Navvye/Olympiad-Problems-CSV huggingface dataset:
1998 Israel National Olympiad
A configuration of several checkers at the centers of squares on a rectangular sheet of grid paper is called boring  if some four checkers occupy the vertices of a rectangle with sides parallel to those of the sheet.

(a) Prove that any configuration of more than $3mn/4$ checkers on an $m\times n$ grid is boring.

(b) Prove that any configuration of $26$ checkers on a $7\times 7$ grid is boring.
1998 Israel National Olympiad
A man has a seven-candle chandellier. The first evening he lighted one candle for one hour, the second evening he lighted two candles, also for one hour, and so on. After one hour the seventh evening, all seven candles simultaneously finished. How did the man choose the candles to light every evening?
1998 Israel National Olympiad
(a) Find two real numebrs $a,b$ such that $|ax+b-\sqrt{x}| \le \frac{1}{24}$ for $1 \le x \le 4$.

(b) Prove that the constant $\frac{1}{24}$ cannot be replaced by a smaller one.
1998 Israel National Olympiad
Find all pairs $(m,n)$ of integers with $m > n > 7$ for which there exists a polynomial $p(x)$ with integer coefficients such that $p(7) = 77, p(m) = 0$, and $p(n) = 85$.
1998 Israel National Olympiad
A polygonal line of the length $1001$ is given in a unit square. Prove that there exists a line parallel to one of the sides of the square that meets the polygonal line in at least $500$ points.
1997 Israel National Olympiad
Find all real solutions to the system of equations

$x^2 +y^2 = 6z$

$y^2 +z^2 = 6x$

$z^2 +x^2 = 6y$.
1997 Israel National Olympiad
We are given a balance with two bowls and a number of weights.

(a) Give an example of four integer weights using which one can measure any weight of $1,2,...,40$ grams.

(b) Are there four weights using which one can measure any weight of $1,2,...,50$ grams?
1997 Israel National Olympiad
Let $n?$ denote the product of all primes smaller than $n$.

Prove that $n? > n$ holds for any natural number $n > 3$.
1997 Israel National Olympiad
Let $f : [0,1] \to [0,1]$ be a continuous, strictly increasing function such that $f(0) = 0$ and $f(1) = 1$. Prove that

$$f\left(\frac{1}{10}\right) + f\left(\frac{2}{10}\right) +...+f\left(\frac{9}{10}\right) +f^{-1}\left(\frac{1}{10}\right) +...+f^{-1}\left(\frac{9}{10}\right) \le \frac{99}{10}$$
1997 Israel National Olympiad
The natural numbers $a_1,a_2,...,a_n, n \ge 12$, are smaller than $9n^2$ and pairwise coprime.

Show that at least one of these numbers is prime.
1997 Israel National Olympiad
In a certain country, every two cities are connected either by an airline route or by a railroad. Prove that one can always choose a type of transportation in such a way that each city can be reached from any other city with at most two transfers.
1997 Israel National Olympiad
A square with side $10^6$, with a corner square with side $10^{-3}$ cut off, is partitioned into $10$ rectangles. Prove that at least one of these rectangles has the ratio of the greater side to the smaller one at least $9$.
1997 Israel National Olympiad
Two equal circles are internally tangent to a larger circle at $A$ and $B$. Let $M$ be a point on the larger circle, and let lines $MA$ and $MB$ intersect the corresponding smaller circles at $A'$ and $B'$. Prove that $A'B'$ is parallel to $AB$.
1996 Israel National Olympiad
Let $a$ be a prime number and $n > 2$ an integer.

Find all integer solutions of the equation $x^n +ay^n = a^2z^n$

.
1996 Israel National Olympiad
Find all polynomials $P(x)$ satisfying $P(x+1)-2P(x)+P(x-1)= x$ for all $x$
1996 Israel National Olympiad
The angles of an acute triangle $ABC$ at $\alpha , \beta, \gamma$. Let $AD$ be a height, $CF$ a median, and $BE$ the bisector of $\angle B$. Show that $AD,CF$ and $BE$ are concurrent if and only if $\cos \gamma \tan\beta = \sin \alpha$ .
1996 Israel National Olympiad
Eight guests arrive to a hotel with four rooms. Each guest dislikes at most three other guests and doesn’t want to share a room with any of them (this feeling is mutual). Show that the guests can reside in the four rooms, with two persons in each room
1996 Israel National Olympiad
Suppose that the circumradius $R$ and the inradius $r$ of a triangle $ABC$ satisfy $R = 2r$. Prove that the triangle is equilateral.
1996 Israel National Olympiad
Let $x,y,z$ be real numbers with $|x|,|y|,|z| > 2$. What is the smallest possible value of $|xyz+2(x+y+z)|$ ?
1996 Israel National Olympiad
Find all positive integers $a,b,c$ such that $a^2 = 4(b+c)$ and $a^3 -2b^3 -4c^3 =\frac12 abc$
1996 Israel National Olympiad
Consider the function $f : N \to N$ given by

