Navvye/Olympiad-Problems-CSV huggingface dataset:
2021 Irish Math Olympiad
A point $C$ lies on a line segment $AB$ between $A$ and $B$ and circles are drawn having $AC$ and $CB$ as diameters. A common tangent to both circles touches the circle with $AC$ as diameter at $P \ne C$ and the circle with $CB$ as diameter at $Q \ne C$.

Prove that $AP, BQ$ and the common tangent to both circles at $C$ all meet at a single point which lies on the circumference of the circle with $AB$ as diameter.
2021 Irish Math Olympiad
Suppose the real numbers $a, A, b, B$ satisfy the inequalities: $$|A - 3a| \le  1 - a\,\,\, , \,\,\, |B -3b| \le  1 - b$$, and $a, b$ are positive. Prove that $$\left|\frac{AB}{3}- 3ab\right | - 3ab \le 1 - ab.$$
2021 Irish Math Olympiad
Let $P_{1}, P_{2}, \ldots, P_{2021}$ be 2021 points in the quarter plane $\{(x, y): x \geq 0, y \geq 0\}$. The centroid of these 2021 points lies at the point $(1,1)$.



Show that there are two distinct points $P_{i}, P_{j}$ such that the distance from $P_{i}$ to $P_{j}$ is no more than $\sqrt{2} / 20$.
2019 Israel National Olympiad
In kindergarden, there are 32 children and three classes: Judo, Agriculture, and Math. Every child is in exactly one class and every class has at least one participant.



One day, the teacher gathered 6 children to clean up the classroom. The teacher counted and found that exactly 1/2 of the Judo students, 1/4 of the Agriculture students and 1/8 of the Math students are cleaning.



How many children are in each class?
2019 Israel National Olympiad
We are given a 5x5 square grid, divided to 1x1 tiles. Two tiles are called linked if they lie in the same row or column, and the distance between their centers is 2 or 3. For example, in the picture the gray tiles are the ones linked to the red tile.







Sammy wants to mark as many tiles in the grid as possible, such that no two of them are linked. What is the maximal number of tiles he can mark?
2019 Israel National Olympiad
Six congruent isosceles triangles have been put together as described in the picture below. Prove that points M, F, C lie on one line.
2019 Israel National Olympiad
In the beginning, the number 1 is written on the board 9999 times. We are allowed to perform the following actions:



Erase four numbers of the form $x,x,y,y$, and instead write the two numbers $x+y,x-y$. (The order or location of the erased numbers does not matter)



Erase the number 0 from the board, if it's there.



Is it possible to reach a state where:



Only one number remains on the board?

At most three numbers remain on the board?
2019 Israel National Olympiad
Guy has 17 cards. Each of them has an integer written on it (the numbers are not necessarily positive, and not necessarily different from each other). Guy noticed that for each card, the square of the number written on it equals the sum of the numbers on the 16 other cards.



What are the numbers on Guy's cards? Find all of the options.
2019 Israel National Olympiad
A set of integers is called legendary if you can reach any integer from it by using the following action multiple times:



If the numbers $x,y$ are in the set, we may add the number $xy-y^2-y+x$ to the set.



Prove that any legendary set contains at least 8 numbers.
2019 Israel National Olympiad
In the plane points $A,B,C$ are marked in blue and points $P,Q$ are marked in red (no 3 marked points lie on a line, and no 4 marked points lie on a circle). A circle is called separating if all points of one color are inside it, and all points of the other color are outside of it. Denote by $O$ the circumcenter of $ABC$ and by $R$ the circumradius of $ABC$.



Prove that exactly one of the following holds:



There exists a separating circle;

There exists a point $X$ on the segment $PQ$ which also lies inside the triangle $ABC$, for which $PX\cdot XQ = R^2-OX^2$.
2018 Israel National Olympiad
$n$ people sit in a circle. Each of them is either a liar (always lies) or a truthteller (always tells the truth). Every person knows exactly who speaks the truth and who lies. In their turn, each person says 'the person two seats to my left is a truthteller'. It is known that there's at least one liar and at least one truthteller in the circle.



Is it possible that $n=2017?$

Is it possible that $n=5778?$
2018 Israel National Olympiad
An arithmetic sequence is an infinite sequence of the form $a_n=a_0+n\cdot d$ with $d\neq 0$.



