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Showing posts from 2022

2014 IMO SL #C2

Problem : We have $2^m$ sheets of paper, with the number $1$ written on each of them. We perform the following operation. In every step we choose two distinct sheets; if the numbers on the two sheets are $a$ and $b$, then we erase these numbers and write the number $a + b$ on both sheets. Prove that after $m2^{m -1}$ steps, the sum of the numbers on all the sheets is at least $4^m$ .

2009 IMO SL #C1

Problem : Consider $2009$ cards, each having one gold side and one black side, lying on parallel on a long table. Initially all cards show their gold sides. Two player, standing by the same long side of the table, play a game with alternating moves. Each move consists of choosing a block of $50$ consecutive cards, the leftmost of which is showing gold, and turning them all over, so those which showed gold now show black and vice versa. The last player who can make a legal move wins. (a) Does the game necessarily end? (b) Does there exist a winning strategy for the starting player?

1998 USAMO #3

Problem : Let $a_0,a_1,\cdots ,a_n$ be numbers from the interval $(0,\pi/2)$ such that$\tan (a_0-\frac{\pi}{4})+ \tan (a_1-\frac{\pi}{4})+\cdots +\tan (a_n-\frac{\pi}{4})\geq n-1.$Prove that$\tan a_0\tan a_1 \cdots \tan a_n\geq n^{n+1}.$

2022 USEMO #3

Point $P$ lies in the interior of a triangle $ABC$. Lines $AP$, $BP$, and $CP$ meet the opposite sides of triangle $ABC$ at $A$', $B'$, and $C'$ respectively. Let $P_A$ the midpoint of the segment joining the incenters of triangles $BPC'$ and $CPB'$, and define points $P_B$ and $P_C$ analogously. Show that if $AB'+BC'+CA'=AC'+BA'+CB'$then points $P,P_A,P_B,$ and $P_C$ are concyclic.

2019 IMO SL #A5

Problem : Let $x_1, x_2, \dots, x_n$ be different real numbers. Prove that $\sum_{1 \leqslant i \leqslant n} \prod_{j \neq i} \frac{1-x_{i} x_{j}}{x_{i}-x_{j}}=\left\{\begin{array}{ll} 0, & \text { if } n \text { is even; } \\ 1, & \text { if } n \text { is odd. } \end{array}\right.$ Solution 1 (induction) : We induct on $n$, and the base case is trivial. Now for any $n > 2,$ define the multivariable rational function$$\sum_{1 \leqslant i \leqslant n} \prod_{j \neq i} \frac{1-x_{i} x_{j}}{x_{i}-x_{j}} = A_n(x_1, \dots x_n)$$and the multivariable polynomial$$B_n(x_1, \dots x_n) = \prod_{i<j}(x_i-x_j)(A_n - (\text{n's remainder mod 2})).$$This polynomial has at most degree $\binom{n}{2} + (n-1).$ But for every $x_i,$ we know $x_i - 1$ and $x_i+1$ divide $B_n$ by the inductive hypothesis, as well as all the terms of the form $x_i-x_j.$ That yields $\binom{n}{2} + 2n$ terms dividing $B_n,$ more than its degree. So $B_n$ is a zero polynomial as desired. $\square$ Solutio

2019 ISL #C7

There are 60 empty boxes $B_1,\ldots,B_{60}$ in a row on a table and an unlimited supply of pebbles. Given a positive integer $n$, Alice and Bob play the following game. In the first round, Alice takes $n$ pebbles and distributes them into the 60 boxes as she wishes. Each subsequent round consists of two steps: (a) Bob chooses an integer $k$ with $1\leq k\leq 59$ and splits the boxes into the two groups $B_1,\ldots,B_k$ and $B_{k+1},\ldots,B_{60}$. (b) Alice picks one of these two groups, adds one pebble to each box in that group, and removes one pebble from each box in the other group. Bob wins if, at the end of any round, some box contains no pebbles. Find the smallest $n$ such that Alice can prevent Bob from winning.

