G
Topic
First Poster
Last Poster
Preparing for Putnam level entrance examinations
Cats_on_a_computer 4
N
by Cats_on_a_computer
Non American high schooler in the equivalent of grade 12 here. Where I live, two the best undergraduates program in the country accepts students based on a common entrance exam. The first half of the exam is “screening”, with 4 options being presented per question, each of which one has to assign a True or False. This first half is about the difficulty of an average AIME, or JEE Adv paper, and it is a requirement for any candidate to achieve at least 24/40 on this half for the examiners to even consider grading the second part. The second part consists of long form questions, and I have, no joke, seen them literally rip off, verbatim, Putnam A6s. Some of the problems are generally standard textbook problems in certain undergrad courses but obviously that doesn’t translate it to being doable for high school students. I’ve effectively got to prepare for a slightly nerfed Putnam, if you will, and so I’ve been looking for resources (not just problems) for Putnam level questions. Does anyone have any suggestions?
4 replies
Romania NMO 2023 Grade 11 P1
DanDumitrescu 15
N
by anudeep
Source: Romania National Olympiad 2023
Determine twice differentiable functions 
which verify relation
![\[
\left( f'(x) \right)^2 + f''(x) \leq 0, \forall x \in \mathbb{R}.
\]](//latex.artofproblemsolving.com/3/a/2/3a235e8be4c61c32f376999e61d64973db21dc75.png)
which verify relation![\[
\left( f'(x) \right)^2 + f''(x) \leq 0, \forall x \in \mathbb{R}.
\]](http://latex.artofproblemsolving.com/3/a/2/3a235e8be4c61c32f376999e61d64973db21dc75.png)
15 replies
Subset Ordered Pairs of {1, 2, ..., 10}
ahaanomegas 11
N
by cappucher
Source: Putnam 1990 A6
If 
is a finite set, let denote the number of elements in . Call an ordered pair 
of subsets of 
if 
for each 
, and 
for each 
. How many admissible ordered pairs of subsets 
are there? Prove your answer.
is a finite set, let denote the number of elements in . Call an ordered pair 
of subsets of 
if 
for each 
, and 
for each 
. How many admissible ordered pairs of subsets 
are there? Prove your answer.
11 replies
Putnam 2000 B4
ahaanomegas 6
N
by mqoi_KOLA
Let 
be a continuous function such that 
for all 
. Show that 
for 
.
be a continuous function such that 
for all 
. Show that 
for 
.
6 replies
Another integral limit
RobertRogo 1
N
by alexheinis
Source: "Traian Lalescu" student contest 2025, Section A, Problem 3
Let 
be a function differentiable at 0 with 
. Find
be a function differentiable at 0 with 
. Find
1 reply
AB=BA if A-nilpotent
KevinDB17 3
N
by loup blanc
Let A,B 2 complex n*n matrices such that AB+I=A+B+BA
If A is nilpotent prove that AB=BA
If A is nilpotent prove that AB=BA
3 replies
Very nice equivalence in matrix equations
RobertRogo 3
N
by Etkan
Source: "Traian Lalescu" student contest 2025, Section A, Problem 4
Let 
Show that the following statements are equivalent:
i) For every
there exist 
such that 
ii) For every there exist 
such that 
Show that the following statements are equivalent:i) For every
there exist 
such that 
ii) For every
there exist 
such that 
3 replies
Miklos Schweitzer 1971_5
ehsan2004 2
N
by pi_quadrat_sechstel
Let 
be a positive sequence and let 
be a constant such that ![\[ \sum_{k=1}^{n-1} \lambda^2_k < K \lambda^2_n \;(n=1,2,...).\]](//latex.artofproblemsolving.com/5/3/c/53ccb27390a368e1a74afae1da4f7566ad86af8a.png)
Prove that there exists a constant 
such that ![\[ \sum_{k=1}^{n-1} \lambda_k < K' \lambda_n \;(n=1,2,...).\]](//latex.artofproblemsolving.com/e/7/4/e744b1b1179f879dbea84a53622afc5ca8e45fe2.png)
L. Leindler
be a positive sequence and let 
be a constant such that ![\[ \sum_{k=1}^{n-1} \lambda^2_k < K \lambda^2_n \;(n=1,2,...).\]](http://latex.artofproblemsolving.com/5/3/c/53ccb27390a368e1a74afae1da4f7566ad86af8a.png)
Prove that there exists a constant 
such that ![\[ \sum_{k=1}^{n-1} \lambda_k < K' \lambda_n \;(n=1,2,...).\]](http://latex.artofproblemsolving.com/e/7/4/e744b1b1179f879dbea84a53622afc5ca8e45fe2.png)
L. Leindler
2 replies
Cute matrix equation
RobertRogo 1
N
by loup blanc
Source: "Traian Lalescu" student contest 2025, Section A, Problem 2
Find all matrices 
such that 
such that 
1 reply
