Navvye/Olympiad-Problems-CSV huggingface dataset:
2015 Paraguay Mathematical Olympiad
A cube is divided into $8$ smaller cubes of the same size, as shown in the figure. Then, each of these small cubes is divided again into $8$ smaller cubes of the same size. This process is done $4$ more times to each resulting cube. What is the ratio between the sum of the total areas of all the small cubes resulting from the last division and the total area of the initial cube?
2015 Paraguay Mathematical Olympiad
The sidelengths of a triangle are natural numbers multiples of $7$, smaller than $40$. How many triangles satisfy these conditions?
2015 Paraguay Mathematical Olympiad
In the figure, the rectangle is formed by $4$ smaller equal rectangles.

If we count the total number of rectangles in the figure we find $10$.

How many rectangles in total will there be in a rectangle that is formed by $n$ smaller equal rectangles?
2014 Paraguay Mathematical Olympiad
Consider a square of side length $12$ centimeters. Irina draws another square that has $8$ centimeters more of perimeter than the original square. What is the area of the square drawn by Irina?
2014 Paraguay Mathematical Olympiad
Clau writes all four-digit natural numbers where $3$ and $7$ are always together. How many digits does she write in total?
2014 Paraguay Mathematical Olympiad
Juan chooses a five-digit positive integer.  Maria erases the ones digit and gets a four-digit number. The sum of this four-digit number and the original five-digit number is $52,713$. What can the sum of the five digits of the original number be?
2014 Paraguay Mathematical Olympiad
Nair and Yuli play the following game:

$1.$ There is a coin to be moved along a horizontal array with $203$ cells.

$2.$ At the beginning, the coin is at the first cell, counting from left to right.

$3.$ Nair plays first.

$4.$ Each of the players, in their turns, can move the coin $1$, $2$, or $3$ cells to the right.

$5.$ The winner is the one who reaches the last cell first.

What strategy does Nair need to use in order to always win the game?
2014 Paraguay Mathematical Olympiad
Let $ABC$ be a triangle with area $92$ square centimeters. Calculate the area of another triangle whose sides have the same lengths as the medians of triangle $ABC$.
2013 Paraguay Mathematical Olympiad
Evaluate the following expression:



$2013^2 + 2011^2 + … + 5^2 + 3^2 -2012^2 -2010^2-…-4^2-2^2$
2013 Paraguay Mathematical Olympiad
Let $ABC$ be a triangle with area $9$, and let $M$ and $N$ be the midpoints of sides $AB$ and $AC$, respectively.

Let $P$ be the point in side $BC$ such that $PC = \frac{1}{3}BC$.  Let $O$ be the intersection point between $PN$ and $CM$.

Find the area of the quadrilateral $BPOM$.
2013 Paraguay Mathematical Olympiad
We divide a natural number $N$, that has $k$ digits, by $19$ and get a residue of $10^{k-2} -Q$, where $Q$ is the quotient and $Q < 101$. Also, $10^{k-2}-Q$ is larger than $0$. How many possible values of $N$ are there?
2013 Paraguay Mathematical Olympiad
Pedro and Juan are playing the following game:

$-$ There are $2$ piles of rocks, with $X$ rocks in one pile and $Y$ rocks in the other pile ($X < 12, Y < 11$).

$-$ Each player can draw:

-- $1$ rock from one of the piles, or

-- $2$ rocks from one of the piles, or

-- $1$ rock from each pile, or

-- $2$ rock from one pile and $1$ from the other pile.

Each player must perform one of these four operations in their turns.

The looser is the one who takes the last rock.

Pedro plays first and has a winning strategy.

What are the three maximum possible values of ($X+Y$)?
2013 Paraguay Mathematical Olympiad
Let $ABC$ be an obtuse triangle, with $AB$ being the largest side.

Draw the angle bisector of $\measuredangle BAC$. Then, draw the perpendiculars to this angle bisector from vertices $B$ and $C$, and call their feet $P$ and $Q$, respectively.

