that gives her a $30\%$ discount. How many more dollars does Pam pay than Susan? $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 15 \qquad \textbf{(E)}\ 20$

2020 Estonia Team Selection Test, 3

Tags: number theory , polynomial

We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.