Help with Competitive Geometry?

by REACHAW, Apr 14, 2025, 11:51 PM

Hi everyone,
I'm struggling a lot with geometry. I've found algebra, number theory, and even calculus to be relatively intuitive. However, when I took geometry, I found it very challenging. I stumbled my way through the class and can do the basic 'textbook' geometry problems, but still struggle a lot with geometry in competitive math. I find myself consistently skipping the geometry problems during contests (even the easier/first ones).

It's difficult for me to see the solution path. I can do the simpler textbook tasks (eg. find congruent triangles) but not more complex ones (eg. draw these two lines to form similar triangles).

Do you have any advice, resources, or techniques I should try?

geometry

number theory

calculus

congruent triangles

Algebra book recomndaitons

by idk12345678, Apr 14, 2025, 10:45 PM

Im currently reading EGMO by Evan Chen and i was wondering if there was a similar book for olympiad algebra. I have egmo for geo and aops intermediate c&p for combo, and the intermediate number thoery transcripts for nt, but i couldnt really find anything for alg

algebra

book recommendation

US Puzzle Championship Scorecards

by djmathman, Apr 14, 2025, 5:52 PM

Some of the discussion in the Contests & Programs SMT thread reminded me of Scorecards puzzles from the US Puzzle Championship. They behave similarly to "24 game" puzzles, but the allowable operations are slightly expanded.

Scorecards rules wrote:

Operations are limited to addition ("+"), subtraction ("-"), multiplication ("x"), division ("/"), and exponentiation ("^").
Decimal points may be used; ...; use minus sign ("-") to indicate negative values. Use parentheses if needed to disambiguate operator precedence.

As an example, the puzzle $11 \leftarrow 5,8,9$ would have answer $8 + 9^{.5} = 11$ .

These have appeared on the USPC three years: 2018, 2022, and 2023. Try your hand at these! Some of them are much more devious than they first appear.

2018
1. $30\leftarrow 4, 5, 8$
2. $25\leftarrow 2,6,9$
3. $23\leftarrow 4,5,9$
2022
1. $13\leftarrow 3,6,7$
2. $24\leftarrow 3,8,8$
3. $25\leftarrow 3,3,4$
2023
1. $26\leftarrow 5,5,6$
2. $15\leftarrow 2,6,8$
3. $11\leftarrow 2,8,9$

This post has been edited 1 time. Last edited by djmathman, Yesterday at 5:54 PM

math puzzle

Algebra Problems

by ilikemath247365, Apr 14, 2025, 4:52 PM

Find all real $(a, b)$ with $a + b = 1$ such that

$(a + \frac{1}{a})^{2} + (b + \frac{1}{b})^{2} = \frac{25}{2}$ .

Find the angle

by pythagorazz, Apr 14, 2025, 9:07 AM

Let $X$ be a point inside equilateral triangle $ABC$ such that $AX=\sqrt{2},BX=3$ , and $CX=\sqrt{5}$ . Find the measure of $\angle{AXB}$ in degrees.

trig

Trigonometry

by pythagorazz, Apr 14, 2025, 9:03 AM

Let $A$ , $B$ , and $C$ be the angles of an acute triangle such that
$\cos^2 A + \cos^2 B + 2 \sin A \sin B \cos C = \frac{34}{25},$ and
$\cos^2 B + \cos^2 C + 2 \sin B \sin C \cos A = \frac{34}{25}.$ Find
$\cos^2 C + \cos^2 A + 2 \sin C \sin A \cos B.$

trigonometry

Vietnam Mock Test

by imnotgoodatmathsorry, Apr 13, 2025, 4:13 PM

Second Entrance Mock test for grade 10 specialized in Mathematics at High School for Gifted Students, HNUE, Vietnam
13/4/2025