$D$ is the point in the line $BC$ such that $AD \perp AP$.

Prove that the lines $AD$, $BQ$ and $PC$ are concurrent.
2012 Paraguay Mathematical Olympiad
Define a list of number with the following properties:

- The first number of the list is a one-digit natural number.

- Each number (since the second) is obtained by adding $9$ to the number before in the list.

- The number $2012$ is in that list.

Find the first number of the list.
2012 Paraguay Mathematical Olympiad
The traveler ant is walking over several chess boards. He only walks vertically and horizontally through the squares of the boards and does not pass two or more times over the same square of a board.

a) In a $4$x$4$ board, from which squares can he begin his travel so that he can pass through all the squares of the board?

b) In a $5$x$5$ board, from which squares can he begin his travel so that he can pass through all the squares of the board?

c) In a $n$x$n$ board, from which squares can he begin his travel so that he can pass through all the squares of the board?
2012 Paraguay Mathematical Olympiad
Let $ABC$ be a triangle (right in $B$) inscribed in a semi-circumference of diameter $AC=10$. Determine the distance of the vertice $B$ to the side $AC$ if the median corresponding to the hypotenuse is the geometric mean of the sides of the triangle.
2012 Paraguay Mathematical Olympiad
Find all four-digit numbers $\overline{abcd}$ such that they are multiples of $3$ and that $\overline{ab}-\overline{cd}=11$.

($\overline{abcd}$ is a four-digit number; $\overline{ab}$ is a two digit-number as $\overline{cd}$ is).
2012 Paraguay Mathematical Olympiad
Let $ABC$ be an equilateral triangle. Let $Q$ be a random point on $BC$, and let $P$ be the meeting point of $AQ$ and the circumscribed circle of $\triangle ABC$.

Prove that $\frac{1}{PQ}=\frac{1}{PB}+\frac{1}{PC}$.
2011 Paraguay Mathematical Olympiad
Find the value of the following expression:



$\frac{1}{2} + (\frac{1}{3} + \frac{2}{3}) + (\frac{1}{4} + \frac{2}{4} + \frac{3}{4}) + \ldots + (\frac{1}{1000} + \frac{2}{1000} + \ldots + \frac{999}{1000})$
2011 Paraguay Mathematical Olympiad
In a triangle $ABC$, let $D$ and $E$ be the midpoints of $AC$ and $BC$ respectively. The distance from the midpoint of $BD$ to the midpoint of $AE$ is $4.5$. What is the length of side $AB$?
2011 Paraguay Mathematical Olympiad
If number $\overline{aaaa}$ is divided by $\overline{bb}$, the quotient is a number between $140$ and $160$ inclusively, and the remainder is equal to $\overline{(a-b)(a-b)}$. Find all pairs of positive integers $(a,b)$ that satisfy this.
2011 Paraguay Mathematical Olympiad
A positive integer $N$ is divided in $n$ parts inversely proportional to the numbers $2, 6, 12, 20, \ldots$

The smallest part is equal to $\frac{1}{400} N$. Find the value of $n$.
2011 Paraguay Mathematical Olympiad
In a rectangle triangle, let $I$ be its incenter and $G$ its geocenter. If $IG$ is parallel to one of the catheti and measures $10 cm$, find the lengths of the two catheti of the triangle.
2010 Paraguay Mathematical Olympiad
The picture below shows the way Juan wants to divide a square field in three regions, so that all three of them share a well at vertex $B$. If the side length of the field is $60$ meters, and each one of the three regions has the same area, how far must the points $M$ and $N$ be from $D$?



Note: the area of each region includes the area the well occupies.



[asy]

pair A=(0,0),B=(60,0),C=(60,-60),D=(0,-60),M=(0,-40),N=(20,-60);

pathpen=black;

D(MP("A",A,W)--MP("B",B,NE)--MP("C",C,SE)--MP("D",D,SW)--cycle);

D(B--MP("M",M,W));

D(B--MP("N",N,S));

D(CR(B,3));[/asy]
2010 Paraguay Mathematical Olympiad
A series of figures is shown in the picture below, each one of them created by following a secret rule. If the leftmost figure is considered the first figure, how many squares will the 21st figure have?