(i) $f(1) = 1$,

(ii) $f(2n) = f(n)$ for any $n \in N$,

(iii) $f(2n+1) = f(2n)+1$ for any $n  \in N$.

(a) Find the maximum value of $f(n)$ for $1 \le  n \le 1995$;

(b) Find all values of $f$ on this interval.
1995 Israel Mathematical Olympiad
Solve the system

$x+\log\left(x+\sqrt{x^2+1}\right)=y$

$y+\log\left(y+\sqrt{y^2+1}\right)=z$

$z+\log\left(z+\sqrt{z^2+1}\right)=x$
1995 Israel Mathematical Olympiad
Let $PQ$ be the diameter of semicircle $H$. Circle $O$ is internally tangent to $H$ and tangent to $PQ$ at $C$. Let $A$ be a point on $H$ and $B$ a point on $PQ$ such that $AB\perp PQ$ and is tangent to $O$. Prove that $AC$ bisects $\angle PAB$
1995 Israel Mathematical Olympiad
If $k$ and $n$ are positive integers, prove the inequality

$$\frac{1}{kn} +\frac{1}{kn+1} +...+\frac{1}{(k+1)n-1} \ge n \left(\sqrt[n]{\frac{k+1}{k}}-1\right)$$
1995 Israel Mathematical Olympiad
Find all integers $m$ and $n$ satisfying $m^3 -n^3 - 9mn = 27$.
1995 Israel Mathematical Olympiad
Let $n$ be an odd positive integer and let $x_1,x_2,...,x_n$ be n distinct real numbers that satisfy $|x_i -x_j| \le 1$ for $1 \le i < j \le n$. Prove that

$$\sum_{i<j} |x_i -x_j| \le \left[\frac{n}{2} \right] \left(\left[\frac{n}{2} \right]-1 \right)$$
1995 Israel Mathematical Olympiad
A $1995 \times 1995$ square board is given. A coloring of the cells of the board is called good  if the cells can be rearranged so as to produce a colored square board that is symmetric with respect to the main diagonal. Find all values of $k$ for which any $k$-coloring of the given board is good.
1995 Israel Mathematical Olympiad
For certain $n$ countries there is an airline connecting any two countries, but some of the airlines are closed. Show that if the number of the closed airlines does not exceed $n-3$, then one can make a round trip using the remaining airlines, starting from one of the countries, visiting every country exactly once and returning to the starting country.
1995 Israel Mathematical Olympiad
A real number $\alpha$ is given. Find all functions $f : R^+ \to R^+$ satisfying

$\alpha x^2f\left(\frac{1}{x}\right) +f(x) =\frac{x}{x+1}$ for all $x > 0$.
2021 ITAMO
A positive integer $m$ is said to be $\emph{zero taker}$ if there exists a positive integer $k$ such that:



$k$ is a perfect square;

$m$ divides $k$;

the decimal expression of $k$ contains at least $2021$ '0' digits, but the last digit of $k$ is not equal to $0$.



Find all positive integers that are zero takers.
2021 ITAMO
Let $ABC$ a triangle and let $I$ be the center of its inscribed circle. Let $D$ be the symmetric point of $I$ with respect to $AB$ and $E$ be the symmetric point of $I$ with respect to $AC$.

Show that the circumcircles of the triangles $BID$ and $CIE$ are eachother tangent.
2021 ITAMO
Given two fractions $a/b$ and $c/d$ we define their pirate sum as:

$\frac{a}{b} \star \frac{c}{d} = \frac{a+c}{b+d}$ where the two initial fractions are simplified the most possible, like the result.

For example, the pirate sum of $2/7$ and $4/5$ is $1/2$.

