New Competition Series: The Million!
by Mathdreams, Apr 2, 2025, 3:57 AM
This post has been edited 2 times. Last edited by Mathdreams, Yesterday at 3:59 AM
2025 USAMO Rubric
by plang2008, Apr 2, 2025, 1:33 AM
1. Let 
and 
be positive integers. Prove that there exists a positive integer 
such that for every odd integer 
, the digits in the base-
representation of 
are all greater than .
2. Let
and be positive integers with 
. Let 
be a polynomial of degree with real coefficients, nonzero constant term, and no repeated roots. Suppose that for any real numbers 
such that the polynomial 
divides , the product 
is zero. Prove that has a nonreal root.
3. Alice the architect and Bob the builder play a game. First, Alice chooses two points
and 
in the plane and a subset 
of the plane, which are announced to Bob. Next, Bob marks infinitely many points in the plane, designating each a city. He may not place two cities within distance at most one unit of each other, and no three cities he places may be collinear. Finally, roads are constructed between the cities as follows: for each pair 
of cities, they are connected with a road along the line segment 
if and only if the following condition holds:
Note:
is directly similar to 
if there exists a sequence of rotations, translations, and dilations sending 
to 
, 
to 
, and 
to 
.
4. Let
be the orthocenter of acute triangle 
, let 
be the foot of the altitude from 
to , and let be the reflection of across 
. Suppose that the circumcircle of triangle 
intersects line at two distinct points and . Prove that is the midpoint of 
.
5. Determine, with proof, all positive integers such that ![\[\frac{1}{n+1} \sum_{i=0}^n \binom{n}{i}^k\]](//latex.artofproblemsolving.com/3/c/5/3c5984d0003c1ca2df2e68841dd9e93ed6c2d27c.png)
is an integer for every positive integer .
6. Let
and be positive integers with 
. There are cupcakes of different flavors arranged around a circle and people who like cupcakes. Each person assigns a nonnegative real number score to each cupcake, depending on how much they like the cupcake. Suppose that for each person , it is possible to partition the circle of cupcakes into groups of consecutive cupcakes so that the sum of 's scores of the cupcakes in each group is at least 
. Prove that it is possible to distribute the cupcakes to the people so that each person receives cupcakes of total score at least with respect to .
and 
be positive integers. Prove that there exists a positive integer 
such that for every odd integer 
, the digits in the base-
representation of 
are all greater than .
.
,
.
and 
to be the remainder when 
is divided by 
.
and 
is odd for 
.
or an equivalent formulation. To prove this construction works, we need to show that
is always an integer between 
and 
inclusive.
is divided by 
.
,
since it telescopes.
,![\[\frac{r_{i+1} n - r_i}{2^i} > \frac{r_{i+1} n}{2^i} - 1 \geq \frac{n}{2^i} - 1 \geq \frac{n}{2^k} - 1.\]](http://latex.artofproblemsolving.com/4/2/0/420d26b6099ca264846885e7eef23b2c9c7b2dd0.png)
Thus, each digit is lower bounded by 
,
fails.2. Let
and be positive integers with 
. Let 
be a polynomial of degree with real coefficients, nonzero constant term, and no repeated roots. Suppose that for any real numbers 
such that the polynomial 
divides , the product 
is zero. Prove that has a nonreal root.
,
such that 
has no nonreal roots. We can also assume 
case is trivial as the constant term must be nonzero, so fix 
.
.
.
which has the 
for all integers 
.
be the coefficient of the 
term of 
for all 
,
for which 
.
and 
.
and 
.
is the coefficient of the 
term in 
and 
.
,
.
,
has 
distinct real roots as well, along with two consecutive nonleading coefficients equal to zero, but one degree lower, by the Power Rule. By doing this repeatedly, we eventually end up with a polynomial where the two consecutive coefficients have degree 
,3. Alice the architect and Bob the builder play a game. First, Alice chooses two points
and 
in the plane and a subset 
of the plane, which are announced to Bob. Next, Bob marks infinitely many points in the plane, designating each a city. He may not place two cities within distance at most one unit of each other, and no three cities he places may be collinear. Finally, roads are constructed between the cities as follows: for each pair 
of cities, they are connected with a road along the line segment 
if and only if the following condition holds:
For every city distinct from 
and 
, there exists 
such
distinct from 
and 
, there exists 
such
that 
is directly similar to either 
or 
.