2019 IMC #3

Problem : Let $f:(-1,1)\to \mathbb{R}$ be a twice differentiable function such that $$2f’(x)+xf''(x)\geqslant 1 \quad \text{ for } x\in (-1,1).$$Prove that $$\int_{-1}^{1}xf(x)dx\geqslant \frac{1}{3}.$$ Solution 1 : The LHS of the original inequality looks like the result of an application of the product rule. Indeed, the derivative of $f'g$ is $g'f'+gf''$. We cannot make $g=x$ and $g'=2$, but we can multiply by $x$ to get $g=x^2, g'=2x$. Now to do this we need to be careful about the sign of $x$. So, suppose that $x > 0$. Then $2xf'(x)+x^2f''(x) \ge x$ and hence $(x^2f')' \ge x$ so that $x^2f'(x) \ge \frac{1}{2} x^2$ and hence $f'(x) \ge \frac{1}{2}$. Similarly, we know that $f'(x)\ge \frac{1}{2}$ for $x < 0$, so we have proved this for all $x$ (for $x=0$ by continuity or by just plugging in $x=0$ into the original inequality). The rest is easy. WLOG, assume that $f(0)=0$ since shifting $f$ by a constant changes

2012 IMO SL #G6

Problem:  Let $ABC$ be a triangle with circumcenter $O$ and incenter $I$. The points $D,E$ and $F$ on the sides $BC,CA$ and $AB$ respectively are such that $BD+BF=CA$ and $CD+CE=AB$. The circumcircles of the triangles $BFD$ and $CDE$ intersect at $P \neq D$. Prove that $OP=OI$. Solution 1:  Let circle with radius $OI$ intersect $BI, CI$ at $S, T \ne I$ respectively. Let $H$ be the foot of $S$ to $AB$ and let $BI$ meet $(ABC)$ at $M \ne A.$ It's known that if any circle passing through $S,B$ intersects $BC,BA$ at $D',F'$ respectively then $BD'+BF' = 2BH.$ But$$BH = BS \cdot \cos \left(\frac{1}{2} \angle ABC \right) = IM \cdot \cos \left(\frac{1}{2} \angle ABC \right) = MA \cdot \cos \left(\frac{1}{2} \angle ABC \right) = \frac{1}{2}AC$$so in fact $BFSD$ cyclic, similarly $CDTE$ cyclic. Then$$\angle TPS = 360^\circ - \angle TPD - \angle SPD = \frac{1}{2} \angle ABC + \frac{1}{2} \angle ACB = \angle TIS$$so $P$ lies on $(TIS)$ centered at $O,$ done. $\square$ Soluti

Occasional Putnam problems

Starting now, I will occasionally post Putnam problems on this blog. However, the main content will still be IMO Problems.  These problems will be under the tag #putnam.

2006 IMO #G3

Let $ABCDE$ be a convex pentagon such that $\angle BAC = \angle CAD = \angle DAE\qquad \text{and}\qquad \angle ABC = \angle ACD = \angle ADE.$The diagonals $BD$ and $CE$ meet at $P$. Prove that the line $AP$ bisects the side $CD$.

Problem request: 2022 AIME II #13

Hi everyone,  I hope you had a great summer!  I was not posting for about a month or so because not many people were viewing the blog in the summer.  I'm going to start to post frequently again, and for now I have a problem request I got a couple days ago.  There is a polynomial $P(x)$ with integer coefficients such that$P(x)=\frac{(x^{2310}-1)^6}{(x^{105}-1)(x^{70}-1)(x^{42}-1)(x^{30}-1)}$holds for every $0<x<1.$ Find the coefficient of $x^{2022}$ in $P(x)$.

2015 IMO SL #A1

Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies$a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}$for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.

2019 IMO SL #C1

The infinite sequence $a_0,a _1, a_2, \dots$ of (not necessarily distinct) integers has the following properties: $0\le a_i \le i$ for all integers $i\ge 0$, and$\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k$for all integers $k\ge 0$. Prove that all integers $N\ge 0$ occur in the sequence (that is, for all $N\ge 0$, there exists $i\ge 0$ with $a_i=N$).

