[66] - Algebraic Systems for and from Contests

Να λυθεί το σύστημα: $$\left\{ \begin{array}{l} ({x^2} + y)\sqrt {y - 2x} - 4 = 2{x^2} + 2x + y\\ {x^3} - {x^2} - y + 6 = 4\sqrt {x + 1} + 2\sqrt {y - 1} \end{array} \right. \quad (x, y \in \mathbb{R}).$$ 2023 Quang Nam Province Math Contest (Grade 11)

Σάρωση για να αποθηκεύσετε ή να κοινοποιήσετε την ανάρτηση

Mathematical Olympiad in China: PROBLEMS and SOLUTIONS

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European Girls' Mathematical Olympiad (EGMO) 2025 - ΤΑ ΠΡΟΒΛΗΜΑΤΑ

1η μέρα - 13 Απριλίου 2025

Πρόβλημα 1. 

Για κάθε θετικό ακέραιο $N$, έστω $c_1 <c_2 < \dots < c_m$ όλοι οι θετικοί ακέραιοι μικρότεροι του $N$ οι οποίοι είναι σχετικά πρώτοι με τον $N$. Να βρείτε όλους τους αριθμούς $N \ge 3$ τέτοιους ώστε $\gcd(N, c_i + c_{i+1}) \ne 1$  για κάθε $1 \le i \le m-1$.

Εδώ ο $\gcd(a,b)$ είναι ο μεγαλύτερος θετικός ακέραιος που διαιρεί και τον $a$ και τον $b$. Οι ακέραιοι $a$ και $b$ είναι σχετικά πρώτοι μεταξύ τους αν $\gcd(a,b)=1$.

Πρόβλημα 2. 

Μια άπειρη γνησίως αύξουσα ακολουθία $a_1 < a_2 < a_3 < \cdots$ θετικών ακέραιων αριθμών αποκαλείται κεντρική αν για κάθε θετικό ακέραιο $n$, ο μέσος όρος των πρώτων $a_n$ όρων της ακολουθίας είναι ίσος με $a_n$.

2025 Indian Math Olympiad — Problem 2

Οι ακέραιοι αριθμοί $1, 2, \ldots, n$ είναι γραμμένοι σε έναν πίνακα. Σε κάθε κίνηση, η Άλις μπορεί να επιλέξει δύο διαφορετικούς αριθμούς $a \ne b$ από τον πίνακα, με την προϋπόθεση ότι το $a + b$ είναι άρτιος αριθμός, να τους σβήσει και να τους αντικαταστήσει με τον αριθμό $\dfrac{a + b}{2}$. 

Να βρείτε όλους τους ακέραιους $n \ge 2$ για τους οποίους η Άλις μπορεί να εκτελέσει μία ακολουθία κινήσεων έτσι ώστε να απομείνει μόνο ένας αριθμός στον πίνακα. 

Παρατήρηση: Όταν $n = 3$, η Άλις μπορεί να μετατρέψει το $(1, 2, 3)$ σε $(2, 2)$, αλλά δεν μπορεί να κάνει άλλες κινήσεις.

China National Mathematical Olympiad 2025 - PROBLEMS

Problem 1

Let $\alpha > 1$ be an irrational number and $L$ be an integer such that $L > \frac{\alpha^2}{\alpha - 1}$. A sequence $x_1, x_2, \dots$ satisfies:

$$x_{n+1} = \begin{cases} \left \lfloor \alpha x_n \right \rfloor & \text{if } x_n \leqslant L \\ \left \lfloor \frac{x_n}{\alpha} \right \rfloor & \text{if } x_n > L \end{cases}$$

Prove that:

1. $\{x_n\}$ is eventually periodic.
2. The eventual fundamental period of $\{x_n\}$ is an odd integer which doesn't depend on the choice of $x_1$.

Problem 2

Let $ABC$ be a triangle with incenter $I$. Denote the midpoints of $AI$, $AC$, and $CI$ by $L$, $M$, and $N$ respectively...

Prove that $PQ$, $LN$, and $EF$ are concurrent.

A Geometry Problem from the 2025 Chinese Olympiad

Let $ABC$ be a triangle with incenter $I$. Denote the midpoints of $AI, AC$ and $CI$ by $L, M$ and $N$ respectively. Point $D$ lies on segment $AM$ such that $BC= BD$. Let the incircle of triangle $ABD$ be tangent to $AD$ and $BD$ at $E$ and $F$ respectively. 

Denote the circumcenter of triangle $AIC$ by $J$, and the circumcircle of triangle $JMD$ by $\omega$. Lines $MN$ and $JL$ meet $\omega$ again at $P$ and $Q$ respectively. Prove that $PQ, LN$ and $EF$ are concurrent.

See solution here.

Bundeswettbewerb Mathematik 2025 - PROBLEMS

Problem 1
Fridolin the frog jumps on the number line: He starts at $0$, then jumps in some order on each of the numbers $1,2,\dots,9$ exactly once and finally returns with his last jump to $0$. Can the total distance he travelled with these $10$ jumps be a) $20$, b) $25$?

Problem 2
For each integer $n \ge 2$ we consider the last digit different from zero in the decimal expansion of $n!$. The infinite sequence of these digits starts with $2,6,4,2,2$. Determine all digits which occur at least once in this sequence, and show that each of those digits occurs in fact infinitely often.

USA Mathematical Olympiad (USAMO) 2025 - THE PROBLEMS

Problem 1

Let k and d be positive integers. Prove that there exists a positive integer N such that for every odd integer n > N, the digits in the base-2n representation of n^k are all greater than d.