Problem 1:
1) Let $a,b$ be positive reals. Prove that: $\frac{a}{a+1} + \frac{b}{b+2} < \frac{\sqrt{a} + \sqrt{b}}{2}$
2) In a small garden there are $3$ rabbits and $3$ carrots. Each rabbit will choose randomly a carrot to eat. Find the probability of a carrot was chose by less than $2$ rabbit.
Problem 2:
1) Solve the equation system: $(x+y)(x^2+y^2)=567$ and $\sqrt{xy}(x+y)^2=243$
2) Let $a,b,c$ be positive rational numbers such that: $a+b+c=2\sqrt{abc}$
Problem 3:
Let triangle $ABC$ ( $\angle A$ , $\angle B$ , $\angle C < 90$ ) with excircle $(O)$ and incircle $(I)$ . Incircle $(I)$ touches $BC,CA,AB$ at $D,E,F$ . The excircle with the diameter of $AI$ cuts excircle $(O)$ at $K$ ( $K \neq A$ ). $KD$ cuts the excircle with the diameter of $AI$ at $P$ ( $P \neq K$ ) and $AK$ cuts $BC$ at $Q$ . Prove that:
1) $\Delta KEC$ ~ $\Delta KFB$ and $KD$ is the bisector of $\angle BKC$
2) $AP \bot BC$
3) $IQ$ is the tangent line of the excircle of $\Delta IBC$
Problem 4,5: (will type tomorrow)

Attachments:

This post has been edited 1 time. Last edited by imnotgoodatmathsorry, Sunday at 4:14 PM

mock test

geometry

how many quadrilaterals ?

by Ecrin_eren, Apr 13, 2025, 4:01 PM

"All the diagonals of an 11-gon are drawn. How many quadrilaterals can be formed using these diagonals as sides? (The vertices of the quadrilaterals are selected from the vertices of the 11-gon.)"

combinatorics

Inequalities

by sqing, Apr 9, 2025, 2:40 PM

Let $a,b,c$ be real numbers so that $a+2b+3c=2$ and $2ab+6bc+3ca =1.$ Show that
$-\frac{1}{6} \leq ab-bc+ ca\leq \frac{1}{2}$ $\frac{5-\sqrt{61}}{9} \leq a-b+c\leq \frac{5+\sqrt{61}}{9}$

This post has been edited 1 time. Last edited by sqing, Apr 9, 2025, 2:41 PM

algebra

inequalities

Probability

by Ecrin_eren, Apr 3, 2025, 11:21 AM

In a board, James randomly writes A , B or C in each cell. What is the probability that, for every row and every column, the number of A 's modulo 3 is equal to the number of B's modulo 3?

combinatorics

probability

Random Random FE

by utkarshgupta, Feb 16, 2017, 2:09 AM

Problem :
Find all functions $f : Let$ $f:\mathbb{R}\to \mathbb{R}$ such that
$f(x^{2}+y+f(y))=2y+f(x)^2$

Okay....
Here it goes...

Solution :
Let $P(x,y)$ be the assertion
$f(x^{2}+y+f(y))=2y+f(x)^2.$
And let $k=\frac{-f(0)^2}{2}$
$P(0,k) \implies f(k)=0$
Now let if possible, there exist some $a \neq k$ such that $f(a)=0$ ,
$P(0,a) \implies f(a) = f(0)^2 + 2a \neq 0$

A contradiction !

Hence $f(t) = 0 \implies t = k$ .

Let $f(0)=c$ then $k = \frac{-c^2}{2}$ .

$P(k,0) \implies f(k^2+f(0)) = f(k)^2 = 0$ $\implies k^2 + c = k$ $\implies c = 0,2$
Let if possible $c=2 \implies k=-2$
Then $P(0,0) \implies f(2) = 4$
$P(0,k) \implies f(2) = 0$

A contradiction !

$\boxed {f(0) = 0}$ And there exists no $a \neq 0$ such that $f(a)=0$

$P(x,\frac{-f(x)^2}{2}) \implies f(x^2 + \frac{-f(x)^2}{2} + f(\frac{-f(x)^2}{2})) = 0$ $\implies x^2 + \frac{-f(x)^2}{2} + f(\frac{-f(x)^2}{2}) = 0$
Thus if $f(a) = f(b)$ then $a^2 = b^2$

Let if possible $f(a) = f(-a)$ for some $a \neq 0$
$f(a + f(a)) = 2a = -f(-a + f(-a))$
Hence $f((a+f(a))^2) = f((-a+f(a))^2)$ Since $x^2 > 0$ ...
We must have $a^2+f(a)^2+2af(a) = a^2 + f(a)^2 - 2af(a)$
That is either $a= 0$ or $f(a) = 0$
A contradiction !

Hence $f$ is injective.