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1995 IMO #2

Let $a$, $b$, $c$ be positive real numbers such that $abc = 1$. Prove that $\frac {1}{a^{3}\left(b + c\right)} + \frac {1}{b^{3}\left(c + a\right)} + \frac {1}{c^{3}\left(a + b\right)}\geq \frac {3}{2}.$

1993 IMO #1

Let $n > 1$ be an integer and let $f(x) = x^n + 5 \cdot x^{n-1} + 3.$ Prove that there do not exist polynomials $g(x),h(x),$ each having integer coefficients and degree at least one, such that $f(x) = g(x) \cdot h(x).$ Solution 1 (overkill):  Since $5>1+3$, from Perron's Criterion this polynomial is irreducible over the integers as desired. Solution 2 (normal solution): Assume FTSOC that there do exist $g(x), h(x)$ such that $f(x)=g(x)\cdot h(x)$. Observe that there are no integer roots of $f(x)$ from the Rational Root Theorem. Thus, $g(x), h(x)$ cannot be linear, and their degree is greater than or equal to $2$. We have $g(0)h(0)=3$. WLOG $g(0)=1$. Let $r_1, r_2, \dots r_j \in\mathbb{C}$ be the roots of $g(x)$. We have $r_1r_2\dots r_n=\pm 1$. Multiplying the equalities $r_i^{n-1}(r_i+5)=-3$ for all $1\leq i\leq j$, we obtain$\vert g(-5)\vert = \vert (r_1+5)(r_2+5)\dots (r_m+5)\vert = 3^m.$But $g(-5)f(-5)=3$, contradiction.

Inequality problem I made

Prove for all positive reals $a,b,c$, $$\sum_{cyc} \frac{a}{\sqrt{3ab+bc}} \ge \frac 32.$$ I'll wait for comments and will post the solution in 2 days! EDIT - Well now that it's been 3 days (oops i forgot to post), I'll show my solution now.  By Holder,$$\left( \sum_{cyc} \frac{a}{\sqrt{3ab+bc}} \right)^2 \left( \sum_{cyc} a(3ab+bc) \right) \ge (a+b+c)^3$$ So it suffices to show$$(a+b+c)^3 \ge \frac 94 \sum_{cyc} a(3ab+bc)$$ Expanding and cancelling terms, we wish to show$$f(a,b,c)=4\sum_{cyc} a^3 + 12\sum_{cyc} b^2a-15\sum_{cyc} a^2b-3abc\ge 0$$ Claim: $f(a,b,c) \le f(a+d, b+d, c+d)$ if $d>0$ Proof:$$f(a+d,b+d,c+d)-f(a,b,c) =d (4\sum_{cyc} 3a^2 + 12\sum_{cyc} (b^2+2ba) - 15\sum_{cyc} (a^2+2ba) - 3(ab+bc+ca))$$ $$+d^2 (4\sum_{cyc} 3a + 12 \sum_{cyc} (2b+a) - 15\sum_{cyc} (2a+b) - 3(a+b+c)$$ The $d^2, d^3$ stuff cancel. So $f(a+d,b+d,c+d) - f(a,b,c) = \frac{9d}{2} \sum_{cyc} (a-b)^2 > 0$. Let $a=\min\{a,b,c\}$, then $f(a,a+b,a+c) \ge f(0,b,c)$. Let $x=\frac bc$, then

2001 IMO SL #A6

Last one. I promise.   Prove that for all positive real numbers $a,b,c$,$\frac{a}{\sqrt{a^2 + 8bc}} + \frac{b}{\sqrt{b^2 + 8ca}} + \frac{c}{\sqrt{c^2 + 8ab}} \geq 1.$

2014 IMO SL #C2

Since the last one was super easy, I have to do another one... right?  We have $2^m$ sheets of paper, with the number $1$ written on each of them. We perform the following operation. In every step we choose two distinct sheets; if the numbers on the two sheets are $a$ and $b$, then we erase these numbers and write the number $a + b$ on both sheets. Prove that after $m2^{m -1}$ steps, the sum of the numbers on all the sheets is at least $4^m$ .

2019 IMO SL #G1

Let $ABC$ be a triangle. Circle $\Gamma$ passes through $A$, meets segments $AB$ and $AC$ again at points $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $BDF$ at $F$ and the tangent to circle $CEG$ at $G$ meet at point $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$.

2013 IMO #4

Let $ABC$ be an acute triangle with orthocenter $H$, and let $W$ be a point on the side $BC$, lying strictly between $B$ and $C$. The points $M$ and $N$ are the feet of the altitudes from $B$ and $C$, respectively. Denote by $\omega_1$ is the circumcircle of $BWN$, and let $X$ be the point on $\omega_1$ such that $WX$ is a diameter of $\omega_1$. Analogously, denote by $\omega_2$ the circumcircle of triangle $CWM$, and let $Y$ be the point such that $WY$ is a diameter of $\omega_2$. Prove that $X,Y$ and $H$ are collinear.