Problem 2

Let n and k be positive integers with k < n. Let P(x) be a polynomial of degree n with real coefficients, nonzero constant term, and no repeated roots. Suppose that for any real numbers a₀, a₁, ..., a_k such that the polynomial a_kx^k + ... + a₁x + a₀ divides P(x), the product a₀a₁...a_k is zero. Prove that P(x) has a nonreal root.

Vietnam National Mathematical Olympiad (VMO) 2025 - PROBLEMS

Problem 1

a) Prove that for all positive real numbers $a$, the polynomial $P(x) = x^3 + ax^2 + (a+1)x + 1$ has a unique positive zero.

b) A sequence $(x_n)_{n \ge 1}$ is defined by $x_1 = 1$ and for all $n \ge 1$, $x_{n+1}$ is the positive zero of the polynomial $P_n(x) = x^3 + x_n x^2 + (x_n+1)x + 1$. Prove that the sequence $(x_n)$ converges, and find the limit of the sequence.

Problem 2

For each non-negative integer $n$, let $a_n = \binom{2n}{n}$.

a) Prove that $a_n$ is a positive integer for all $n \ge 0$. When $n$ changes, what is the largest possible remainder when $a_n$ is divided by 3?

b) Find all pairs of positive integers $k, m$ such that $k \ge m$ and for all odd positive integers $n$, $a_n \equiv m \pmod{k}$.

Romanian Master of Mathematics (RMM) 2025 - THE PROBLEMS

Problem 1

Let $n$ be an integer, and let $P_1, P_2, \dots, P_n$ be distinct points in the plane such that the distances between the points are pairwise different. Define $d(P_i, P_j)$ to be the distance between points $P_i$ and $P_j$. For each point $P_i$, define $r(P_i)$ to be the 10th smallest of the distances from $P_i$ to $P_j$, excluding $j=i$. Suppose that for all $i$ and $j$ satisfying $r(P_i) < r(P_j)$, we have $i < j$. Prove that $i \le 10$ for all $i$ in the range $1 \le i \le n$.

Proposed by Morteza Saghafian, Iran

Problem 2

Consider an infinite sequence of positive integers $a_1, a_2, a_3, \dots$ such that $a_1$ is a square and $a_{n+1} = a_n + \sqrt{a_n}$ is a square for all positive integers $n$. Is it possible for two terms of such a sequence to be equal?

Proposed by Pavel Kozlov, Russia

Norwegian Mathematical Olympiad 2025 - Abelkonkurransen Finale | PROBLEMS

Problem 1

1a. Peer and Solveig play a game with $n$ coins. Peer flips one or more non-adjacent coins, and Solveig flips exactly two adjacent coins. Peer wins if all coins show $K$. For which values of $n \geq 2$ can Solveig prevent Peer from winning?

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1b. In Duckville, the trophy is passed around among inhabitants. Gregers gets it back exactly every 2025 days. Hedvig has it today. For which values of $n > 0$ is it impossible for Hedvig to receive the trophy again in $n$ days?

Problem 2

2a. Eleven pupils each write a different positive integer on a sticky note. Prove that the teacher can always choose two or more notes so that the average of their numbers is not an integer.

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2b. Find the integers $a$ such that $n! - a$ is a perfect square for infinitely many $n$.

Bangladesh Mathematical Olympiad 2025 - PROBLEMS

Problem 1

One day in a room there were several inhabitants of an island where only truth-tellers and liars live. Three of them made the following statements:

• There are no more than three of us here. We are all liars.
• There are no more than four of us here. Not all of us are liars.
• There are five of us here. At least three of us are liars.

How many people are in the room and how many of them are liars?

Problem 2

Find all real solutions to the equation $(x^2 - 9x + 19)^{(x^2 - 3x + 2)} = 1$.

Polish Mathematical Olympiad Finals 2024

 

2022 Hong Kong International Math Olympiad Question

 

Εύρεση Ισότητας μέσω Ανισοτήτων

Austrian - Polish Math Competition 1994

India National Olympiad 2025 - PROBLEMS

                      

Problem 1

Consider the sequence defined by $a_1 = 2$, $a_2 = 3$, and

$$a_{2k+1} = 2 + 2a_k, \quad a_{2k+2} = 2 + a_k + a_{k+1},$$

for all integers $k \geq 1$. Determine all positive integers $n$ such that

$$\frac{a_n}{n}$$

is an integer.

Proposed by: Niranjan Balachandran, SS Krishnan, and Prithwijit De

ΒΙΒΛΙΟ: Διεθνείς Μαθηματικές Ολυμπιάδες 1959 – 2000 (pdf)

Ιστορικά στοιχεία - Συμμετοχές - Βαθμολογίες - Κατάταξη χωρών

Κάντε κλι στην εικόνα.

Ελληνική Μαθηματική Εταιρεία: Προκριματικός Διαγωνισμός Νέων 2019 (jumior)

42η ΕΘΝΙΚΗ ΜΑΘΗΜΑΤΙΚΗ ΟΛΥΜΠΙΑΔΑ «ΑΡΧΙΜΗΔΗΣ» 2025 - ΘΕΜΑΤΑ ΚΑΙ ΛΥΣΕΙΣ

ΑΡΧΙΜΗΔΗΣ_2025_ΛΥΚΕΙΟΥ.pdf

ΑΡΧΙΜΗΔΗΣ_2025_ΓΥΜΝΑΣΙΟΥ.pdf

Κάντε κλικ εδώ, για να δείτε τις λύσεις από το mathematica.

[55] - Algebraic Systems from and for Contests