Alice wins the game if (i) the resulting roads allow for travel between any pair of cities via a finite sequence of roads and (ii) no two roads cross. Otherwise, Bob wins. Determine, with proof, which player has a winning strategy.
is directly similar to either 
or 
.
Note:
is directly similar to 
if there exists a sequence of rotations, translations, and dilations sending 
to 
, 
to 
, and 
to 
.
to be the set of points strictly outside the circle with diameter 
.
is always acute. Suppose two roads 
and 
cross. Then, 
is a convex quadrilateral. Since 
.
is not acute but there does not exist an 
such that 
are connected by some path. This is trivially true for 
,
are also connected.
is obtuse, we have 
.
,
,
is not mentioned. Do not reward points if the induction argument is incomplete or incorrect.4. Let
be the orthocenter of acute triangle 
, let 
be the foot of the altitude from to , and let be the reflection of across 
. Suppose that the circumcircle of triangle 
intersects line at two distinct points and . Prove that is the midpoint of 
.![[asy]
size(300);
defaultpen(linewidth(0.6)+fontsize(11));
pair A = (0,6), B = (-3,0), C = (5,0), H = orthocenter(A,B,C), D = foot(A,B,C), P = 2*D - H, F = foot(C,A,B), Xx = 3*C-2*B;
path circ = circumcircle(A,F,P);
pair[] oops = intersectionpoints(circ, Xx--B);
pair X = oops[0], Y = oops[1];
draw(A--B--C--cycle, rgb(0.8,0.1,0.1));
draw(C--F^^A--P, rgb(0.9,0.4,0.1));
draw(circ,rgb(0.1,0.5,0.1));
draw(circumcircle(A,B,C),rgb(0.1,0.1,0.7));
draw(F--X--P--C--Y,rgb(0.8,0.1,0.8));
dot("$A$",A,N,linewidth(3.3));
dot("$B$",B,SW,linewidth(3.3));
dot("$C$",C,SE,linewidth(3.3));
dot("$D$",D,NE,linewidth(3.3));
dot("$F$",F,NW,linewidth(3.3));
dot("$X$",X,SW,linewidth(3.3));
dot("$Y$",Y,SE,linewidth(3.3));
dot("$P$",P,S,linewidth(3.3));
dot("$H$",H,NE,linewidth(3.3));
[/asy]](http://latex.artofproblemsolving.com/5/4/b/54b45b4ebef7f8e82abfa7c1daaeee177d6ff3dd.png)
be the center of 
.
be the intersection of 
and the line through 
,
.
be the foot of the altitude from 
and 
,
is a midline and thus 
.
is a midline of 
,
and thus 
.
(by definition), by either HL congruence or the Pythagorean theorem, we must have 
.
.
and 
to attempt to identify 
.
or 
,
.
.5. Determine, with proof, all positive integers
such that ![\[\frac{1}{n+1} \sum_{i=0}^n \binom{n}{i}^k\]](http://latex.artofproblemsolving.com/3/c/5/3c5984d0003c1ca2df2e68841dd9e93ed6c2d27c.png)
is an integer for every positive integer .
is divisible by 
,
be a prime power dividing 
.
.
,
if 
and 
otherwise, so for each term, either both the numerator are divisible by 
,
,
,
.
has an inverse modulo 
.![\[\binom ni \equiv \pm 1 \binom{\lfloor n/p \rfloor}{\lfloor i/p \rfloor} \pmod{p^a}.\]](http://latex.artofproblemsolving.com/0/1/c/01c00beb6efc07ed028fd0273f262b8bfb965d33.png)
modulo 
.
,![\[\sum_{i=0}^n \binom ni^k \equiv \sum_{i=0}^n (\pm 1)^k \equiv p \equiv 0 \pmod p.\]](http://latex.artofproblemsolving.com/2/e/c/2ececb6afa8a10e116056cdc57a6c948298d0c29.png)
where 
,
satisfies the induction hypothesis for the prime 
.![\[\sum_{i=0}^n \binom ni^k \equiv \sum_{i=0}^n (\pm 1)^k \binom{\lfloor n/p \rfloor}{\lfloor i/p \rfloor}^k \equiv p\sum_{i=0}^d \binom di^k \pmod {p^a}\]](http://latex.artofproblemsolving.com/6/c/9/6c9a4b6aac62b1befef1511451c9fa30993d2cf7.png)
By the induction hypothesis, 
divides the inner binomial sum, so since we are multiplying it by 
.
