Olympiad problems

2015 Thailand TSTST, 2

Tags: combinatorics

Let $C$ be the set of all 100-digit numbers consisting of only the digits $1$ and $2$. Given a number in $C$, we may transform the number by considering any $10$ consecutive digits $x_0x_1x_2 \dots x_9$ and transform it into $x_5x_6\dots x_9x_0x_1\dots x_4$. We say that two numbers in $C$ are similar if one of them can be reached from the other by performing finitely many such transformations. Let $D$ be a subset of $C$ such that any two numbers in $D$ are not similar. Determine the maximum possible size of $D$.

2023 Israel National Olympiad, P7

Tags: number theory , digit , guessing game

Ana and Banana are playing a game. Initially, Ana secretly picks a number $1\leq A\leq 10^6$. In each subsequent turn of the game, Banana may pick a positive integer $B$, and Ana will reveal to him the most common digit in the product $A\cdot B$ (written in decimal notation). In the case when at least two digits are tied for being the most common, Ana will reveal all of them to Banana. For example, if $A\cdot B=2022$, Ana will tell Banana that the digit $2$ is the most common, while if $A\cdot B=5783783$, Ana will reveal that $3, 7$ and $8$ are the most common. Banana's goal is to determine with certainty the number $A$ after some number of turns. Does he have a winning strategy?

2020 Israel Olympic Revenge, P1

Tags: functional equation , algebra , function , symmetry

Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for all $x,y\in \mathbb{R}$ one has \[f(f(x)+y)=f(x+f(y))\] and in addition the set $f^{-1}(a)=\{b\in \mathbb{R}\mid f(b)=a\}$ is a finite set for all $a\in \mathbb{R}$.

2024 Serbia JBMO TST, 4

Tags: geometry

Let $I$ be the incenter of a triangle $ABC$ with $AB \neq AC$. Let $M$ be the midpoint of $BC$, $M' \in BC$ be such that $IM'=IM$ and $K$ be the midpoint of the arc $BAC$. If $AK \cap BC=L$, show that $KLIM'$ is cyclic.

2016 Bulgaria EGMO TST, 1

Tags: partition , number theory , combinatorics

Is it possible to partition the set of integers into three disjoint sets so that for every positive integer $n$ the numbers $n$, $n-50$ and $n+1987$ belong to different sets?

1999 National Olympiad First Round, 9

Tags: geometry , trigonometry , rectangle

Find the area of inscribed convex octagon, if the length of four sides is $2$, and length of other four sides is $ 6\sqrt {2}$. $\textbf{(A)}\ 120 \qquad\textbf{(B)}\ 24 \plus{} 68\sqrt {2} \qquad\textbf{(C)}\ 88\sqrt {2} \qquad\textbf{(D)}\ 124 \qquad\textbf{(E)}\ 72\sqrt {3}$

1993 Putnam, B3

Tags: probability and stats

$x$ and $y$ are chosen at random (with uniform density) from the interval $(0, 1)$. What is the probability that the closest integer to $x/y$ is even?

2023 HMNT, 8

Tags: number theory

Call a number [i]feared [/i] if it contains the digits $13$ as a contiguous substring and [i]fearless [/i] otherwise. (For example, $132$ is feared, while $123$ is fearless.) Compute the smallest positive integer $n$ such that there exists a positive integer $a < 100$ such that $n$ and $n + 10a$ are fearless while $n +a$, $n + 2a$, $. . . $, $n + 9a$ are all feared.

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 $5$.

1953 Moscow Mathematical Olympiad, 253

Tags: inequalities , root , trinomial , algebra

Given the equations (1) $ax^2 + bx + c = 0$ (2)$ -ax^2 + bx + c = 0$ prove that if $x_1$ and $x_2$ are some roots of equations (1) and (2), respectively, then there is a root $x_3$ of the equation $$\frac{a}{2}x^2 + bx + c = 0$$ such that either $x_1 \le x_3 \le x_2$ or $x_1 \ge x_3 \ge x_2$.

2021 Belarusian National Olympiad, 9.6

Tags: ratio , geometry

The medians of a right triangle $ABC$ ($\angle C = 90^{\circ}$) intersect at $M$. Point $L$ lies on the $AC$ such that $\angle ABL=\angle CBL$. It turned out that $\angle BML = 90^{\circ}$. Find the ration $AB : BC$.

2014 Iran MO (2nd Round), 3

Tags: floor function , function , combinatorics

Members of "Professionous Riddlous" society have been divided into some groups, and groups are changed in a special way each weekend: In each group, one of the members is specified as the best member, and the best members of all groups separate from their previous group and form a new group. If a group has only one member, that member joins the new group and the previous group will be removed. Suppose that the society has $n$ members at first, and all the members are in one group. Prove that a week will come, after which number of members of each group will be at most $1+\sqrt{2n}$.