Given an integer $n \ge 3$, initially on a blackboard there are the fractions:

$\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, ..., \frac{1}{n}$.

At each step we choose two fractions written on the blackboard, we delete them and write at their place their pirate sum. Continue doing the same thing until on the blackboard there is only one fraction.

Determine, in function of $n$, the maximum and the minimum possible value for the last fraction.
2021 ITAMO
Let $ABC$ be an acute-angled triangle, let $M$ be the midpoint of $BC$ and let $H$ be the foot of the $B$-altitude. Let $Q$ be the circumcenter of $ABM$ and let $X$ be the intersection point between $BH$ and the axis of $BC$.



Show that the circumcircles of the two triangles $ACM$, $AXH$ and the line $CQ$ pass through a same point if and only if $BQ$ is perpendicular to $CQ$.
2021 ITAMO
A sequence $x_1, x_2, ..., x_n, ...$ consists of an initial block of $p$ positive distinct integers that then repeat periodically. This means that $\{x_1, x_2, \dots, x_p\}$ are $p$ distinct positive integers and $x_{n+p}=x_n$ for every positive integer $n$. The terms of the sequence are not known and the goal is to find the period $p$. To do this, at each move it possible to reveal the value of a term of the sequence at your choice.



(a) Knowing that $1 \le p \le 10$, find the least $n$ such that there is a strategy which allows to find $p$ revealing at most $n$ terms of the sequence.

(b) Knowing that $p$ is one of the first $k$ prime numbers, find for which values of $k$ there exist a strategy that allows to find $p$ revealing at most $5$ terms of the sequence.
2020 ITAMO
Let $\omega$ be a circle and let $A,B,C,D,E$  be five points on $\omega$  in this order. Define $F=BC\cap DE$, such that the points $F$ and $A$ are on opposite sides, with regard to the line $BE$ and the line $AE$ is tangent to the circumcircle of the triangle $BFE$.

a) Prove that the lines $AC$ and $DE$ are parallel

b) Prove that $AE=CD$
2020 ITAMO
Determine all the pairs $(a,b)$ of positive integers, such that all the following three conditions are satisfied:

1- $b>a$ and $b-a$ is a prime number

2- The last digit of the number $a+b$ is $3$

3- The number $ab$ is a square of an integer.
2020 ITAMO
Let $a_1, a_2, \dots, a_{2020}$ and $b_1, b_2, \dots, b_{2020}$ be real numbers(not necessarily distinct). Suppose that the set of positive integers $n$ for which the following equation:

$|a_1|x-b_1|+a_2|x-b_2|+\dots+a_{2020}|x-b_{2020}||=n$ (1) has exactly two real solutions, is a finite set. Prove that the set of positive integers $n$ for which the equation (1) has at least one real solution, is also a finite set.
2020 ITAMO
Let $ABC$ be an acute-angled triangle with $AB=AC$, let $D$ be the foot of perpendicular, of the point $C$, to the line $AB$ and the point $M$ is the midpoint of $AC$. Finally, the point $E$ is the second intersection of the line $BC$ and the circumcircle of $\triangle CDM$. Prove that the lines $AE, BM$ and $CD$ are concurrents if and only if $CE=CM$.
2020 ITAMO
Le $S$ be the set of positive integers greater than or equal to $2$. A function $f: S\rightarrow S$ is italian if $f$ satifies all the following three conditions:

1) $f$ is surjective

2) $f$ is increasing in the prime numbers(that is, if $p_1<p_2$ are prime numbers, then $f(p_1)<f(p_2)$)

3) For every $n\in S$ the number $f(n)$ is the product of $f(p)$, where $p$ varies among all the primes which divide $n$ (For instance, $f(360)=f(2^3\cdot 3^2\cdot 5)=f(2)\cdot f(3)\cdot f(5)$).