.
is incorrect, reward up to 6. Let
and be positive integers with 
. There are cupcakes of different flavors arranged around a circle and people who like cupcakes. Each person assigns a nonnegative real number score to each cupcake, depending on how much they like the cupcake. Suppose that for each person , it is possible to partition the circle of cupcakes into groups of consecutive cupcakes so that the sum of 's scores of the cupcakes in each group is at least 
. Prove that it is possible to distribute the cupcakes to the people so that each person receives cupcakes of total score at least with respect to .
between the set of people except Evan 
if the total score of the cupcakes in that bubble with respect to 
.
denotes the set of neighbors of 
people.
.
from the graph, noting that no person in AMC 10/AIME Study Forum
by PatTheKing806, Mar 27, 2025, 11:34 PM
Me (PatTheKing806) and EaZ_Shadow have created a AMC 10/AIME Study Forum! Hopefully, this forum wont die quickly. To signup, do /join or \join.
Click here to join! (or do some pushups)
People should join this forum if they are wanting to do well on the AMC 10 next year, trying get into AIME, or loves math!
This post has been edited 2 times. Last edited by PatTheKing806, Mar 27, 2025, 11:35 PM
LMT Spring 2025 and Girls' LMT 2025
by vrondoS, Mar 27, 2025, 1:55 AM
Attachments:
This post has been edited 1 time. Last edited by vrondoS, Mar 27, 2025, 5:18 AM
waiting,
by Helena_Liang, Feb 16, 2025, 1:41 AM
I. The Anatomy of Anticipation
we're always waiting. waiting for something.
we exist in the perpetual suspension of waiting. for caterpillars to bloom into butterflies, for stormclouds to unleash the daylight, for the sand in the hourglass to reverse gravity, for the arithmetic of dawn to disprove night's theorem.
even the apocalypse becomes a scheduled event in the theater, constructing countdown clocks when the horizon coughs up the swallowed sun. humans cheer for an incoming asteroid to smash their civilization — the suffocating loop of waiting is too much to bear.
the wait stitches itself into quantum mechanics: schrodinger's cat oscillating between life and death, light years of history collapsing into paper cranes that unfold into different dimensions.
all this waiting crystallizes into fractured light, electrons caught mid-leap between energy levels, our silence etching probabilities into the colors of now.
II. The Fractures of Now
why are we forever trapped between waiting and having waited?
the window dissects the palette of dusk. the glass stained with streaks of rain carves the cityscape into shattered prisms. beyond the glass, trembling lights smear into bleeding watercolors, as if to exaggerate the uncertainty in the unfinished sentence of life. each raindrop contains a microcosmic tornado. the pulse of windshield wipers traces the butterfly's path from summers ago, that fragile choreography turning into tomorrow's tempest, each beat scraping away milliseconds from nightfall.
here, in the display glass, shadows press against the fogged windows. temporary constellations in this liquid cosmos trapping us in the amber night, fossilizing within the translucent rain.
life itself is a paradox: to wait is to exist, yet existence demands us to keep waiting.
III. The Mirage of Arrival
waiting, a juxtaposition of evanescence and hope.
the moon awaits the sun, the world awaits peace, and humans await change. yet the moon is destined to remain apart from the sun, and the world will never cease its conflicts. change, too, is like a mirage: forever parallel to tomorrow, an idea we chase yet can never touch. that is the essence of waiting.
as children, we wait to grow up, but after we experience fragments of life and human nature, we clamor for a rebirth. for a remedy for regret.
we are all actors in life's tragicomedy, costumed in borrowed desires and emotions. our scripts are written in superficial vanity, as we mask the emptiness with grand gestures. we rehearse for a grand finale that rewrites itself nightly.
time slips through our fingers — a comet's tail of almosts, leaving too many regrets and resentment. we pocket fragments of moments like stolen diamonds and hoard the joy of anticipation. yet when silence falls, we realize: what we await is not beyond the horizon, but the selves we buried under the layers of waiting.