2007 Peru Iberoamerican Team Selection Test, P3

Tags: geometry

We have an acute triangle $ABC$. Consider the square $A_1A_2A_3A_4$ which has one vertex in $AB$, one vertex in $AC$ and two vertices ($A_1$ and $A_2$) in $BC$ and let $x_A=\angle A_1AA_2$. Analogously we define $x_B$ and $x_C$. Prove that $x_A+x_B+x_C=90$

1996 Tournament Of Towns, (514) 1

Tags: geometry , 3d geometry , locus

Consider three edges $a, b, c$ of a cube such that no two of these edges lie in one plane. Find the locus of points inside the cube which are equidistant from $a$, $b$ and $c$. (V Proizvolov,)

2002 Switzerland Team Selection Test, 7

Tags: geometry , linear algebra , area , triangle

Let $ABC$ be a triangle and $P$ an exterior point in the plane of the triangle. Suppose the lines $AP$, $BP$, $CP$ meet the sides $BC$, $CA$, $AB$ (or extensions thereof) in $D$, $E$, $F$, respectively. Suppose further that the areas of triangles $PBD$, $PCE$, $PAF$ are all equal. Prove that each of these areas is equal to the area of triangle $ABC$ itself.

1999 Switzerland Team Selection Test, 7

Tags: rectangle , ratio , square , combinatorial geometry , combinatorics , geometry

A square is dissected into rectangles with sides parallel to the sides of the square. For each of these rectangles, the ratio of its shorter side to its longer side is considered. Show that the sum of all these ratios is at least $1$.

2022 Baltic Way, 13

Tags: geometry

Let $ABCD$ be a cyclic quadrilateral with $AB < BC$ and $AD < DC$. Let $E$ and $F$ be points on the sides $BC$ and $CD$, respectively, such that $AB = BE$ and $AD = DF$. Let further M denote the midpoint of the segment $EF$. Prove that $\angle BMD = 90^o$.

2003 China Team Selection Test, 1

Tags: trigonometry , calculus , geometry

Let $ ABCD$ be a quadrilateral which has an incircle centered at $ O$. Prove that \[ OA\cdot OC\plus{}OB\cdot OD\equal{}\sqrt{AB\cdot BC\cdot CD\cdot DA}\]

2005 Sharygin Geometry Olympiad, 10.5

Tags: tangent , distance , circles , geometry

Two circles of radius $1$ intersect at points $X, Y$, the distance between which is also equal to $1$. From point $C$ of one circle, tangents $CA, CB$ are drawn to the other. Line $CB$ will cross the first circle a second time at point $A'$. Find the distance $AA'$.

2011 BMO TST, 1

Tags: analytic geometry , parabola , function , algebra , conic

The given parabola $y=ax^2+bx+c$ doesn't intersect the $X$-axis and passes from the points $A(-2,1)$ and $B(2,9)$. Find all the possible values of the $x$ coordinates of the vertex of this parabola.

1988 Balkan MO, 1

Tags: trigonometry , ratio , angle bisector , geometry

Let $ABC$ be a triangle and let $M,N,P$ be points on the line $BC$ such that $AM,AN,AP$ are the altitude, the angle bisector and the median of the triangle, respectively. It is known that $\frac{[AMP]}{[ABC]}=\frac{1}{4}$ and $\frac{[ANP]}{[ABC]}=1-\frac{\sqrt{3}}{2}$. Find the angles of triangle $ABC$.

1999 Romania National Olympiad, 4

Tags: geometry , ratio

In the triangle $ABC$, let $D \in (BC)$, $E \in (AB)$, $EF \parallel BC$, $F \in (AC)$, $EG\parallel AD$, $G\in (BC)$ and $M,N$ be the midpoints of $(AD)$ and $(BC)$, respectively. Prove that: a) $\frac{EF}{BC}+\frac{EG}{AD}=1$ b) the midpoint of $[FG]$ lies on the line $ MN$.

2016 Junior Regional Olympiad - FBH, 1

Tags: inequalities

If $a>b>c$ are real numbers prove that $$\frac{1}{a-b}+\frac{1}{b-c}>\frac{2}{a-c}$$

2023 Brazil Cono Sur TST, 3

Tags:

The integers from $1$ to $2022$ are written on cards placed in a row on a table. Each number appears only once and each card shows exactly one number. Esmeralda performs consecutively the following operations $1011$ times: • She chooses a card on the table and puts it in a box on her right. • Right after it, she picks the leftmost card on the table and puts it in a box on her left. At the end of the process, she calculates the sum of the numbers in the left box. For each initial configuration $P$ of the cards, let $S(P)$ be the maximum sum Esmeralda can achieve. Determine the number of initial configurations $P$ for which $S(P)$ achieves its least value.

2017 Czech-Polish-Slovak Junior Match, 3

Tags: inequalities , algebra

Prove that for all real numbers $x, y$ holds $(x^2 + 1)(y^2 + 1) \ge 2(xy - 1)(x + y)$. For which integers $x, y$ does equality occur?