2017 IMO SL #N6

Find the smallest positive integer $n$ or show no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $(a_1, a_2, \ldots, a_n)$ such that both $$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$are integers.

2014 IMO SL #G7

Let $ABC$ be a triangle with circumcircle $\Omega$ and incentre $I$. Let the line passing through $I$ and perpendicular to $CI$ intersect the segment $BC$ and the arc $BC$ (not containing $A$) of $\Omega$ at points $U$ and $V$ , respectively. Let the line passing through $U$ and parallel to $AI$ intersect $AV$ at $X$, and let the line passing through $V$ and parallel to $AI$ intersect $AB$ at $Y$ . Let $W$ and $Z$ be the midpoints of $AX$ and $BC$, respectively. Prove that if the points $I, X,$ and $Y$ are collinear, then the points $I, W ,$ and $Z$ are also collinear.

2018 IMO #C2 (#4)

A site is any point $(x, y)$ in the plane such that $x$ and $y$ are both positive integers less than or equal to 20. Initially, each of the 400 sites is unoccupied. Amy and Ben take turns placing stones with Amy going first. On her turn, Amy places a new red stone on an unoccupied site such that the distance between any two sites occupied by red stones is not equal to $\sqrt{5}$. On his turn, Ben places a new blue stone on any unoccupied site. (A site occupied by a blue stone is allowed to be at any distance from any other occupied site.) They stop as soon as a player cannot place a stone. Find the greatest $K$ such that Amy can ensure that she places at least $K$ red stones, no matter how Ben places his blue stones.

2021 IMO #C3 (#5)

Two squirrels, Bushy and Jumpy, have collected 2021 walnuts for the winter. Jumpy numbers the walnuts from 1 through 2021, and digs 2021 little holes in a circular pattern in the ground around their favorite tree. The next morning Jumpy notices that Bushy had placed one walnut into each hole, but had paid no attention to the numbering. Unhappy, Jumpy decides to reorder the walnuts by performing a sequence of 2021 moves. In the $k$-th move, Jumpy swaps the positions of the two walnuts adjacent to walnut $k$. Prove that there exists a value of $k$ such that, on the $k$-th move, Jumpy swaps some walnuts $a$ and $b$ such that $a<k<b$.

Blog was private (problem fixed)

Hello,  I just noticed that the blog was private for a while. I fixed the problem now. Sorry for the inconvenience.

2005 IMO SL #A3

Four real numbers $p$, $q$, $r$, $s$ satisfy $p+q+r+s = 9$ and $p^{2}+q^{2}+r^{2}+s^{2}= 21$. Prove that there exists a permutation $\left(a,b,c,d\right)$ of $\left(p,q,r,s\right)$ such that $ab-cd \geq 2$.

2020 Peru EGMO TST #3

As I said in an earlier post, I will also post problems that are not from the IMO but I believe that they are of the same level. Let $ABC$ be a triangle with $AB<AC$ and $I$ be your incenter. Let $M$ and $N$ be the midpoints of the sides $BC$ and $AC$, respectively. If the lines $AI$ and $IN$ are perpendicular, prove that the line $AI$ is tangent to the circumcircle of $\triangle IMC$.

Twitch streaming update

From now on, I will only stream on Twitch if someone wants one-on-one detailed help on a solution they don't understand on my blog.

Thonks released!

This was the huge surprise I was talking about in the other post! Now, we have a site like this for Physics as well, which you can find at  thinkphysics1.blogspot.com .  I have combined these two to form my organization named Thonks, and its website can be found at  thonks123123.github.io/Thonks .  I hope you like it!

2017 IMO SL #C2

Let $n$ be a positive integer. Define a chameleon to be any sequence of $3n$ letters, with exactly $n$ occurrences of each of the letters $a, b,$ and $c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon $X$ , there exists a chameleon $Y$ such that $X$ cannot be changed to $Y$ using fewer than $3n^2/2$ swaps.