A geometric sequence is an infinite sequence of the form $b_n=b_0 \cdot q^n$ where $q\neq 1,0,-1$.



Does every arithmetic sequence of integers have an infinite subsequence which is geometric?

Does every arithmetic sequence of real numbers have an infinite subsequence which is geometric?
2018 Israel National Olympiad
Determine the minimal and maximal values the expression $\frac{|a+b|+|b+c|+|c+a|}{|a|+|b|+|c|}$ can take, where $a,b,c$ are real numbers.
2018 Israel National Olympiad
The three-digit number 999 has a special property: It is divisible by 27, and its digit sum is also divisible by 27. The four-digit number 5778 also has this property, as it is divisible by 27 and its digit sum is also divisible by 27. How many four-digit numbers have this property?
2018 Israel National Olympiad
The sequence $a_n$ is defined for any $n\geq 10$ by the following inductive rule:



$a_{10}=5778$

If $a_n=0$ then $a_{n+1}=0$.

If $a_n\neq0$ then $a_{n+1}$ is the number whose base-$(n+1)$ representation equals the base $n$ representation of the number $a_n -1$.



For example,

$a_{11}=5\cdot11^3+7\cdot11^2+7\cdot11^1+7\cdot11^0=7586$

$a_{12}=5\cdot12^3+7\cdot12^2+7\cdot12^1+6\cdot12^0=9738$



Does there exist $n\geq10$ for which $a_n=0$?

Is $a_{1,000,000}=0$?

Is $a_{100^{100^{100}}}=0$?
2018 Israel National Olympiad
In the corners of triangle $ABC$ there are three circles with the same radius. Each of them is tangent to two of the triangle's sides. The vertices of triangle $MNK$ lie on different sides of triangle $ABC$, and each edge of $MNK$ is also tangent to one of the three circles. Likewise, the vertices of triangle $PQR$ lie on different sides of triangle $ABC$, and each edge of $PQR$ is also tangent to one of the three circles (see picture below). Prove that triangles $MNK,PQR$ have the same inradius.
2018 Israel National Olympiad
A uniform covering of the integers $1,2,...,n$ is a finite multiset of subsets of $\{1,2,...,n\}$, so that each number lies in the same amount of sets from the covering. A covering may contain the same subset multiple times, it must contain at least one subset, and it may contain the empty subset. For example, $(\{1\},\{1\},\{2,3\},\{3,4\},\{2,4\})$ is a uniform covering of $1,2,3,4$ (every number occurs in two sets). The covering containing only the empty set is also uniform (every number occurs in zero sets).



Given two uniform coverings, we define a new uniform covering, their sum (denoted by $\oplus$), by adding the sets from both coverings. For example:



$(\{1\},\{1\},\{2,3\},\{3,4\},\{2,4\})\oplus(\{1\},\{2\},\{3\},\{4\})=$

$(\{1\},\{1\},\{1\},\{2\},\{3\},\{4\},\{2,3\},\{3,4\},\{2,4\})$



A uniform covering is called non-composite if it's not a sum of two uniform coverings.



Prove that for any $n\geq1$, there are only finitely many non-composite uniform coverings of $1,2,...,n$.
2017 Israel National Olympiad




In the right picture there is a square with four congruent circles inside it. Each circle is tangent to two others, and to two of the edges of the square. Evaluate the ratio between the blue part and white part of the square's area.

In the left picture there is a regular hexagon with six congruent circles inside it. Each circle is tangent to two others, and to one of the edges on the hexagon in its midpoint. Evaluate the ratio between the blue part and white part of the hexagon's area.
2017 Israel National Olympiad
Denote by $P(n)$ the product of the digits of a positive integer $n$. For example, $P(1948)=1\cdot9\cdot4\cdot8=288$.



Evaluate the sum $P(1)+P(2)+\dots+P(2017)$.

Determine the maximum value of $\frac{P(n)}{n}$ where $2017\leq n\leq5777$.
2017 Israel National Olympiad
A large collection of congruent right triangles is given, each with side length 3,4,5. Find the maximal number of such triangles you can place inside a 20x20 square, with no two triangles intersecting (in their interiors).
2017 Israel National Olympiad
Three rational number $x,p,q$ satisfy $p^2-xq^2$=1. Prove that there are integers $a,b$ such that $p=\frac{a^2+xb^2}{a^2-xb^2}$ and $q=\frac{2ab}{a^2-xb^2}$.
2017 Israel National Olympiad
A regular pentagon $ABCDE$ is given. The point $X$ is on his circumcircle, on the arc $\overarc{AE}$. Prove that $|AX|+|CX|+|EX|=|BX|+|DX|$.