day after day, in the fading light, I wait for a version of me.
she flickers in the theater of tomorrow.
those were some of my thoughts.
inspirations behind this piece, if you're interested.
one is a piece of art that I made that I'll put on my blog later. vio3 suggested that the woman in my art looks like she is waiting, which then inspired me to create this. vio3 is truly the best person when it comes to ideas (and art fixing :
))
two is allusions to asteroid 2024 yr4 and butterfly effect
and an allusion to tempest sonata (peak beethoven moment)
the comma after the "waiting" in the title is also intentional if you couldn't notice. it references to "beyond the glass, trembling lights smear into bleeding watercolors, as if to exaggerate the uncertainty in the unfinished sentence of life."
we're always waiting. waiting for something.
we exist in the perpetual suspension of waiting. for caterpillars to bloom into butterflies, for stormclouds to unleash the daylight, for the sand in the hourglass to reverse gravity, for the arithmetic of dawn to disprove night's theorem.
even the apocalypse becomes a scheduled event in the theater, constructing countdown clocks when the horizon coughs up the swallowed sun. humans cheer for an incoming asteroid to smash their civilization — the suffocating loop of waiting is too much to bear.
the wait stitches itself into quantum mechanics: schrodinger's cat oscillating between life and death, light years of history collapsing into paper cranes that unfold into different dimensions.
all this waiting crystallizes into fractured light, electrons caught mid-leap between energy levels, our silence etching probabilities into the colors of now.
II. The Fractures of Now
why are we forever trapped between waiting and having waited?
the window dissects the palette of dusk. the glass stained with streaks of rain carves the cityscape into shattered prisms. beyond the glass, trembling lights smear into bleeding watercolors, as if to exaggerate the uncertainty in the unfinished sentence of life. each raindrop contains a microcosmic tornado. the pulse of windshield wipers traces the butterfly's path from summers ago, that fragile choreography turning into tomorrow's tempest, each beat scraping away milliseconds from nightfall.
here, in the display glass, shadows press against the fogged windows. temporary constellations in this liquid cosmos trapping us in the amber night, fossilizing within the translucent rain.
life itself is a paradox: to wait is to exist, yet existence demands us to keep waiting.
III. The Mirage of Arrival
waiting, a juxtaposition of evanescence and hope.
the moon awaits the sun, the world awaits peace, and humans await change. yet the moon is destined to remain apart from the sun, and the world will never cease its conflicts. change, too, is like a mirage: forever parallel to tomorrow, an idea we chase yet can never touch. that is the essence of waiting.
as children, we wait to grow up, but after we experience fragments of life and human nature, we clamor for a rebirth. for a remedy for regret.
we are all actors in life's tragicomedy, costumed in borrowed desires and emotions. our scripts are written in superficial vanity, as we mask the emptiness with grand gestures. we rehearse for a grand finale that rewrites itself nightly.
time slips through our fingers — a comet's tail of almosts, leaving too many regrets and resentment. we pocket fragments of moments like stolen diamonds and hoard the joy of anticipation. yet when silence falls, we realize: what we await is not beyond the horizon, but the selves we buried under the layers of waiting.
day after day, in the fading light, I wait for a version of me.
she flickers in the theater of tomorrow.
those were some of my thoughts.
inspirations behind this piece, if you're interested.
one is a piece of art that I made that I'll put on my blog later. vio3 suggested that the woman in my art looks like she is waiting, which then inspired me to create this. vio3 is truly the best person when it comes to ideas (and art fixing :
))two is allusions to asteroid 2024 yr4 and butterfly effect
and an allusion to tempest sonata (peak beethoven moment)
the comma after the "waiting" in the title is also intentional if you couldn't notice. it references to "beyond the glass, trembling lights smear into bleeding watercolors, as if to exaggerate the uncertainty in the unfinished sentence of life."