Note: only the little squares are to be counted (i.e., the $2 \times 2$ squares, $3 \times 3$ squares, $\dots$ should not be counted)





Extra (not part of the original problem): How many squares will the 21st figure have, if we consider all $1 \times 1$ squares, all $2 \times 2$ squares, all $3 \times 3$ squares, and so on?.
2010 Paraguay Mathematical Olympiad
In a triangle $ABC$, let $M$ be the midpoint of $AC$. If $BC = \frac{2}{3} MC$ and $\angle{BMC}=2 \angle{ABM}$, determine $\frac{AM}{AB}$.
2010 Paraguay Mathematical Olympiad
Find all 4-digit numbers $\overline{abcd}$ that are multiples of $11$, such that the 2-digit number $\overline{ac}$ is a multiple of $7$ and $a + b + c + d = d^2$.
2010 Paraguay Mathematical Olympiad
In a triangle $ABC$, let $D$, $E$ and $F$ be the feet of the altitudes from $A$, $B$ and $C$ respectively. Let $D'$, $E'$ and $F'$ be the second intersection of lines $AD$, $BE$ and $CF$ with the circumcircle of $ABC$. Show that the triangles $DEF$ and $D'E'F'$ are similar.
2009 Paraguay Mathematical Olympiad
Find the value of the following expression:



$2 + 33 + 6 + 35 + 10 + 37 + \ldots + 1194 + 629 + 1198 + 631$
2009 Paraguay Mathematical Olympiad
In a triangle $ABC$ ($\angle{BCA} = 90^{\circ}$), let $D$ be the intersection of $AB$ with a circumference with diameter $BC$. Let $F$ be the intersection of $AC$ with a line tangent to the circumference. If $\angle{CAB} = 46^{\circ}$, find the measure of $\angle{CFD}$.
2009 Paraguay Mathematical Olympiad
Find out how many positive integers $n$ not larger than $2009$ exist such that the last digit of $n^{20}$ is $1$.
2009 Paraguay Mathematical Olympiad
Let $a_1, a_2, ..., a_n $ be a sequence such that the arithmetic mean of the $n$ terms is $n$. Consider $n = 2009$. Determine the sum of the $2009$ terms of the sequence.
2009 Paraguay Mathematical Olympiad
In a triangle $ABC$, let $I$ be its incenter. The distance from $I$ to the segment $BC$ is $4 cm$ and the distance from that point to vertex $B$ is $12 cm$. Let $D$ be a point in the plane region between segments $AB$ and $BC$ such that $D$ is the center of a circumference that is tangent to lines $AB$ and $BC$ and passes through $I$. Find all possible values of the length $BD$.
2008 Paraguay Mathematical Olympiad
How many positive integers $n < 500$ exist such that its prime factors are exclusively $2$, $7$, $11$, or a combination of these?
2008 Paraguay Mathematical Olympiad
Find for which values of $n$, an integer larger than $1$ but smaller than $100$, the following expression has its minimum value:



$S = |n-1| + |n-2| + \ldots + |n-100|$
2008 Paraguay Mathematical Olympiad
Let $ABC$ be a triangle, where $AB = AC$ and $BC = 12$. Let $D$ be the midpoint of $BC$. Let $E$ be a point in $AC$ such that $DE \perp AC$. Let $F$ be a point in $AB$ such that $EF \parallel BC$. If $EC = 4$, determine the length of $EF$.
2008 Paraguay Mathematical Olympiad
Let $\Gamma$ be a circumference and $A$ a point outside it. Let $B$ and $C$ be points in $\Gamma$ such that $AB$ and $AC$ are tangent to $\Gamma$. Let $P$ be a point in $\Gamma$. Let $D$, $E$ and $F$ be points in $BC$, $AC$ and $AB$ respectively, such that $PD \perp BC$, $PE \perp AC$, and $PF \perp AB$.