Remark: Here's a more general version of the problem: Prove that for any point $X$ in the plane, $|AX|+|CX|+|EX|\ge|BX|+|DX|$, with equality only on the arc $\overarc{AE}$.
2017 Israel National Olympiad
Let $f:\mathbb{Q}\times\mathbb{Q}\to\mathbb{Q}$ be a function satisfying:



For any $x_1,x_2,y_1,y_2 \in \mathbb Q$, $$f\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right) \leq \frac{f(x_1,y_1)+f(x_2,y_2)}{2}.$$

$f(0,0) \leq 0$.

For any $x,y \in \mathbb Q$ satisfying $x^2+y^2>100$, the inequality $f(x,y)>1$ holds.\


Prove that there is some positive rational number $b$ such that for all rationals $x,y$, $$f(x,y) \ge b\sqrt{x^2+y^2} - \frac{1}{b}.$$
2017 Israel National Olympiad
A table with $m$ rows and $n$ columns is given. In each cell of the table an integer is written. Heisuke and Oscar play the following game: at the beginning of each turn, Heisuke may choose to swap any two columns. Then he chooses some rows and writes down a new row at the bottom of the table, with each cell consisting the sum of the corresponding cells in the chosen rows. Oscar then deletes one row chosen by Heisuke (so that at the end of each turn there are exactly $m$ rows). Then the next turn begins and so on. Prove that Heisuke can assure that, after some finite amount of turns, no number in the table is smaller than the number to the number on his right.

Example: If we begin with $(1,1,1),(6,5,4),(9,8,7)$, Heisuke may choose to swap the first and third column to get $(1,1,1),(4,5,6),(7,8,9)$. Then he chooses the first and second rows to obtain $(1,1,1),(4,5,6),(7,8,9),(5,6,7)$. Then Oscar has to delete either the first or the second row, let's say the second. We get $(1,1,1),(7,8,9),(5,6,7)$ and Heisuke wins.
2016 Israel National Olympiad
Nina and Meir are walking on a $3$ km path towards grandma's house. They start walking at the same time from the same point. Meir's speed is $4$ km/h and Nina's speed is $3$ km/h.



Along the path there are several benches. Whenever Nina or Meir reaches a bench, they sit on it for some time and eat a cookie. Nina always takes $t$ minutes to eat a cookie, and Meir always takes $2t$ minutes to eat a cookie, where $t$ is a positive integer.



It turns out that Nina and Meir reached grandma's house at the same time. How many benches were there? Find all of the options.
2016 Israel National Olympiad
We are given a cone with height 6, whose base is a circle with radius $\sqrt{2}$. Inside the cone, there is an inscribed cube: Its bottom face on the base of the cone, and all of its top vertices lie on the cone. What is the length of the cube's edge?
2016 Israel National Olympiad
Denote by $S(n)$ the sum of digits of $n$. Given a positive integer $N$, we consider the following process: We take the sum of digits $S(N)$, then take its sum of digits $S(S(N))$, then its sum of digits $S(S(S(N)))$... We continue this until we are left with a one-digit number.



We call the number of times we had to activate $S(\cdot)$ the depth of $N$.



For example, the depth of 49 is 2, since $S(49)=13\rightarrow S(13)=4$, and the depth of 45 is 1, since $S(45)=9$.



Prove that every positive integer $N$ has a finite depth, that is, at some point of the process we get a one-digit number.

Define $x(n)$ to be the minimal positive integer with depth $n$. Find the residue of $x(5776)\mod 6$.

Find the residue of $x(5776)-x(5708)\mod 2016$.
2016 Israel National Olympiad
In the beginning, there is a circle with three points on it. The points are colored (clockwise): Green, blue, red. Jonathan may perform the following actions, as many times as he wants, in any order:



Choose two adjacent points with different colors, and add a point between them with one of the two colors only.

Choose two adjacent points with the same color, and add a point between them with any of the three colors.

Choose three adjacent points, at least two of them having the same color, and delete the middle point.