2018 Silk Road #1

In an acute-angled triangle $ABC$ on the sides $AB$, $BC$, $AC$ the points $H$, $L$, $K$ so that $CH \perp AB$, $HL \parallel AC$, $HK \parallel BC$. Let $P$ and $Q$ feet of altitudes of a triangle $HBL$, drawn from the vertices $H$ and $B$ respectively. Prove that the feet of the altitudes of the triangle $AKH$, drawn from the vertices $A$ and $H$ lie on the line $PQ$.

No stream today

Hi,  I'm sorry but I won't be able to stream today.  I have a little pain in neck and it hurts a bit when I talk. I will stream again next week (Friday, April 15th) at 7:00 PM

Widespread all over the world

Now, we are officially spread all throughout the world! We have views from every continent (well other than Antarctica, of course).  We already had it spread in all continents other than Africa but now we have many views from Africa as well. Before, it was technically "spread" in Africa, but there were minimal views from there.  I'm really happy about this :) Thank you so much to everyone who views my blog :D I hope it continues to help you.

Choosing non-IMO problems sometimes

Hi,  I have decided that sometimes, I will pick problems that are not from the IMO shortlists but I believe that it is an IMO-level problem (in other words, I think that the problem could have appeared in the IMO as well). Because after all, there are a limited number of problems.

1989 IMO SL #6

A permutation $\{x_1, x_2, \ldots, x_{2n}\}$ of the set $\{1,2, \ldots, 2n\}$ where $n$ is a positive integer, is said to have property $T$ if $|x_i - x_{i + 1}| = n$ for at least one $i$ in $\{1,2, \ldots, 2n - 1\}.$ Show that, for each $n$, there are more permutations with property $T$ than without.

2000 IMO SL #N4

Find all triplets of positive integers $(a,m,n)$ such that $a^m + 1 \mid (a + 1)^n$.

2020 ISL #A2

Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as \begin{align*} (x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z) \end{align*}with $P, Q, R \in \mathcal{A}$. Find the smallest non-negative integer $n$ such that $x^i y^j z^k \in \mathcal{B}$ for all non-negative integers $i, j, k$ satisfying $i + j + k \geq n$.

Twitch streaming

In  this  post, Anonymous suggested that I should start a Twitch stream. Thanks for the suggestion! I will stream every Friday at 7:00 PM EST, starting March 25th. You can find my Twitch account here .  You can submit problems that you want me to solve during the stream at my email, imomath@imomath.xyz, or at my discord handle, math4l#4750.  If I get no problem requests, then I will pick a random problem to solve and explain.

CUSTOM DOMAIN!

Due to the request of a few users, I got a custom domain at  imomath.xyz . The  math4l.blogspot.com  link will still work, but you can just go to  imomath.xyz  now.

2006 IMO SL #A5

If $a,b,c$ are the sides of a triangle, prove that $\frac{\sqrt{b+c-a}}{\sqrt{b}+\sqrt{c}-\sqrt{a}}+\frac{\sqrt{c+a-b}}{\sqrt{c}+\sqrt{a}-\sqrt{b}}+\frac{\sqrt{a+b-c}}{\sqrt{a}+\sqrt{b}-\sqrt{c}}\leq 3$

2015 IMO SL #G4

Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values of $\frac{BT}{BM}$.

2007 IMO SL #C7

Let $\alpha < \frac {3 - \sqrt {5}}{2}$ be a positive real number. Prove that there exist positive integers $n$ and $p > \alpha \cdot 2^n$ for which one can select $2 \cdot p$ pairwise distinct subsets $S_1, \ldots, S_p, T_1, \ldots, T_p$ of the set $\{1,2, \ldots, n\}$ such that $S_i \cap T_j \neq \emptyset$ for all $1 \leq i,j \leq p$.

2011 IMO SL #A2

Determine all sequences $(x_1,x_2,\ldots,x_{2011})$ of positive integers, such that for every positive integer $n$ there exists an integer $a$ with$\sum^{2011}_{j=1} j x^n_j = a^{n+1} + 1$

2003 IMO SL #G4

Let $\Gamma_1$, $\Gamma_2$, $\Gamma_3$, $\Gamma_4$ be distinct circles such that $\Gamma_1$, $\Gamma_3$ are externally tangent at $P$, and $\Gamma_2$, $\Gamma_4$ are externally tangent at the same point $P$. Suppose that $\Gamma_1$ and $\Gamma_2$; $\Gamma_2$ and $\Gamma_3$; $\Gamma_3$ and $\Gamma_4$; $\Gamma_4$ and $\Gamma_1$ meet at $A$, $B$, $C$, $D$, respectively, and that all these points are different from $P$. Prove that $\frac{AB\cdot BC}{AD\cdot DC}=\frac{PB^2}{PD^2}.$

50K views!!