Show that $PD^2 = PE \cdot PF$
2008 Paraguay Mathematical Olympiad
Let $m,n,p$ be rational numbers such that $\sqrt{m} + \sqrt{n} + \sqrt{p}$ is a rational number. Prove that $\sqrt{m}, \sqrt{n}, \sqrt{p}$ are also rational numbers
2007 Paraguay Mathematical Olympiad
A list with $2007$ positive integers is written on a board, such that the arithmetic mean of all the numbers is $12$. Then, seven consecutive numbers are erased from the board. The arithmetic mean of the remaining numbers is $11.915$.

The seven erased numbers have this property: the sixth number is half of the seventh, the fifth number is half of the sixth, and so on. Determine the $7$ erased numbers.
2007 Paraguay Mathematical Olympiad
Let $ABCD$ be a square, such that the length of its sides are integers. This square is divided in $89$ smaller squares, $88$ squares that have sides with length $1$, and $1$ square that has sides with length $n$, where $n$ is an integer larger than $1$. Find all possible lengths for the sides of $ABCD$.
2007 Paraguay Mathematical Olympiad
Let $ABCD$ be a square, $E$ and $F$ midpoints of $AB$ and $AD$ respectively, and $P$ the intersection of $CF$ and $DE$.

a) Show that $DE \perp CF$.

b) Determine the ratio $CF : PC : EP$
2007 Paraguay Mathematical Olympiad
Each number from the set $\{1, 2, 3, 4, 5, 6, 7\}$ must be written in each circle of the diagram, so that the sum of any three aligned numbers is the same (e.g., $A+D+E = D+C+B$). What number cannot be placed on the circle $E$?
2007 Paraguay Mathematical Olympiad
Let $A, B, C,$ be points in the plane, such that we can draw $3$ equal circumferences in which the first one passes through $A$ and $B$, the second one passes through $B$ and $C$, the last one passes through $C$ and $A$, and all $3$ circumferences share a common point $P$.

Show that the radius of each of these circumferences is equal to the circumradius of triangle $ABC$, and that $P$ is the orthocenter of triangle $ABC$.
2006 Paraguay Mathematical Olympiad
What are the last two digits of the decimal representation of $21^{2006}$?
2006 Paraguay Mathematical Olympiad
Consider all right triangles with integer sides such that the length of the hypotenuse and one of the two sides are consecutive. How many such triangles exist?
2006 Paraguay Mathematical Olympiad
Let $\Gamma_A$, $\Gamma_B$, $\Gamma_C$ be circles such that $\Gamma_A$ is tangent to $\Gamma_B$ and $\Gamma_B$ is tangent to $\Gamma_C$. All three circles are tangent to lines $L$ and $M$, with $A$, $B$, $C$ being the tangency points of $M$ with $\Gamma_A$, $\Gamma_B$, $\Gamma_C$, respectively. Given that $12=r_A<r_B<r_C=75$, calculate:



a) the length of $r_B$.

b) the distance between point $A$ and the point of intersection of lines $L$ and $M$.
2006 Paraguay Mathematical Olympiad
Consider all pairs of positive integers $(a,b)$, with $a<b$, such that



$\sqrt{a} +\sqrt{b} = \sqrt{2.160}$



Determine all posible values of $a$.
2006 Paraguay Mathematical Olympiad
Let $ABC$ be a triangle, and let $P$ be a point on side $BC$ such that $\frac{BP}{PC}=\frac{1}{2}$. If $\measuredangle ABC$ $=$ $45^{\circ}$ and $\measuredangle APC$ $=$ $60^{\circ}$, determine $\measuredangle ACB$ without trigonometry.
2020 Philippine MO
A T-tetromino is formed by adjoining three unit squares to form a $1 \times 3$ rectangle, and adjoining on top of the middle square a fourth unit square.