Can Jonathan reach a state where only three points remain on the circle, colored (clockwise): Blue, green, red?
2016 Israel National Olympiad
The Fibonacci sequence $F_n$ is defined by $F_1=F_2=1$ and the recurrence relation $F_n=F_{n-1}+F_{n-2}$ for all integers $n\geq3$.



Let $m,n\geq1$ be integers. Find the minimal degree $d$ for which there exists a polynomial $f(x)=a_dx^d+a_{d-1}x^{d-1}+\dots+a_1x+a_0$, which satisfies $f(k)=F_{m+k}$ for all $k=0,1,...,n$.
2016 Israel National Olympiad
Points $A_1,A_2,A_3,...,A_{12}$ are the vertices of a regular polygon in that order. The 12 diagonals $A_1A_6,A_2A_7,A_3A_8,...,A_{11}A_4,A_{12}A_5$ are marked, as in the picture below. Let $X$ be some point in the plane. From $X$, we draw perpendicular lines to all 12 marked diagonals. Let $B_1,B_2,B_3,...,B_{12}$ be the feet of the perpendiculars, so that $B_1$ lies on $A_1A_6$, $B_2$ lies on $A_2A_7$ and so on.



Evaluate the ratio $\frac{XA_1+XA_2+\dots+XA_{12}}{B_1B_6+B_2B_7+\dots+B_{12}B_5}$.
2016 Israel National Olympiad
Find all functions $f:\mathbb{Z}\rightarrow\mathbb{C}$ such that $f(x(2y+1))=f(x(y+1))+f(x)f(y)$ holds for any two integers $x,y$.
2015 Israel National Olympiad




Find an example of three positive integers $a,b,c$ satisfying $31a+30b+28c=365$.

Prove that any triplet $a,b,c$  satisfying the above condition, also satisfies $a+b+c=12$.
2015 Israel National Olympiad
A triangle is given whose altitudes' lengths are $\frac{1}{5},\frac{1}{5},\frac{1}{8}$. Evaluate the triangle's area.
2015 Israel National Olympiad
Prove that the number $\left(\frac{76}{\frac{1}{\sqrt[3]{77}-\sqrt[3]{75}}-\sqrt[3]{5775}}+\frac{1}{\frac{76}{\sqrt[3]{77}+\sqrt[3]{75}}+\sqrt[3]{5775}}\right)^3$ is an integer.
2015 Israel National Olympiad
Let $k,m,n$ be positive integers such that $n^m$ is divisible by $m^n$, and $m^k$ is divisible by $k^m$.



Prove that $n^k$ is divisible by $k^n$.

Find an example of $k,m,n$ satisfying the above conditions, where all three numbers are distinct and bigger than 1.
2015 Israel National Olympiad
Let $ABCD$ be a tetrahedron. Denote by $S_1$ the inscribed sphere inside it, which is tangent to all four faces. Denote by $S_2$ the outer escribed sphere outside $ABC$, tangent to face $ABC$ and to the planes containing faces $ABD,ACD,BCD$. Let $K$ be the tangency point of $S_1$ to the face $ABC$, and let $L$ be the tangency point of $S_2$ to the face $ABC$. Let $T$ be the foot of the perpendicular from $D$ to the face $ABC$.



Prove that $L,T,K$ lie on one line.
2015 Israel National Olympiad
Let $n\geq1$ be a positive integer. $n$ lamps are placed in a line. At minute 0, some lamps are on (maybe all of them). Every minute the state of the lamps changes: A lamp is on at minute $t+1$ if and only if at minute $t$, exactly one of its neighbors is on (the two lamps at the ends have one neighbor each, all other lamps have two neighbors).



For which values of $n$ can we guarantee that all lamps will be off after some time?
2015 Israel National Olympiad
The Fibonacci sequence $F_n$ is defined by $F_0=0,F_1=1$ and the recurrence relation $F_n=F_{n-1}+F_{n-2}$ for all integers $n\geq2$. Let $p\geq3$ be a prime number.



Prove that $F_{p-1}+F_{p+1}-1$ is divisible by $p$.

Prove that $F_{p^{k+1}-1}+F_{p^{k+1}+1}-\left(F_{p^k-1}+F_{p^k+1}\right)$ is divisible by $p^{k+1}$ for any positive integer $k$.
2014 Israel National Olympiad
Consider the number $\left(101^2-100^2\right)\cdot\left(102^2-101^2\right)\cdot\left(103^2-102^2\right)\cdot...\cdot\left(200^2-199^2\right)$.