I.  Am.  Completely.  Shocked.  I couldn't even think, in my dreams, for this blog to blow up so much. I'm glad to know that my solutions are helping people! Keep visiting this blog, more solutions are to come! As of now, I have no plans of closing this blog, and I plan to continue posting daily.  Btw, I won't be posting much of these "X views" anymore, since they're getting so frequent lol. I'll only post at some major number of views, like 100K, 150K, 200K, etc.

1960 IMO SL #6

An isosceles trapezoid with bases $a$ and $c$ and altitude $h$ is given.  a) On the axis of symmetry of this trapezoid, find all points $P$ such that both legs of the trapezoid subtend right angles at $P$;  b) Calculate the distance of $p$ from either base;  c) Determine under what conditions such points $P$ actually exist. Discuss various cases that might arise.

2011 IMO SL #C5

Let $m$ be a positive integer, and consider a $m\times m$ checkerboard consisting of unit squares. At the centre of some of these unit squares there is an ant. At time $0$, each ant starts moving with speed $1$ parallel to some edge of the checkerboard. When two ants moving in the opposite directions meet, they both turn $90^{\circ}$ clockwise and continue moving with speed $1$. When more than $2$ ants meet, or when two ants moving in perpendicular directions meet, the ants continue moving in the same direction as before they met. When an ant reaches one of the edges of the checkerboard, it falls off and will not re-appear.  Considering all possible starting positions, determine the latest possible moment at which the last ant falls off the checkerboard, or prove that such a moment does not necessarily exist.

1985 IMO SL #11

Find a method by which one can compute the coefficients of $P(x) = x^6 + a_1x^5 + \cdots+ a_6$ from the roots of $P(x) = 0$ by performing not more than $15$ additions and $15$ multiplications.

2014 IMO SL #C9

Consider a prism with pentagons $A_1A_2A_3A_4A_5$ and $B_1B_2B_3B_4B_5$ as the top and bottom faces is given. Each side of the two pentagons and each of the 25 segments $A_iB_j$ is colored red or green. Every triangle whose vertices are vertices of the prism and whose sides have all been colored has two sides of a different color. Prove that all 10 sides of the top and bottom faces have the same color.

2003 IMO SL #C1

Let $A$ be a $101$-element subset of the set $S=\{1,2,\ldots,10^6\}$. Prove that there exist numbers $t_1$, $t_2, \ldots, t_{100}$ in $S$ such that the sets$A_j=\{x+t_j\mid x\in A\},\qquad j=1,2,\ldots,100$are pairwise disjoint.

10K views!!!

I am speechless. Thank you so much for 10K views! I will try to keep posting interesting content every day. Keep visiting! By the way, I have recently added a new feature in this blog. If you have a specific contest math problem you would like me to solve, please use the "Problem/Advice request" form on the left side of the homepage. It doesn't have to be an olympiad problem, but as long as it is a contest problem you can submit it through the form. You can use the same form if you want to ask for any advice. I will make a blog post regarding your problem/advice request.

2003 IMO SL #A6

Let $n$ be a positive integer and let $(x_1,\ldots,x_n)$, $(y_1,\ldots,y_n)$ be two sequences of positive real numbers. Suppose $(z_2,\ldots,z_{2n})$ is a sequence of positive real numbers such that $z_{i+j}^2 \geq x_iy_j$ for all $1\le i,j \leq n$. Let $M=\max\{z_2,\ldots,z_{2n}\}$. Prove that$\left( \frac{M+z_2+\dots+z_{2n}}{2n} \right)^2 \ge \left( \frac{x_1+\dots+x_n}{n} \right) \left( \frac{y_1+\dots+y_n}{n} \right).$

1986 IMO SL #15

Let $ABCD$ be a convex quadrilateral whose vertices do not lie on a circle. Let $A'B'C'D'$ be a quadrilateral such that $A',B', C',D'$ are the centers of the circumcircles of triangles $BCD,ACD,ABD$, and $ABC$. We write $T (ABCD) = A'B'C'D'$. Let us define $A''B''C''D'' = T (A'B'C'D') = T (T (ABCD)).$  (a) Prove that $ABCD$ and $A''B''C''D''$ are similar.  (b) The ratio of similitude depends on the size of the angles of $ABCD$. Determine this ratio.