Determine the least number of unit squares that must be removed from a $202 \times 202$ grid so that it can be tiled using T-tetrominoes.
2020 Philippine MO
Determine all positive integers $k$ for which there exist positive integers $r$ and $s$ that satisfy the equation $$(k^2-6k+11)^{r-1}=(2k-7)^{s}.$$
2020 Philippine MO
Define the sequence $\{a_i\}$ by $a_0=1$, $a_1=4$, and $a_{n+1}=5a_n-a_{n-1}$ for all $n\geq 1$. Show that all terms of the sequence are of the form $c^2+3d^2$ for some integers $c$ and $d$.
2020 Philippine MO
Let $\triangle ABC$ be an acute triangle with circumcircle $\Gamma$ and $D$ the foot of the altitude from $A$. Suppose that $AD=BC$. Point $M$ is the midpoint of $DC$, and the bisector of $\angle ADC$ meets $AC$ at $N$. Point $P$ lies on $\Gamma$ such that lines $BP$ and $AC$ are parallel. Lines $DN$ and $AM$ meet at $F$, and line $PF$ meets $\Gamma$ again at $Q$. Line $AC$ meets the circumcircle of $\triangle PNQ$ again at $E$. Prove that $\angle DQE = 90^{\circ}$.
2019 Philippine MO
Find all functions $f : R \to R$ such that $f(2xy) + f(f(x + y)) = xf(y) + yf(x) + f(x + y)$ for all real numbers $x$ and $y$.
2019 Philippine MO
Twelve students participated in a theater festival consisting of $n$ different performances. Suppose there were six students in each performance, and each pair of performances had at most two students in common. Determine

the largest possible value of $n$.
2019 Philippine MO
Find all triples $(a, b, c)$ of positive integers such that

$a^2 + b^2 = n\cdot lcm(a, b) + n^2$

$b^2 + c^2 = n \cdot lcm(b, c) + n^2$

$c^2 + a^2 = n \cdot lcm(c, a) + n^2$

for some positive integer $n$.
2019 Philippine MO
In acute triangle $ABC $with $\angle BAC > \angle BCA$, let $P$ be the point on side $BC$ such that $\angle PAB = \angle BCA$. The circumcircle of triangle $AP B$ meets side $AC$ again at $Q$. Point $D$ lies on segment $AP$ such that $\angle QDC = \angle CAP$.

Point $E$ lies on line $BD$ such that $CE = CD$. The circumcircle of triangle $CQE$ meets segment $CD$ again at $F$, and line $QF$ meets side $BC$ at $G$. Show that $B, D, F$, and $G$ are concyclic
2018 Philippine MO
In triangle $ABC$ with $\angle ABC = 60^{\circ}$ and $5AB = 4BC$, points $D$ and $E$ are the feet of the altitudes from $B$ and $C$, respectively. $M$ is the midpoint of $BD$ and the circumcircle of triangle $BMC$ meets line $AC$ again at $N$. Lines $BN$ and $CM$ meet at $P$. Prove that $\angle EDP = 90^{\circ}$.
2018 Philippine MO
Suppose $a_1, a_2, \ldots$ is a sequence of integers, and $d$ is some integer. For all natural numbers $n$,

\begin{align*}\text{(i)} |a_n| \text{ is prime;} && \text{(ii)} a_{n+2} = a_{n+1} + a_n + d. \end{align*}Show that the sequence is constant.
2018 Philippine MO
Let $n$ be a positive integer. An $n \times n$ matrix (a rectangular array of numbers with $n$ rows and $n$ columns) is said to be a platinum matrix if:



the $n^2$ entries are integers from $1$ to $n$;

each row, each column, and the main diagonal (from the upper left corner to the lower right corner) contains each integer from $1$ to $n$ exactly once; and

there exists a collection of $n$ entries containing each of the numbers from $1$ to $n$, such that no two entries lie on the same row or column, and none of which lie on the main diagonal of the matrix.