Determine the maximum and the minimum possible value of $f(2020)$, when $f$ varies among all italian functions.
2020 ITAMO
In each cell of a table $8\times 8$ lives a knight or a liar. By the tradition, the knights always say the truth and the liars always lie. All the inhabitants of the table say the following statement "The number of liars in my column is (strictly) greater than the number of liars in my row". Determine how many possible configurations are compatible with the statement.
2019 ITAMO
Let $ABCDEF$ be a hexagon inscribed in a circle such that $AB=BC,$ $CD=DE$ and $EF=AF.$ Prove that segments $AD,$ $BE$ and $CF$ are concurrent$.$
2019 ITAMO
Let $p,q$ be prime numbers$.$ Prove that if $p+q^2$ is a perfect square$,$ then $p^2+q^n$ is not a perfect square for any positive integer $n.$
2019 ITAMO
Let $n>2$ be an integer$.$ We want to color in red exactly $n+1$ of the numbers $1,2,\cdots,2n-1, 2n$ so that there do not exists three distinct red integers $x,y,z$ satisfying $x+y=z.$ Prove that there is one and one only way to color the red numbers according to the given condition$.$
2019 ITAMO
Let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x.$



Let $\lambda \geq 1$ be a real number and $n$ be a positive integer with the property that $\lfloor \lambda^{n+1}\rfloor, \lfloor \lambda^{n+2}\rfloor ,\cdots, \lfloor \lambda^{4n}\rfloor$ are all perfect squares$.$ Prove that $\lfloor \lambda \rfloor$ is a perfect square$.$
2019 ITAMO
Let $ABC$ be an acute angled triangle$.$ Let $D$ be the foot of the internal angle bisector of $\angle BAC$ and let $M$ be the midpoint of $AD.$ Let $X$ be a point on segment $BM$ such that $\angle MXA=\angle DAC.$ Prove that $AX$ is perpendicular to $XC.$
2019 ITAMO
Alberto and Barbara are sitting one next to each other in front of a table onto which they arranged in a line $15$ chocolates. Some of them are milk chocolates, while the others are dark chocolates. Starting from Alberto, they play the following game: during their turn, each player eats a positive number of consecutive chocolates, starting from the leftmost of the remaining ones, so that the number of chocolates eaten that are of the same type as the first one is odd (for example, if after some turns the sequence of the remaining chocolates is $\text{MMDMD},$ where $\text{M}$ stands for $\emph{milk}$ and $\text{D}$ for $\emph{dark},$ the player could either eat the first chocolate, the first $4$ chocolates or all $5$ of them). The player eating the last chocolate wins.



Among all $2^{15}$ possible initial sequences of chocolates, how many of them allow Barbara to have a winning strategy?
2018 ITAMO
$1.$A bottle in the shape of a cone lies on its base. Water is poured into the bottle until its level reaches a distance of 8 centimeters from the vertex of the cone (measured vertically). We now turn the bottle upside down without changing the amount of water it contains; This leaves an empty space in the upper part of the cone that is 2 centimeters high.



Find the height of the bottle.
2018 ITAMO
$2.$Let $ABC$ be an acute-angeled triangle , non-isosceles and with barycentre $G$ (which is , in fact , the intersection of the medians).Let $M$ be the midpoint of $BC$ , and let Ω be the circle with centre $G$ and radius $GM$ , and let $N$ be the point of intersection between Ω and $BC$ that is distinct from $M$.Let $S$ be the symmetric point of $A$ with respect to $N$ , that is , the point on the line $AN$ such that $AN=NS$.



Prove that $GS$ is perpendicular to $BC$
2018 ITAMO
Let $x_1,x_2, ... , x_n$ be positive integers,Asumme that in their decimal representations no $x_i$ "prolongs" $x_j$.For instance , $123$ prolongs $12$ , $459$ prolongs $4$ , but $124$ does not prolog $123$.

Prove that :

$\frac {1}{x_1}+\frac {1}{x_2}+...+\frac {1}{x_n} < 3$.
2018 ITAMO
$4.$ Let $N$ be an integer greater than $1$.Denote by $x$ the smallest positive integer with the following property:there exists a positive integer $y$ strictly less than $x-1$ , such that $x$ divides $N+y$.Prove that x is either $p^n$ or $2p$ , where $p$ is a prime number and $n$ is a positive integer
2018 ITAMO
$5.$Let x be a real number with $0<x<1$ and let $0.c_1c_2c_3...$ be the decimal expansion of x.Denote by $B(x)$ the set of all subsequences of $c_1c_2c_3$ that consist of 6 consecutive digits.