This post has been edited 2 times. Last edited by Helena_Liang, Feb 18, 2025, 6:26 AM
Double dose of cyanide on day 2
by brianzjk, Mar 23, 2023, 10:20 PM
Let 
be an integer. We say that an arrangement of the numbers , 
, 
, 
in a 
table is row-valid if the numbers in each row can be permuted to form an arithmetic progression, and column-valid if the numbers in each column can be permuted to form an arithmetic progression. For what values of is it possible to transform any row-valid arrangement into a column-valid arrangement by permuting the numbers in each row?
be an integer. We say that an arrangement of the numbers , 
, 
, 
in a 
table is row-valid if the numbers in each row can be permuted to form an arithmetic progression, and column-valid if the numbers in each column can be permuted to form an arithmetic progression. For what values of is it possible to transform any row-valid arrangement into a column-valid arrangement by permuting the numbers in each row?This post has been edited 1 time. Last edited by brianzjk, Mar 23, 2023, 11:35 PM
Moving P(o)in(t)s
by bobthegod78, Apr 15, 2021, 5:08 PM
Carina has three pins, labeled 
, and , respectively, located at the origin of the coordinate plane. In a move, Carina may move a pin to an adjacent lattice point at distance away. What is the least number of moves that Carina can make in order for triangle to have area 2021?
(A lattice point is a point
in the coordinate plane where 
and 
are both integers, not necessarily positive.)
, and , respectively, located at the origin of the coordinate plane. In a move, Carina may move a pin to an adjacent lattice point at distance away. What is the least number of moves that Carina can make in order for triangle to have area 2021?(A lattice point is a point
in the coordinate plane where 
and 
are both integers, not necessarily positive.)This post has been edited 2 times. Last edited by bobthegod78, Apr 15, 2021, 6:40 PM
elements are removed from the set 
,Permutations Part 1: 2010 USAJMO #1
by tenniskidperson3, Apr 29, 2010, 3:36 PM
A permutation of the set of positive integers ![$[n] = \{1, 2, . . . , n\}$](//latex.artofproblemsolving.com/4/f/f/4ff58e6e89a225ffa8db0ab50038fdadd9d4e2bc.png)
is a sequence 
such that each element of ![$[n]$](//latex.artofproblemsolving.com/0/d/e/0deff9086fd7840ef23860f5bc924f1d4d2d2310.png)
appears precisely one time as a term of the sequence. For example, 
is a permutation of ![$[5]$](//latex.artofproblemsolving.com/c/4/2/c4216c69e041fca0904e8c5366d4480d1b2ca8b8.png)
. Let 
be the number of permutations of for which 
is a perfect square for all 
. Find with proof the smallest such that is a multiple of 
.
![$[n] = \{1, 2, . . . , n\}$](http://latex.artofproblemsolving.com/4/f/f/4ff58e6e89a225ffa8db0ab50038fdadd9d4e2bc.png)
is a sequence 
such that each element of ![$[n]$](http://latex.artofproblemsolving.com/0/d/e/0deff9086fd7840ef23860f5bc924f1d4d2d2310.png)
appears precisely one time as a term of the sequence. For example, 
is a permutation of ![$[5]$](http://latex.artofproblemsolving.com/c/4/2/c4216c69e041fca0904e8c5366d4480d1b2ca8b8.png)
. Let 
be the number of permutations of for which 
is a perfect square for all 
. Find with proof the smallest such that is a multiple of 
.This post has been edited 2 times. Last edited by tenniskidperson3, Dec 22, 2015, 2:55 AM
Reason: Lolol latex was wrong for 5 years and nobody caught it
Reason: Lolol latex was wrong for 5 years and nobody caught it
Isosceles Triangulation
by worthawholebean, May 1, 2008, 4:57 PM
Let 
be a convex polygon with 
sides, 
. Any set of 
diagonals of that do not intersect in the interior of the polygon determine a triangulation of into 
triangles. If is regular and there is a triangulation of consisting of only isosceles triangles, find all the possible values of .
be a convex polygon with 
sides, 
. Any set of 
diagonals of that do not intersect in the interior of the polygon determine a triangulation of into 
triangles. If is regular and there is a triangulation of consisting of only isosceles triangles, find all the possible values of .This post has been edited 1 time. Last edited by worthawholebean, May 1, 2008, 8:54 PM
flec and helena's aime progress | **supposedly** a math blog but who know what this is turning into
March 2025
February 2025
January 2025
December 2024
November 2024
August 2024
July 2024
June 2024
May 2024
April 2024
March 2024
February 2024
January 2024
December 2023
November 2023