Determine all values of $n$ for which there exists an $n \times n$ platinum matrix.
2018 Philippine MO
Determine all ordered pairs $(x, y)$ of nonnegative integers that satisfy the equation $$3x^2 + 2 \cdot 9^y = x(4^{y+1}-1).$$
2017 Philippine MO
Given \(n \in \mathbb{N}\), let \(\sigma (n)\) denote the sum of the divisors of \(n\) and \(\phi (n)\) denote the number of integers \(n \geq m\) for which \(\gcd(m,n) = 1\). Show that for all \(n \in \mathbb{N}\),



$$\large \frac{1}{\sigma (n)} + \frac{1}{\phi (n)} \geq \frac{2}{n}$$

and determine when equality holds.
2017 Philippine MO
Find all positive real numbers \((a,b,c) \leq 1\) which satisfy



$$ \huge \min \Bigg\{ \sqrt{\frac{ab+1}{abc}}\, \sqrt{\frac{bc+1}{abc}}, \sqrt{\frac{ac+1}{abc}} \Bigg \} = \sqrt{\frac{1-a}{a}} + \sqrt{\frac{1-b}{b}} + \sqrt{\frac{1-c}{c}}$$
2017 Philippine MO
Each of the numbers in the set \(A = \{1,2, \cdots, 2017\}\) is colored either red or white. Prove that for \(n \geq 18\), there exists a coloring of the numbers in \(A\) such that any of its n-term arithmetic sequences contains both colors.
2017 Philippine MO
Circles \(\mathcal{C}_1\) and \(\mathcal{C}_2\) with centers at \(C_1\) and \(C_2\) respectively, intersect at two points \(A\) and \(B\). Points \(P\) and \(Q\) are varying points on \(\mathcal{C}_1\) and \(\mathcal{C}_2\), respectively, such that \(P\), \(Q\) and \(B\) are collinear and \(B\) is always between \(P\) and \(Q\). Let lines \(PC_1\) and \(QC_2\) intersect at \(R\), let \(I\) be the incenter of \(\Delta PQR\), and let \(S\) be the circumcenter of \(\Delta PIQ\). Show that as \(P\) and \(Q\) vary, \(S\) traces the arc of a circle whose center is concyclic with \(A\), \(C_1\) and \(C_2\).
2016 Philippine MO
The operations below can be applied on any expression of the form \(ax^2+bx+c\).



$(\text{I})$ If \(c \neq 0\), replace \(a\) by \(4a-\frac{3}{c}\) and \(c\) by \(\frac{c}{4}\).



$(\text{II})$ If \(a \neq 0\), replace \(a\) by \(-\frac{a}{2}\) and \(c\) by \(-2c+\frac{3}{a}\).



$(\text{III}_t)$ Replace \(x\) by \(x-t\), where \(t\) is an integer. (Different values of \(t\) can be used.)



Is it possible to transform \(x^2-x-6\) into each of the following by applying some sequence of the above operations?



$(\text{a})$ \(5x^2+5x-1\)



$(\text{b})$ \(x^2+6x+2\)
2016 Philippine MO
Prove that the arithmetic sequence $5, 11, 17, 23, 29, \ldots$ contains infinitely many primes.
2016 Philippine MO
Let \(n\) be any positive integer. Prove that $$\sum^n_{i=1} \frac{1}{(i^2+i)^{3/4}} > 2-\frac{2}{\sqrt{n+1}}$$.
2016 Philippine MO
Two players, \(A\) (first player) and \(B\), take alternate turns in playing a game using 2016 chips as follows: the player whose turn it is, must remove \(s\) chips from the remaining pile of chips, where \(s \in \{ 2,4,5 \}\). No one can skip a turn. The player who at some point is unable to make a move (cannot remove chips from the pile) loses the game. Who among the two players can force a win on this game?
2016 Philippine MO
Pentagon \(ABCDE\) is inscribed in a circle. Its diagonals \(AC\) and \(BD\) intersect at \(F\). The bisectors of \(\angle BAC\) and \(\angle CDB\) intersect at \(G\). Let \(AG\) intersect \(BD\) at \(H\), let \(DG\) intersect \(AC\) at \(I\), and let \(EG\) intersect \(AD\) at \(J\). If \(FHGI\) is cyclic and $$JA \cdot FC \cdot GH = JD \cdot FB \cdot GI,$$prove that \(G\), \(F\) and \(E\) are collinear.
2013 Philippine MO
1.	Determine, with proof, the least positive integer $n$ for which there exist $n$ distinct positive integers,