1990 IMO SL #18

Let $a, b \in \mathbb{N}$ with $1 \leq a \leq b,$ and $M = \left[\frac {a + b}{2} \right]$. Define a function $f: \mathbb{Z} \mapsto \mathbb{Z}$ by $f(n) = \begin{cases} n + a, & \text{if } n \leq M, \\ n - b, & \text{if } n >M. \end{cases}$Let $f^1(n) = f(n),$ $f_{i + 1}(n) = f(f^i(n)),$ $i = 1, 2, \ldots$ Find the smallest natural number $k$ such that $f^k(0) = 0$

2007 IMO SL #A6

Let $a_1, a_2, \ldots, a_{100}$ be nonnegative real numbers such that $a^2_1 + a^2_2 + \ldots + a^2_{100} = 1.$ Prove that $a^2_1 \cdot a_2 + a^2_2 \cdot a_3 + \ldots + a^2_{100} \cdot a_1 < \frac {12}{25}.$

3K views!

Thanks everyone for 3K views on this blog! I didn't expect it to blow up this much. Keep viewing it, more interesting content is to come!

2013 IMO SL #G5

Let $ABCDEF$ be a convex hexagon with $AB=DE$, $BC=EF$, $CD=FA$, and $\angle A-\angle D = \angle C -\angle F = \angle E -\angle B$. Prove that the diagonals $AD$, $BE$, and $CF$ are concurrent.

2018 IMO SL #A1

Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying$$f(x^2f(y)^2)=f(x)^2f(y)$$for all $x,y\in\mathbb{Q}_{>0}$

PROBLEMS I'M GOING TO BE POSTING [UPDATE #2]

At this point,  I have decided that I am only going to do IMO Shortlist problems.  Once I finish EGMO, I will keep doing this if I don't feel like the problems have started to repeat.  If I do feel that way, then I will then go towards research and most likely close this blog.  These plans might change in the future.

2017 IMO SL #A1

Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that $$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$If $M>1$, prove that the polynomial $$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$has no positive roots.

School...

Since winter break is now over for me, although I will still be posting daily, I will post 3-6 problems instead of more.  However, the content will still be high quality! You won't ever run out of posts to view, and you won't ever be bored.

2016 India IMO Training Camp #3

Let $a,b,c,d$ be real numbers satisfying $|a|,|b|,|c|,|d|>1$ and $abc+abd+acd+bcd+a+b+c+d=0$. Prove that $\frac {1} {a-1}+\frac {1} {b-1}+ \frac {1} {c-1}+ \frac {1} {d-1} >0$

2019 EGMO #2

Let $n$ be a positive integer. Dominoes are placed on a $2n \times 2n$ board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each $n$, determine the largest number of dominoes that can be placed in this way. (A domino is a tile of size $2 \times 1$ or $1 \times 2$. Dominoes are placed on the board in such a way that each domino covers exactly two cells of the board, and dominoes do not overlap. Two cells are said to be adjacent if they are different and share a common side.)

Which problems will I do going forward?

From now on, I will post one IMO shortlist, then two of these below.        1 : "EGMO" ,     2 : "Romanian Masters of Mathematics" ,     3 : "Tournament of Towns" ,     4 : "USA ELMO Shortlist" ,     5 : "USA Math Prize for Girls Olympiad" ,     7 : "USAJMO" ,     8 : "Canada National Olympiad" ,     9 : "China TST" ,     10 : "India IMO Training Camp" ,     11 : "India National Olympiad" For example, it could go like: 2021 IMO #1, 2019 EGM0 #2, China TST #3.

1600 views!

Thanks for 1600 views, everybody! Keep viewing this blog!

I don't know how to solve it... what to do?