Determine its units digit.

Determine its tens digit.
2014 Israel National Olympiad
Let $\Delta A_1A_2A_3, \Delta B_1B_2B_3, \Delta C_1C_2C_3$ be three equilateral triangles. The vertices in each triangle are numbered clockwise. It is given that $A_3=B_3=C_3$. Let $M$ be the center of mass of $\Delta A_1B_1C_1$, and let $N$ be the center of mass of $\Delta A_2B_2C_2$.



Prove that $\Delta A_3MN$ is an equilateral triangle.
2014 Israel National Olympiad
Let $ABCDEF$ be a convex hexagon. In the hexagon there is a point $K$, such that $ABCK,DEFK$ are both parallelograms. Prove that the three lines connecting $A,B,C$ to the midpoints of segments $CE,DF,EA$ meet at one point.
2014 Israel National Olympiad
We are given a row of $n\geq7$ tiles. In the leftmost 3 tiles, there is a white piece each, and in the rightmost 3 tiles, there is a  black piece each. The white and black players play in turns (the white starts). In each move, a player may take a piece of their color, and move it to an adjacent tile, so long as it's not occupied by a piece of the same color. If the new tile is empty, nothing happens. If the tile is occupied by a piece of the opposite color, both pieces are destroyed (both white and black). The player who destroys the last two pieces wins the game.



Which player has a winning strategy, and what is it? (The answer may depend on $n$)
2014 Israel National Olympiad
Let $p$ be a polynomial with integer coefficients satisfying $p(16)=36,p(14)=16,p(5)=25$. Determine all possible values of $p(10)$.
2014 Israel National Olympiad
Let $n$ be a positive integer. Find the maximal real number $k$, such that the following holds:



For any $n$ real numbers $x_1,x_2,...,x_n$, we have $\sqrt{x_1^2+x_2^2+\dots+x_n^2}\geq k\cdot\min(|x_1-x_2|,|x_2-x_3|,...,|x_{n-1}-x_n|,|x_n-x_1|)$
2014 Israel National Olympiad
Find one real value of $x$ satisfying $\frac{x^7}{7}=1+\sqrt[7]{10}x\left(x^2-\sqrt[7]{10}\right)^2$.
2013 Israel National Olympiad
In the picture there are six coins, each with radius 1cm. Each coin is tangent to exactly two other coins next to it (as in the picture). Between the coins, there is an empty area whose boundary is a star-like shape. What is the perimeter of this shape?
2013 Israel National Olympiad
Let $A=\{n\in\mathbb{Z}\mid 0<n<2013\}$. A subset $B\subseteq A$ is called reduced if for any two numbers $x,y\in B$, we must have $x\cdot y \notin B$. For example, any subset containing the numbers $3,5,15$ cannot be reduced, and same for a subset containing $4,16$.



Find the maximal size of a reduced subset of $A$.

How many reduced subsets are there with that maximal size?
2013 Israel National Olympiad
Let $p(x)=x^4-5773x^3-46464x^2-5773x+46$. Determine the sum of $\arctan$-s of its real roots.
2013 Israel National Olympiad
Determine the number of positive integers $n$ satisfying:



$n<10^6$

$n$ is divisible by 7

$n$ does not contain any of the digits 2,3,4,5,6,7,8.
2013 Israel National Olympiad
A point in the plane is called integral if both its $x$ and $y$ coordinates are integers. We are given a triangle whose vertices are integral. Its sides do not contain any other integral points. Inside the triangle, there are exactly 4 integral points. Must those 4 points lie on one line?
2013 Israel National Olympiad
Let $x_1,...,x_n$ be positive real numbers, satisfying $x_1+\dots+x_n=n$. Prove that



$\frac{x_1}{x_2}+\frac{x_2}{x_3}+\dots+\frac{x_{n-1}}{x_n}+\frac{x_n}{x_1}\leq\frac{4}{x_1\cdot x_2\cdot\dots\cdot x_n}+n-4$.
2012 Israel National Olympiad
In the picture below, the circles are tangent to each other and to the edges of the rectangle. The larger circle's radius equals 1. Determine the area of the rectangle.
2012 Israel National Olympiad
In some foreign country, there is a secret object, guarded by seven guards. Each guard has a guarding shift of 7 consecutive hours every day, in fixed hours. There is always at least one guard guarding the secret object at any given time.