For instance , $B(\frac{1}{22})={045454,454545,545454}$

Find the minimum number of elements of $B(x)$ as $x$ varies among all irrational numbers with $0<x<1$
2018 ITAMO
Let $ABC$ be a triangle with $AB=AC$ and let $I$ be its incenter. Let $\Gamma$ be the circumcircle of $ABC$. Lines $BI$ and $CI$ intersect $\Gamma$ in two new points, $M$ and $N$ respectively. Let $D$ be another point on $\Gamma$ lying on arc $BC$ not containing $A$, and let $E,F$ be the intersections of $AD$ with $BI$ and $CI$, respectively. Let $P,Q$ be the intersections of $DM$ with $CI$ and of $DN$ with $BI$ respectively.



(i) Prove that $D,I,P,Q$ lie on the same circle $\Omega$

(ii) Prove that lines $CE$ and $BF$ intersect on $\Omega$
2017 ITAMO
Let $a$ and $b$ be positive real numbers. Consider a regular hexagon of side $a$, and build externally on its sides six rectangles of sides $a$ and $b$. The new twelve vertices lie on a circle. Now repeat the same construction, but this time exchanging the roles of $a$ and $b$; namely; we start with a regular hexagon of side $b$ and we build externally on its sides six rectangles of sides $a$ and $b$. The new twelve vertices lie on another circle.

Show that the two circles have the same radius.
2017 ITAMO
Let $n\geq 2$ be an integer. Consider the solutions of the system

$$\begin{cases}

      n=a+b-c \\
      n=a^2+b^2-c^2

    \end{cases}$$where $a,b,c$ are integers. Show that there is at least one solution and that the solutions are finitely many.
2017 ITAMO
Madam Mim has a deck of $52$ cards, stacked in a pile with their backs facing up. Mim separates the small pile consisting of the seven cards on the top of the deck, turns it upside down, and places it at the bottom of the deck. All cards are again in one pile, but not all of them face down; the seven cards at the bottom do, in fact, face up. Mim repeats this move until all cards have their backs facing up again. In total, how many moves did Mim make $?$
2017 ITAMO
Let $ABCD$ be a thetraedron with the following propriety: the four lines connecting a vertex and the incenter of opposite face are concurrent. Prove $AB \cdot CD= AC \cdot BD = AD\cdot BC$.
2017 ITAMO
Let $ x_1 , x_2, x_3 ...$ a succession of positive integers such that for every couple of positive integers $(m,n)$ we have $ x_{mn} \neq x_{m(n+1)}$ . Prove that there exists a positive integer $i$ such that $x_i \ge 2017 $.
2017 ITAMO
Prove that there are infinitely many positive integers $m$ such that the number of odd distinct prime factor of $m(m+3)$ is a multiple of $3$.
2016 ITAMO
Let $ABC$ be a triangle, and let $D$ and $E$ be the orthogonal projections of $A$ onto the internal bisectors from $B$ and $C$. Prove that $DE$ is parallel to $BC$.
2016 ITAMO
A mathematical contest had $3$ problems, each of which was given a score between $0$ and $7$ ($0$ and $7$ included). It is known that, for any two contestants, there exists at most one problem in which they have obtained the same score (for example, there are no two contestants whose ordered scores are $7,1,2$ and $7,1,5$, but there might be two contestants whose ordered scores are $7,1,2$ and $7,2,1$). Find the maximum number of contestants.
2016 ITAMO
Let $\Gamma$ be the excircle of triangle $ABC$ opposite to the vertex $A$ (namely, the circle tangent to $BC$ and to the prolongations of the sides $AB$ and $AC$ from the part $B$ and $C$). Let $D$ be the center of $\Gamma$ and $E$, $F$, respectively, the points in which $\Gamma$ touches the prolongations of $AB$ and $AC$. Let $J$ be the intersection between the segments $BD$ and $EF$.



Prove that $\angle CJB$ is a right angle.
2016 ITAMO
Determine all pairs of positive integers $(a,n)$ with $a\ge n\ge 2$ for which $(a+1)^n+a-1$ is a power of $2$.
2016 ITAMO
Let $x_0,x_1,x_2,\ldots$ be a sequence of rational numbers defined recursively as follows: $x_0$ can be any rational number and, for $n\ge 0$,

$$

x_{n+1}=\begin{cases} \left|\frac{x_n}2-1\right| & \text{if the numerator of }x_n\text{ is even}, \\
\left|\frac1{x_n}-1\right| & \text{if the numerator of }x_n\text{ is odd},\end{cases}

$$where by numerator of a rational number we mean the numerator of the fraction in its lowest terms. Prove that for any value of $x_0$:

(a) the sequence contains only finitely many distinct terms;