$\left(1-\frac{1}{x_1}\right)\left(1-\frac{1}{x_2}\right)......\left(1-\frac{1}{x_n}\right)=\frac{15}{2013}$
2013 Philippine MO
2.	Let P be a point in the interior of triangle ABC . Extend AP, BP, and CP to meet BC, AC, and AB at D, E, and F, respectively. If triangle APF, triangle BPD and triangle CPE  have equal areas, prove that P is the centroid of triangle ABC  .
2013 Philippine MO
3.	Let n be a positive integer. The numbers 1, 2, 3,....., 2n  are randomly assigned to 2n distinct points on a circle. To each chord joining two of these points, a value is assigned equal to the absolute value of the difference between the assigned numbers at its endpoints.

Show that one can choose  n pairwise non-intersecting chords such that the sum of the values assigned to them is $n^2$ .
2013 Philippine MO
4.	Let $a$, $p$ and $q$  be positive integers with  $p \le q$. Prove that if one of the numbers $a^p$ and $a^q$ is divisible by $p$  , then the other number must also be divisible by $p$ .
2013 Philippine MO
Let $r$ and $s$ be positive real numbers such that $(r+s-rs)(r+s+rs)=rs$. Find the minimum value of $r+s-rs$ and $r+s+rs$
2012 Philippine MO
A computer generates even integers half of the time and another computer generates even integers a third of the time. If $a_i$ and $b_i$ are the integers generated by the computers, respectively, at time $i$, what is the probability that $a_1b_1 +a_2b_2 +\cdots + a_kb_k$ is an even integer.
2012 Philippine MO
Let $f$ be a polynomial function with integer coefficients and $p$ be a prime number. Suppose there are at least four distinct integers satisfying $f(x) = p$. Show that $f$ does not have integer zeros.
2012 Philippine MO
If $ab>0$ and $\displaystyle 0<x<\frac{\pi}{2}$, prove that $$ \left ( 1+\frac{a^2}{\sin x} \right ) \left ( 1+\frac{b^2}{\cos x} \right ) \geq \frac{(1+\sqrt{2}ab)^2 \sin 2x}{2}. $$
2012 Philippine MO
Let $\star$ be an operation defined in the set of nonnegative integers with the following properties: for any nonnegative integers $x$ and $y$,

(i) $(x + 1)\star 0 = (0\star x) + 1$

(ii) $0\star (y + 1) = (y\star 0) + 1$

(iii) $(x + 1)\star (y + 1) = (x\star y) + 1$.

If $123\star 456 = 789$, find $246\star 135$.
2012 Philippine MO
There are exactly $120$ Twitter subscribers from National Science High School. Statistics show that each of $10$ given celebrities has at least $85$ followers from National Science High School. Prove that there must be two students such that each of the $10$ celebrities is being followed in Twitter by at least one of these students.
2011 Philippine MO
Find all nonempty finite sets $X$ of real numbers such that for all $x\in X$, $x+|x| \in X$.
2011 Philippine MO
In triangle $ABC$, let $X$ and $Y$ be the midpoints of $AB$ and $AC$, respectively. On segment $BC$, there is a point $D$, different from its midpoint, such that $\angle{XDY}=\angle{BAC}$. Prove that $AD\perp BC$.
2011 Philippine MO
The $2011$th prime number is $17483$ and the next prime is $17489$.

Does there exist a sequence of $2011^{2011}$ consecutive positive integers that contain exactly $2011$ prime numbers?