Printed positive marks are preserved per question. The source uses different blank-answer and incorrect-answer rules by section, including an optional bonus penalty; the import does not reproduce that complete official scoring scheme.
What is the value of $\dfrac{2^{20}+2^{18}}{2^{19}+2^{18}}$?
Triangle $ABC$ has $AB=AC$. Point $D$ is on $AB$, with $BA=3BD$; point $E$ is on $AC$, with $CA=3CE$. Which congruence condition directly proves triangles $EBC$ and $DCB$ congruent? The conditions are SSS, ASA, AAS, SAS, HL: S means side, A means angle, H means hypotenuse and L means leg. HL applies only to right triangles.
$ABCD$ is a square and $O$ is the intersection of its diagonals. Triangle $PQR$ has base $QR=20$ cm. The ratio $[ABCD]:[PQR]$ is $8:5$, and $[AOB]=256$ cm². Find the height from $P$ in cm.
Jack and Jill each sell the same number of eggs daily. Jack sells 2 eggs for 5 baht and earns $a$ baht; Jill sells 3 for 7 baht and earns $b$. One day Jill sells both equal stocks together at 5 eggs for 12 baht, earning $c$. If $a:b:c=x:y:z$ for positive integers $x,y,z$, find the minimum $x+y+z$.
In a marathon, 32% of participants are adult men, 38% adult women and 20% boys; the others are girls. There are 960 adult men. Which statement is true?
At the 2018 FIFA World Cup, 32 teams are divided into eight groups A–H of four teams. In the first round each pair of teams within a group plays exactly one match. A winner gets 3 points and a loser 0; a draw gives each team 1 point. The two highest-scoring teams in each group advance to the second round.
Group E results are: Costa Rica–Serbia 0–1; Brazil–Switzerland 1–1; Brazil–Costa Rica 2–0; Serbia–Switzerland 1–2; Serbia–Brazil 0–2; Switzerland–Costa Rica 2–2. What is the total points of all four teams after the first round?
At the 2018 FIFA World Cup, 32 teams are divided into eight groups A–H of four teams. In the first round each pair of teams within a group plays exactly one match. A winner gets 3 points and a loser 0; a draw gives each team 1 point. The two highest-scoring teams in each group advance to the second round.
Belgium led Tunisia 2–1 at half-time. In the second half the two teams scored four more goals in total. Which final outcome is impossible?
At the 2018 FIFA World Cup, 32 teams are divided into eight groups A–H of four teams. In the first round each pair of teams within a group plays exactly one match. A winner gets 3 points and a loser 0; a draw gives each team 1 point. The two highest-scoring teams in each group advance to the second round.
The total points in groups A–H are respectively 18, 15, 16, 17, 16, 18, 18, 17. How many first-round matches in all groups ended in a draw?
In isosceles triangle $ABC$, $AB=AC$ and $\angle BAC=20^\circ$. $D$ is the midpoint of $BC$ and $E$ the midpoint of $AD$. How many claims are true? (1) $\angle EBC=\angle ECB=60^\circ$; (2) $\angle ABC=\angle ACB=80^\circ$; (3) $\angle EAB=\angle EAC=10^\circ$; (4) $\angle AEB=\angle AEC=150^\circ$.