(b) the sequence contains exactly one of the numbers $0$ and $2/3$ (namely, either there exists an index $k$ such that $x_k=0$, or there exists an index $m$ such that $x_m=2/3$, but not both).
2016 ITAMO
A mysterious machine contains a secret combination of $2016$ integer numbers $x_1,x_2,\ldots,x_{2016}$. It is known that all the numbers in the combination are equal but one. One may ask questions to the machine by giving to it a sequence of $2016$ integer numbers $y_1,\ldots,y_{2016}$, and the machine answers by telling the value of the sum

$$

x_1y_1+\dots+x_{2016}y_{2016}.

$$After answering the first question, the machine accepts a second question and then a third one, and so on.



Determine how many questions are necessary to determine the combination:

(a) knowing that the number which is different from the others is equal to zero;

(b) not knowing what the number different from the others is.
2015 ITAMO
Let ABCDA'B'C'D' be a rectangular parallelipiped, where ABCD is the lower face and A, B, C and D' are below A', B', C' and D', respectively. The parallelipiped is divided into eight parts by three planes parallel to its faces. For each vertex P, let V P denote the volume of the part containing P. Given that V A= 40, V C = 300 , V B' = 360 and V C'= 90, find the volume of ABCDA'B'C'D'.
2015 ITAMO
A music streaming service proposes songs classified in $10$ musical genres, so that each song belong to one and only one gender. The songs are played one after the other: the first $17$ are chosen by the user, but starting from the eighteenth the service automatically determines which song to play. Elisabetta has noticed that, if one makes the classification of which genres they appear several times during the last $17$ songs played, the new song always belongs to the genre at the top of the ranking or, in case of same merit, at one of the first genres.

Prove that, however, the first $17$ tracks are chosen, from a certain point onwards the songs proposed are

all of the same kind.
2015 ITAMO
Let ABC a triangle, let K be the foot of the bisector relative to BC and J be the foot of the trisectrix relative to BC closer to the side AC (3* m(JAC)=m(CAB) ). Let C' and B' be two point on the line  AJ on the side of J with respect to A, such that AC'=AC and AB=AB'. Prove that ABB'C is cyclic if and only if lines C'K and BB' are parallel.
2015 ITAMO
Determine all pairs of integers $(a, b)$  that solve the equation $a^3 + b^3 + 3ab = 1$.
2015 ITAMO
Let $AB$ be a chord of a circle $\Gamma$ and let $C$ be a point on the segment $AB$. Let $r$ be a line through $C$ which intersects $\Gamma$ at the points $D,E$; suppose that $D,E$ lie on different sides with respect to the perpendicular bisector of $AB$.

Let $\Gamma_D$ be the circumference which is externally tangent to $\Gamma$ at $D$ and touches the line $AB$ at $F$. Let $\Gamma_E$ be the circumference which is externally tangent to $\Gamma$ at $E$ and touches the line $AB$ at $G$.

Prove that $CA=CB$ if and only if $CF=CG$.
2015 ITAMO
Ada and Charles play the following game:at the beginning, an integer n>1 is written on the blackboard.In turn, Ada and Charles remove the number k that they find on the blackboard.In turn Ad and Charles remove the number k that they find on the blackboard and they replace it :

1 -either with a positive divisor k different from 1 and k

2- or with k+1

At the beginning each players have a thousand points each.When a player choses move 1, he/she gains one point;when a player choses move 2, he/she loses one point.The game ends when one of the tho players is left with zero points and this player loses the game.Ada moves first.For what values Chares has a winning strategy?
2014 ITAMO
For every $3$-digit natural number $n$ (leading digit of $n$ is nonzero), we consider the number $n_0$ obtained from $n$ eliminating all possible digits that are zero. For example, if $n = 207$, then $n_0 = 27$. Determine the number of three-digit positive integers $n$, for which $n_0$ is a divisor of $n$ different from $n$.
2014 ITAMO
Let $ABC$ be a triangle. Let $H$ be the foot of the altitude from $C$ on $AB$. Suppose that $AH = 3HB$. Suppose in addition we are given that



(a) $M$ is the midpoint of $AB$;

(b) $N$ is the midpoint of $AC$;

(c) $P$ is a point on the opposite side of $B$ with respect to the line $AC$ such that $NP = NC$ and $PC = CB$.



Prove that $\angle APM = \angle PBA$.