Prove that one of the guards can be fired, and there will still be at least one guard guarding at any given time (without changing the schedule of the other guards).
2012 Israel National Olympiad
Let $a,b,c$ be real numbers such that $a^3(b+c)+b^3(a+c)+c^3(a+b)=0$. Prove that $ab+bc+ca\leq0$.
2012 Israel National Olympiad
We are given a 7x7 square board. In each square, one of the diagonals is traced, and then one of the two formed triangles is colored blue. What is the largest area a continuous blue component can have?



(Note: continuous blue component means a set of blue triangles connected via their edges, passing through corners is not permitted)
2012 Israel National Olympiad
Find all integer solutions of the equation $a^3+3ab^2+7b^3=2011$.
2012 Israel National Olympiad
Let $A,B,C,O$ be points in the plane such that angles $\angle AOB,\angle BOC, \angle COA$ are obtuse. On $OA,OB,OC$ points $X,Y,Z$ respectively are chosen, such that $OX=OY=OZ$. On segments $OX,OY,OZ$ points $K,L,M$ respectively are chosen.



The lines $AL$ and $BK$ intersect at point $R$, which isn't on $XY$. The segment $XY$ intersects $AL,BK$ at points $R_1,R_2$.



The lines $BM$ and $CL$ intersect at point $P$, which isn't on $YZ$. The segment $YZ$ intersects $BM,CL$ at points $P_1,P_2$.



The lines $CK$ and $AM$ intersect at point $Q$, which isn't on $ZX$. The segment $ZX$ intersects $CK,AM$ at points $Q_1,Q_2$.



Suppose that $PP_1=PP_2$ and $QQ_1=QQ_2$. Prove that $RR_1=RR_2$.
2011 Israel National Olympiad
We are given 5771 weights weighing 1,2,3,...,5770,5771. We partition the weights into $n$ sets of equal weight. What is the maximal $n$ for which this is possible?
2011 Israel National Olympiad
Evaluate the sum $\sqrt{1-\frac{1}{2}\cdot\sqrt{1\cdot3}}+\sqrt{2-\frac{1}{2}\cdot\sqrt{3\cdot5}}+\sqrt{3-\frac{1}{2}\cdot\sqrt{5\cdot7}}+\dots+\sqrt{40-\frac{1}{2}\cdot\sqrt{79\cdot81}}$.
2011 Israel National Olympiad
In some foreign country's government, there are 12 ministers. Each minister has 5 friends and 6 enemies in the government (friendship/enemyship is a symmetric relation). A triplet of ministers is called uniform if all three of them are friends with each other, or all three of them are enemies. How many uniform triplets are there?
2011 Israel National Olympiad
Let $\alpha_1,\alpha_2,\alpha_3$ be three congruent circles that are tangent to each other. A third circle $\beta$ is tangent to them at points $A_1,A_2,A_3$ respectively. Let $P$ be a point on $\beta$ which is different from $A_1,A_2,A_3$. For $i=1,2,3$, let $B_i$ be the second intersection point of the line $PA_i$ with circle $\alpha_i$. Prove that $\Delta B_1B_2B_3$ is equilateral.
2011 Israel National Olympiad
We have two lists of numbers: One initially containing 1,6,11,...,96, and the other initially containing 4,9,14,...,99. In every turn, we erase two numbers from one of the lists, and write $\frac{1}{3}$ of their sum (not necessarily an integer) in the other list. We continue this process until there are no possible moves.



Prove that at the end of the process, there is exactly one number in each list.

Prove that those two numbers are not equal.
2011 Israel National Olympiad
There are $N$ red cards and $N$ blue cards. Each card has a positive integer between $1$ and $N$ (inclusive) written on it. Prove that we can choose a (non-empty) subset of the red cards and a (non-empty) subset of the blue cards, so that the sum of the numbers on the chosen red cards equals the sum of the numbers on the chosen blue cards.
1998 Israel National Olympiad
In space are given $n$ segments $A_iB_i$ and a point $O$ not lying on any segment, such that the sum of the angles $A_iOB_i$ is less than $180^o$ . Prove that there exists a plane passing through $O$ and not intersecting any of the segments.
1998 Israel National Olympiad
Show that there is a multiple of $2^{1998}$ whose decimal representation consists only of the digits $1$ and $2$.