$P(x,0) \implies f(x^2) = f(x)^2 \ge 0$ $P(0,y) \implies f(y+f(y)) = 2y$
Let $z^2 - y^2 + f(z^2) - f(y^2) = x^2$ (assume greater than zero or else we can assume it to be equal to $-x^2$ )
$P(x^2,y^2) \implies f(x^2+y^2 + f(y^2)) = f(x^2) + 2y^2$ $P(0,z^2) \implies f(z^2+f(z^2)) = 2z^2$ $\implies f(x^2 + f(z^2) - f(y^2)) = 2(y^2-z^2) = f(x^2 + f(x^2))$ Since $f$ is injective
$f(x^2+y^2) = f(x^2) + f(y^2)$ .
We also have $-f(x) = f(-x)$ and $f(x^2) > 0$
So by Cauchy we are done !

That is $\boxed {f(x)=x}$ is the only solution !

This post has been edited 2 times. Last edited by utkarshgupta, Feb 16, 2017, 4:31 AM

functional equation

AoPS

Submit

Here goes first post of 2025! Great blog.
by math_holmes15, Jan 14, 2025, 8:53 AM
First post of 2024
by Yiyj1, Feb 8, 2024, 5:40 AM
First post of 2023
by HoRI_DA_GRe8, Jul 22, 2023, 7:45 AM
Nice blog ! Your isogonality lemma is really powerful !
by 554183, Oct 14, 2021, 8:55 AM
Post plss....
by samrocksnature, Apr 11, 2021, 10:12 PM
alas,this is ded
by Hamroldt, Mar 18, 2021, 4:13 PM
Thanks for the nice blog.
by Feridimo, Mar 6, 2020, 4:17 PM
I think this might be silly but ... when should we expect to have another post ?? I am very keen to see it
by gamerrk1004, Nov 4, 2019, 4:54 PM
Let's all echo what's written in the blog description - Stay Insane / 'Cause it's your labor, will and pain/ That takes you to the top of soda fountain
by Kayak, Oct 2, 2017, 7:18 PM
hey utkarsh jee is over now ... continue your elementary blog pleaseeeeeee!
by kk108, Jun 17, 2017, 11:19 AM
Congrats on becoming a contest moderator!
by Ankoganit, Mar 9, 2017, 5:22 AM
INTERSTING BLOG
by kk108, Feb 19, 2017, 2:04 PM
I have no plans for this blog right now....
No time here people !
But lets see....
I may try some combinatorics
by utkarshgupta, Feb 15, 2017, 12:47 PM
Thanks for the nice blog!
by Orkhan-Ashraf_2002, Feb 13, 2017, 6:34 PM
Revive it!!!
Best blog out there, for sure!
by rmtf1111, Jan 12, 2017, 6:02 PM
Oh you certainly will..
Bu I don't think I will
by achilles04, Feb 11, 2015, 6:24 PM
Thanks
Hope you are there too ...

And hope I get selected
by utkarshgupta, Feb 11, 2015, 11:14 AM
Nice blog ..
keep it up and you might one day become a mathematician lIke me
( just joking )
Btw don't forget us after going to imotc.
by achilles04, Feb 10, 2015, 4:49 PM
nice blog!!!!!
by pradhit, Feb 7, 2015, 11:51 AM
No ideas about the cut off but should be less than 60 according to me...
So you never know
by utkarshgupta, Feb 6, 2015, 4:06 PM
Hi utkarsh,
what do u expect the cutoff to be this year??

by kgdpsdurg, Feb 6, 2015, 12:41 PM
Hi utkarsh
How many did you clear in INMO 2015?
by moldevort, Feb 1, 2015, 3:59 PM
@ Anonymous Bunny
If we both (that should have included me) are lucky enough !
by utkarshgupta, Sep 17, 2014, 2:44 AM
Nice blog. Great job!

If I am lucky enough, hopefully we shall meet at IMOTC.
by AnonymousBunny, Sep 16, 2014, 8:08 PM
Thanx all !!

If anyone wish to contribute anything just pm !

I will add u to the contributors' list !
by utkarshgupta, Sep 16, 2014, 6:35 AM
doing good progress!! btw nice blog.
by rachitgoel, Sep 15, 2014, 4:21 PM
Nice blog.
by Kezer, Aug 5, 2014, 4:04 PM
Hi Utkarsh! Wonderful blog. Keep it up.
by YESMAths, Jul 20, 2014, 1:36 PM
I need some shouts and characters

by utkarshgupta, Jul 20, 2014, 4:22 AM