Everyone has trouble with math olympiad problems, but what makes the difference is what you do  after .  Below are steps you should do (in order) if you do not know how to solve a problem.  1. Keep thinking about the problem until you completely run out of ideas, and you have tried every single one and it hasn't worked. Don't let any idea go; e 2. Okay, now it's time to give up. Read the solution carefully and understand every step of it. Don't think about the motivation for it yet.  3. When you have finished reading the solution and have verified that it is correct, in your mind summarize what they did, yet without thinking about the motivation. 4. Now, think about what the 'key' in the solution was, or what the 'heart of the problem' was. Basically, think about the main insight in the solution after which it got easy. Make sure to remember this 'insight', because it will apply to many, many other problems than just this one. 5. After a couple d

2007 IMO Sl #A1

Real numbers $a_{1}$, $a_{2}$, $\ldots$, $a_{n}$ are given. For each $i$, $(1 \leq i \leq n )$, define $d_{i} = \max \{ a_{j}\mid 1 \leq j \leq i \} - \min \{ a_{j}\mid i \leq j \leq n \}$ and let $d = \max \{d_{i}\mid 1 \leq i \leq n \}$.  (a) Prove that, for any real numbers $x_{1}\leq x_{2}\leq \cdots \leq x_{n}$, $\max \{ |x_{i} - a_{i}| \mid 1 \leq i \leq n \}\geq \frac {d}{2}. \quad \quad (*)$  (b) Show that there are real numbers $x_{1}\leq x_{2}\leq \cdots \leq x_{n}$ such that the equality holds in (*).

2000 IMO #2

Let $a, b, c$ be positive real numbers so that $abc = 1$. Prove that $\left( a - 1 + \frac 1b \right) \left( b - 1 + \frac 1c \right) \left( c - 1 + \frac 1a \right) \leq 1.$

2003 USA TST #4

Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that$f(m+n)f(m-n) = f(m^2)$for $m,n \in \mathbb{N}$.

2018 USA ELMO Shortlist #G2

Let $ABC$ be a scalene triangle with orthocenter $H$ and circumcenter $O$. Let $P$ be the midpoint of $\overline{AH}$ and let $T$ be on line $BC$ with $\angle TAO=90^{\circ}$. Let $X$ be the foot of the altitude from $O$ onto line $PT$. Prove that the midpoint of $\overline{PX}$ lies on the nine-point circle* of $\triangle ABC$. *The nine-point circle of $\triangle ABC$ is the unique circle passing through the following nine points: the midpoint of the sides, the feet of the altitudes, and the midpoints of $\overline{AH}$, $\overline{BH}$, and $\overline{CH}$

Update

I will be posting higher quality math olympiad problems from now on. Using the random math olympiad problem generator Python program (which I had posted before), most problems are not high quality.  So, in the Python code, I have now removed the olympiads that do not have many good problems.  My updated list from which the computer randomizes looks like this:    1 : "IMO Shortlist" ,   2 : "IMO Shortlist" ,   3 : "APMO" ,   4 : "Austrian-Polish" ,     5 : "Balkan MO Shortlist" ,     6 : "Balkan MO Shortlist" ,     7 : "Baltic Way" ,     8 : "Benelux" ,     9 : "EGMO" ,     10 : "Iranian Geometry Olympiad" ,     11 : "Romanian Masters of Mathematics" ,     12 : "Silk Road" ,     13 : "Tournament of Towns" ,     14 : "USA BAMO" ,     15 : "USA ELMO Shortlist" ,     16 : "USA ELMO Shortlist" ,     17 : "USA USEMO" ,     1

1999 Bulgaria National Olympiad #2

Let $\{a_n\}$ be a sequence of integers satisfying $(n-1)a_{n+1}=(n+1)a_n-2(n-1) \forall n\ge 1$. If $2000|a_{1999}$, find the smallest $n\ge 2$ such that $2000|a_n$.

2018 Silk Road #1

In an acute-angled triangle $ABC$ on the sides $AB$, $BC$, $AC$ the points $H$, $L$, $K$ so that $CH \perp AB$, $HL \parallel AC$, $HK \parallel BC$. Let $P$ and $Q$ feet of altitudes of a triangle $HBL$, drawn from the vertices $H$ and $B$ respectively. Prove that the feet of the altitudes of the triangle $AKH$, drawn from the vertices $A$ and $H$ lie on the line $PQ$.

Welcome, 2022

It is now officially the very first moment of 2022 (in EST).  Welcome, 2022! I hope you bring great happiness and health to everyone.  Time to eat cake.  via GIPHY